代码拉取完成，页面将自动刷新
同步操作将从 scruel/Notes-ML-AndrewNg 强制同步，此操作会覆盖自 Fork 仓库以来所做的任何修改，且无法恢复！！！
确定后同步将在后台操作，完成时将刷新页面，请耐心等待。
<!doctype html>
<html>
<head>
<meta charset='UTF-8'><meta name='viewport' content='width=device-width initial-scale=1'>
<title>week1</title><link href='https://fonts.loli.net/css?family=Open+Sans:400italic,700italic,700,400&subset=latin,latin-ext' rel='stylesheet' type='text/css' /><style type='text/css'>html {overflow-x: initial !important;}:root { --bg-color:#ffffff; --text-color:#333333; --select-text-bg-color:#B5D6FC; --select-text-font-color:auto; --monospace:"Lucida Console",Consolas,"Courier",monospace; --title-bar-height:20px; }
.mac-os-11 { --title-bar-height:28px; }
html { font-size: 14px; background-color: var(--bg-color); color: var(--text-color); font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; -webkit-font-smoothing: antialiased; }
body { margin: 0px; padding: 0px; height: auto; bottom: 0px; top: 0px; left: 0px; right: 0px; font-size: 1rem; line-height: 1.42857; overflow-x: hidden; background: inherit; tab-size: 4; }
iframe { margin: auto; }
a.url { word-break: break-all; }
a:active, a:hover { outline: 0px; }
.in-text-selection, ::selection { text-shadow: none; background: var(--select-text-bg-color); color: var(--select-text-font-color); }
#write { margin: 0px auto; height: auto; width: inherit; word-break: normal; overflow-wrap: break-word; position: relative; white-space: normal; overflow-x: visible; padding-top: 36px; }
#write.first-line-indent p { text-indent: 2em; }
#write.first-line-indent li p, #write.first-line-indent p * { text-indent: 0px; }
#write.first-line-indent li { margin-left: 2em; }
.for-image #write { padding-left: 8px; padding-right: 8px; }
body.typora-export { padding-left: 30px; padding-right: 30px; }
.typora-export .footnote-line, .typora-export li, .typora-export p { white-space: pre-wrap; }
.typora-export .task-list-item input { pointer-events: none; }
@media screen and (max-width: 500px) {
body.typora-export { padding-left: 0px; padding-right: 0px; }
#write { padding-left: 20px; padding-right: 20px; }
.CodeMirror-sizer { margin-left: 0px !important; }
.CodeMirror-gutters { display: none !important; }
}
#write li > figure:last-child { margin-bottom: 0.5rem; }
#write ol, #write ul { position: relative; }
img { max-width: 100%; vertical-align: middle; image-orientation: from-image; }
button, input, select, textarea { color: inherit; font: inherit; }
input[type="checkbox"], input[type="radio"] { line-height: normal; padding: 0px; }
*, ::after, ::before { box-sizing: border-box; }
#write h1, #write h2, #write h3, #write h4, #write h5, #write h6, #write p, #write pre { width: inherit; }
#write h1, #write h2, #write h3, #write h4, #write h5, #write h6, #write p { position: relative; }
p { line-height: inherit; }
h1, h2, h3, h4, h5, h6 { break-after: avoid-page; break-inside: avoid; orphans: 4; }
p { orphans: 4; }
h1 { font-size: 2rem; }
h2 { font-size: 1.8rem; }
h3 { font-size: 1.6rem; }
h4 { font-size: 1.4rem; }
h5 { font-size: 1.2rem; }
h6 { font-size: 1rem; }
.md-math-block, .md-rawblock, h1, h2, h3, h4, h5, h6, p { margin-top: 1rem; margin-bottom: 1rem; }
.hidden { display: none; }
.md-blockmeta { color: rgb(204, 204, 204); font-weight: 700; font-style: italic; }
a { cursor: pointer; }
sup.md-footnote { padding: 2px 4px; background-color: rgba(238, 238, 238, 0.7); color: rgb(85, 85, 85); border-radius: 4px; cursor: pointer; }
sup.md-footnote a, sup.md-footnote a:hover { color: inherit; text-transform: inherit; text-decoration: inherit; }
#write input[type="checkbox"] { cursor: pointer; width: inherit; height: inherit; }
figure { overflow-x: auto; margin: 1.2em 0px; max-width: calc(100% + 16px); padding: 0px; }
figure > table { margin: 0px; }
tr { break-inside: avoid; break-after: auto; }
thead { display: table-header-group; }
table { border-collapse: collapse; border-spacing: 0px; width: 100%; overflow: auto; break-inside: auto; text-align: left; }
table.md-table td { min-width: 32px; }
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.CodeMirror-linenumber { user-select: none; }
.CodeMirror { text-align: left; }
.CodeMirror-placeholder { opacity: 0.3; }
.CodeMirror pre { padding: 0px 4px; }
.CodeMirror-lines { padding: 0px; }
div.hr:focus { cursor: none; }
#write pre { white-space: pre-wrap; }
#write.fences-no-line-wrapping pre { white-space: pre; }
#write pre.ty-contain-cm { white-space: normal; }
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.md-diagram-panel { width: 100%; margin-top: 10px; text-align: center; padding-top: 0px; padding-bottom: 8px; overflow-x: auto; }
#write .md-fences.mock-cm { white-space: pre-wrap; }
.md-fences.md-fences-with-lineno { padding-left: 0px; }
#write.fences-no-line-wrapping .md-fences.mock-cm { white-space: pre; overflow-x: auto; }
.md-fences.mock-cm.md-fences-with-lineno { padding-left: 8px; }
.CodeMirror-line, twitterwidget { break-inside: avoid; }
.footnotes { opacity: 0.8; font-size: 0.9rem; margin-top: 1em; margin-bottom: 1em; }
.footnotes + .footnotes { margin-top: 0px; }
.md-reset { margin: 0px; padding: 0px; border: 0px; outline: 0px; vertical-align: top; background: 0px 0px; text-decoration: none; text-shadow: none; float: none; position: static; width: auto; height: auto; white-space: nowrap; cursor: inherit; -webkit-tap-highlight-color: transparent; line-height: normal; font-weight: 400; text-align: left; box-sizing: content-box; direction: ltr; }
li div { padding-top: 0px; }
blockquote { margin: 1rem 0px; }
li .mathjax-block, li p { margin: 0.5rem 0px; }
li blockquote { margin: 1rem 0px; }
li { margin: 0px; position: relative; }
blockquote > :last-child { margin-bottom: 0px; }
blockquote > :first-child, li > :first-child { margin-top: 0px; }
.footnotes-area { color: rgb(136, 136, 136); margin-top: 0.714rem; padding-bottom: 0.143rem; white-space: normal; }
#write .footnote-line { white-space: pre-wrap; }
@media print {
body, html { border: 1px solid transparent; height: 99%; break-after: avoid; break-before: avoid; font-variant-ligatures: no-common-ligatures; }
#write { margin-top: 0px; padding-top: 0px; border-color: transparent !important; }
.typora-export * { -webkit-print-color-adjust: exact; }
.typora-export #write { break-after: avoid; }
.typora-export #write::after { height: 0px; }
.is-mac table { break-inside: avoid; }
}
.footnote-line { margin-top: 0.714em; font-size: 0.7em; }
a img, img a { cursor: pointer; }
pre.md-meta-block { font-size: 0.8rem; min-height: 0.8rem; white-space: pre-wrap; background: rgb(204, 204, 204); display: block; overflow-x: hidden; }
p > .md-image:only-child:not(.md-img-error) img, p > img:only-child { display: block; margin: auto; }
#write.first-line-indent p > .md-image:only-child:not(.md-img-error) img { left: -2em; position: relative; }
p > .md-image:only-child { display: inline-block; width: 100%; }
#write .MathJax_Display { margin: 0.8em 0px 0px; }
.md-math-block { width: 100%; }
.md-math-block:not(:empty)::after { display: none; }
.MathJax_ref { fill: currentcolor; }
[contenteditable="true"]:active, [contenteditable="true"]:focus, [contenteditable="false"]:active, [contenteditable="false"]:focus { outline: 0px; box-shadow: none; }
.md-task-list-item { position: relative; list-style-type: none; }
.task-list-item.md-task-list-item { padding-left: 0px; }
.md-task-list-item > input { position: absolute; top: 0px; left: 0px; margin-left: -1.2em; margin-top: calc(1em - 10px); border: none; }
.math { font-size: 1rem; }
.md-toc { min-height: 3.58rem; position: relative; font-size: 0.9rem; border-radius: 10px; }
.md-toc-content { position: relative; margin-left: 0px; }
.md-toc-content::after, .md-toc::after { display: none; }
.md-toc-item { display: block; color: rgb(65, 131, 196); }
.md-toc-item a { text-decoration: none; }
.md-toc-inner:hover { text-decoration: underline; }
.md-toc-inner { display: inline-block; cursor: pointer; }
.md-toc-h1 .md-toc-inner { margin-left: 0px; font-weight: 700; }
.md-toc-h2 .md-toc-inner { margin-left: 2em; }
.md-toc-h3 .md-toc-inner { margin-left: 4em; }
.md-toc-h4 .md-toc-inner { margin-left: 6em; }
.md-toc-h5 .md-toc-inner { margin-left: 8em; }
.md-toc-h6 .md-toc-inner { margin-left: 10em; }
@media screen and (max-width: 48em) {
.md-toc-h3 .md-toc-inner { margin-left: 3.5em; }
.md-toc-h4 .md-toc-inner { margin-left: 5em; }
.md-toc-h5 .md-toc-inner { margin-left: 6.5em; }
.md-toc-h6 .md-toc-inner { margin-left: 8em; }
}
a.md-toc-inner { font-size: inherit; font-style: inherit; font-weight: inherit; line-height: inherit; }
.footnote-line a:not(.reversefootnote) { color: inherit; }
.md-attr { display: none; }
.md-fn-count::after { content: "."; }
code, pre, samp, tt { font-family: var(--monospace); }
kbd { margin: 0px 0.1em; padding: 0.1em 0.6em; font-size: 0.8em; color: rgb(36, 39, 41); background: rgb(255, 255, 255); border: 1px solid rgb(173, 179, 185); border-radius: 3px; box-shadow: rgba(12, 13, 14, 0.2) 0px 1px 0px, rgb(255, 255, 255) 0px 0px 0px 2px inset; white-space: nowrap; vertical-align: middle; }
.md-comment { color: rgb(162, 127, 3); opacity: 0.8; font-family: var(--monospace); }
code { text-align: left; vertical-align: initial; }
a.md-print-anchor { white-space: pre !important; border-width: initial !important; border-style: none !important; border-color: initial !important; display: inline-block !important; position: absolute !important; width: 1px !important; right: 0px !important; outline: 0px !important; background: 0px 0px !important; text-decoration: initial !important; text-shadow: initial !important; }
.md-inline-math .MathJax_SVG .noError { display: none !important; }
.html-for-mac .inline-math-svg .MathJax_SVG { vertical-align: 0.2px; }
.md-math-block .MathJax_SVG_Display { text-align: center; margin: 0px; position: relative; text-indent: 0px; max-width: none; max-height: none; min-height: 0px; min-width: 100%; width: auto; overflow-y: hidden; display: block !important; }
.MathJax_SVG_Display, .md-inline-math .MathJax_SVG_Display { width: auto; margin: inherit; display: inline-block !important; }
.MathJax_SVG .MJX-monospace { font-family: var(--monospace); }
.MathJax_SVG .MJX-sans-serif { font-family: sans-serif; }
.MathJax_SVG { display: inline; font-style: normal; font-weight: 400; line-height: normal; zoom: 90%; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; }
.MathJax_SVG * { transition: none 0s ease 0s; }
.MathJax_SVG_Display svg { vertical-align: middle !important; margin-bottom: 0px !important; margin-top: 0px !important; }
.os-windows.monocolor-emoji .md-emoji { font-family: "Segoe UI Symbol", sans-serif; }
.md-diagram-panel > svg { max-width: 100%; }
[lang="flow"] svg, [lang="mermaid"] svg { max-width: 100%; height: auto; }
[lang="mermaid"] .node text { font-size: 1rem; }
table tr th { border-bottom: 0px; }
video { max-width: 100%; display: block; margin: 0px auto; }
iframe { max-width: 100%; width: 100%; border: none; }
.highlight td, .highlight tr { border: 0px; }
mark { background: rgb(255, 255, 0); color: rgb(0, 0, 0); }
.md-html-inline .md-plain, .md-html-inline strong, mark .md-inline-math, mark strong { color: inherit; }
mark .md-meta { color: rgb(0, 0, 0); opacity: 0.3 !important; }
@media print {
.typora-export h1, .typora-export h2, .typora-export h3, .typora-export h4, .typora-export h5, .typora-export h6 { break-inside: avoid; }
}
.md-diagram-panel .messageText { stroke: none !important; }
.md-diagram-panel .start-state { fill: var(--node-fill); }
.md-diagram-panel .edgeLabel rect { opacity: 1 !important; }
.md-require-zoom-fix foreignobject { font-size: var(--mermaid-font-zoom); }
.CodeMirror { height: auto; }
.CodeMirror.cm-s-inner { background: inherit; }
.CodeMirror-scroll { overflow: auto hidden; z-index: 3; }
.CodeMirror-gutter-filler, .CodeMirror-scrollbar-filler { background-color: rgb(255, 255, 255); }
.CodeMirror-gutters { border-right: 1px solid rgb(221, 221, 221); background: inherit; white-space: nowrap; }
.CodeMirror-linenumber { padding: 0px 3px 0px 5px; text-align: right; color: rgb(153, 153, 153); }
.cm-s-inner .cm-keyword { color: rgb(119, 0, 136); }
.cm-s-inner .cm-atom, .cm-s-inner.cm-atom { color: rgb(34, 17, 153); }
.cm-s-inner .cm-number { color: rgb(17, 102, 68); }
.cm-s-inner .cm-def { color: rgb(0, 0, 255); }
.cm-s-inner .cm-variable { color: rgb(0, 0, 0); }
.cm-s-inner .cm-variable-2 { color: rgb(0, 85, 170); }
.cm-s-inner .cm-variable-3 { color: rgb(0, 136, 85); }
.cm-s-inner .cm-string { color: rgb(170, 17, 17); }
.cm-s-inner .cm-property { color: rgb(0, 0, 0); }
.cm-s-inner .cm-operator { color: rgb(152, 26, 26); }
.cm-s-inner .cm-comment, .cm-s-inner.cm-comment { color: rgb(170, 85, 0); }
.cm-s-inner .cm-string-2 { color: rgb(255, 85, 0); }
.cm-s-inner .cm-meta { color: rgb(85, 85, 85); }
.cm-s-inner .cm-qualifier { color: rgb(85, 85, 85); }
.cm-s-inner .cm-builtin { color: rgb(51, 0, 170); }
.cm-s-inner .cm-bracket { color: rgb(153, 153, 119); }
.cm-s-inner .cm-tag { color: rgb(17, 119, 0); }
.cm-s-inner .cm-attribute { color: rgb(0, 0, 204); }
.cm-s-inner .cm-header, .cm-s-inner.cm-header { color: rgb(0, 0, 255); }
.cm-s-inner .cm-quote, .cm-s-inner.cm-quote { color: rgb(0, 153, 0); }
.cm-s-inner .cm-hr, .cm-s-inner.cm-hr { color: rgb(153, 153, 153); }
.cm-s-inner .cm-link, .cm-s-inner.cm-link { color: rgb(0, 0, 204); }
.cm-negative { color: rgb(221, 68, 68); }
.cm-positive { color: rgb(34, 153, 34); }
.cm-header, .cm-strong { font-weight: 700; }
.cm-del { text-decoration: line-through; }
.cm-em { font-style: italic; }
.cm-link { text-decoration: underline; }
.cm-error { color: red; }
.cm-invalidchar { color: red; }
.cm-constant { color: rgb(38, 139, 210); }
.cm-defined { color: rgb(181, 137, 0); }
div.CodeMirror span.CodeMirror-matchingbracket { color: rgb(0, 255, 0); }
div.CodeMirror span.CodeMirror-nonmatchingbracket { color: rgb(255, 34, 34); }
.cm-s-inner .CodeMirror-activeline-background { background: inherit; }
.CodeMirror { position: relative; overflow: hidden; }
.CodeMirror-scroll { height: 100%; outline: 0px; position: relative; box-sizing: content-box; background: inherit; }
.CodeMirror-sizer { position: relative; }
.CodeMirror-gutter-filler, .CodeMirror-hscrollbar, .CodeMirror-scrollbar-filler, .CodeMirror-vscrollbar { position: absolute; z-index: 6; display: none; }
.CodeMirror-vscrollbar { right: 0px; top: 0px; overflow: hidden; }
.CodeMirror-hscrollbar { bottom: 0px; left: 0px; overflow: hidden; }
.CodeMirror-scrollbar-filler { right: 0px; bottom: 0px; }
.CodeMirror-gutter-filler { left: 0px; bottom: 0px; }
.CodeMirror-gutters { position: absolute; left: 0px; top: 0px; padding-bottom: 30px; z-index: 3; }
.CodeMirror-gutter { white-space: normal; height: 100%; box-sizing: content-box; padding-bottom: 30px; margin-bottom: -32px; display: inline-block; }
.CodeMirror-gutter-wrapper { position: absolute; z-index: 4; background: 0px 0px !important; border: none !important; }
.CodeMirror-gutter-background { position: absolute; top: 0px; bottom: 0px; z-index: 4; }
.CodeMirror-gutter-elt { position: absolute; cursor: default; z-index: 4; }
.CodeMirror-lines { cursor: text; }
.CodeMirror pre { border-radius: 0px; border-width: 0px; background: 0px 0px; font-family: inherit; font-size: inherit; margin: 0px; white-space: pre; overflow-wrap: normal; color: inherit; z-index: 2; position: relative; overflow: visible; }
.CodeMirror-wrap pre { overflow-wrap: break-word; white-space: pre-wrap; word-break: normal; }
.CodeMirror-code pre { border-right: 30px solid transparent; width: fit-content; }
.CodeMirror-wrap .CodeMirror-code pre { border-right: none; width: auto; }
.CodeMirror-linebackground { position: absolute; left: 0px; right: 0px; top: 0px; bottom: 0px; z-index: 0; }
.CodeMirror-linewidget { position: relative; z-index: 2; overflow: auto; }
.CodeMirror-wrap .CodeMirror-scroll { overflow-x: hidden; }
.CodeMirror-measure { position: absolute; width: 100%; height: 0px; overflow: hidden; visibility: hidden; }
.CodeMirror-measure pre { position: static; }
.CodeMirror div.CodeMirror-cursor { position: absolute; visibility: hidden; border-right: none; width: 0px; }
.CodeMirror div.CodeMirror-cursor { visibility: hidden; }
.CodeMirror-focused div.CodeMirror-cursor { visibility: inherit; }
.cm-searching { background: rgba(255, 255, 0, 0.4); }
@media print {
.CodeMirror div.CodeMirror-cursor { visibility: hidden; }
}
:root {
--side-bar-bg-color: #fafafa;
--control-text-color: #777;
}
@include-when-export url(https://fonts.loli.net/css?family=Open+Sans:400italic,700italic,700,400&subset=latin,latin-ext);
/* open-sans-regular - latin-ext_latin */
/* open-sans-italic - latin-ext_latin */
/* open-sans-700 - latin-ext_latin */
/* open-sans-700italic - latin-ext_latin */
html {
font-size: 16px;
}
body {
font-family: "Open Sans","Clear Sans", "Helvetica Neue", Helvetica, Arial, sans-serif;
color: rgb(51, 51, 51);
line-height: 1.6;
}
#write {
max-width: 860px;
margin: 0 auto;
padding: 30px;
padding-bottom: 100px;
}
@media only screen and (min-width: 1400px) {
#write {
max-width: 1024px;
}
}
@media only screen and (min-width: 1800px) {
#write {
max-width: 1200px;
}
}
#write > ul:first-child,
#write > ol:first-child{
margin-top: 30px;
}
a {
color: #4183C4;
}
h1,
h2,
h3,
h4,
h5,
h6 {
position: relative;
margin-top: 1rem;
margin-bottom: 1rem;
font-weight: bold;
line-height: 1.4;
cursor: text;
}
h1:hover a.anchor,
h2:hover a.anchor,
h3:hover a.anchor,
h4:hover a.anchor,
h5:hover a.anchor,
h6:hover a.anchor {
text-decoration: none;
}
h1 tt,
h1 code {
font-size: inherit;
}
h2 tt,
h2 code {
font-size: inherit;
}
h3 tt,
h3 code {
font-size: inherit;
}
h4 tt,
h4 code {
font-size: inherit;
}
h5 tt,
h5 code {
font-size: inherit;
}
h6 tt,
h6 code {
font-size: inherit;
}
h1 {
font-size: 2.25em;
line-height: 1.2;
border-bottom: 1px solid #eee;
}
h2 {
font-size: 1.75em;
line-height: 1.225;
border-bottom: 1px solid #eee;
}
/*@media print {
.typora-export h1,
.typora-export h2 {
border-bottom: none;
padding-bottom: initial;
}
.typora-export h1::after,
.typora-export h2::after {
content: "";
display: block;
height: 100px;
margin-top: -96px;
border-top: 1px solid #eee;
}
}*/
h3 {
font-size: 1.5em;
line-height: 1.43;
}
h4 {
font-size: 1.25em;
}
h5 {
font-size: 1em;
}
h6 {
font-size: 1em;
color: #777;
}
p,
blockquote,
ul,
ol,
dl,
table{
margin: 0.8em 0;
}
li>ol,
li>ul {
margin: 0 0;
}
hr {
height: 2px;
padding: 0;
margin: 16px 0;
background-color: #e7e7e7;
border: 0 none;
overflow: hidden;
box-sizing: content-box;
}
li p.first {
display: inline-block;
}
ul,
ol {
padding-left: 30px;
}
ul:first-child,
ol:first-child {
margin-top: 0;
}
ul:last-child,
ol:last-child {
margin-bottom: 0;
}
blockquote {
border-left: 4px solid #dfe2e5;
padding: 0 15px;
color: #777777;
}
blockquote blockquote {
padding-right: 0;
}
table {
padding: 0;
word-break: initial;
}
table tr {
border-top: 1px solid #dfe2e5;
margin: 0;
padding: 0;
}
table tr:nth-child(2n),
thead {
background-color: #f8f8f8;
}
table th {
font-weight: bold;
border: 1px solid #dfe2e5;
border-bottom: 0;
margin: 0;
padding: 6px 13px;
}
table td {
border: 1px solid #dfe2e5;
margin: 0;
padding: 6px 13px;
}
table th:first-child,
table td:first-child {
margin-top: 0;
}
table th:last-child,
table td:last-child {
margin-bottom: 0;
}
.CodeMirror-lines {
padding-left: 4px;
}
.code-tooltip {
box-shadow: 0 1px 1px 0 rgba(0,28,36,.3);
border-top: 1px solid #eef2f2;
}
.md-fences,
code,
tt {
border: 1px solid #e7eaed;
background-color: #f8f8f8;
border-radius: 3px;
padding: 0;
padding: 2px 4px 0px 4px;
font-size: 0.9em;
}
code {
background-color: #f3f4f4;
padding: 0 2px 0 2px;
}
.md-fences {
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<div id='write' class=''><div class='md-toc' mdtype='toc'><p class="md-toc-content" role="list"><span role="listitem" class="md-toc-item md-toc-h1" data-ref="n2"><a class="md-toc-inner" href="#1-引言introduction">1. 引言(Introduction)</a></span><span role="listitem" class="md-toc-item md-toc-h2" data-ref="n3"><a class="md-toc-inner" href="#11-welcome">1.1 Welcome</a></span><span role="listitem" class="md-toc-item md-toc-h2" data-ref="n40"><a class="md-toc-inner" href="#12-什么是机器学习what-is-machine-learning">1.2 什么是机器学习(What is Machine Learning)</a></span><span role="listitem" class="md-toc-item md-toc-h2" data-ref="n79"><a class="md-toc-inner" href="#13-监督学习supervised-learning">1.3 监督学习(Supervised Learning)</a></span><span role="listitem" class="md-toc-item md-toc-h2" data-ref="n96"><a class="md-toc-inner" href="#14-无监督学习unsupervised-learning">1.4 无监督学习(Unsupervised Learning)</a></span><span role="listitem" class="md-toc-item md-toc-h1" data-ref="n130"><a class="md-toc-inner" href="#2-单变量线性回归linear-regression-with-one-variable">2 单变量线性回归(Linear Regression with One Variable)</a></span><span role="listitem" class="md-toc-item md-toc-h2" data-ref="n131"><a class="md-toc-inner" href="#21-模型表示model-representation">2.1 模型表示(Model Representation)</a></span><span role="listitem" class="md-toc-item md-toc-h2" data-ref="n165"><a class="md-toc-inner" href="#22-代价函数cost-function">2.2 代价函数(Cost Function)</a></span><span role="listitem" class="md-toc-item md-toc-h2" data-ref="n189"><a class="md-toc-inner" href="#23-代价函数---直观理解1cost-function---intuition-i">2.3 代价函数 - 直观理解1(Cost Function - Intuition I)</a></span><span role="listitem" class="md-toc-item md-toc-h2" data-ref="n203"><a class="md-toc-inner" href="#24-代价函数---直观理解2cost-function---intuition-ii">2.4 代价函数 - 直观理解2(Cost Function - Intuition II)</a></span><span role="listitem" class="md-toc-item md-toc-h2" data-ref="n216"><a class="md-toc-inner" href="#25-梯度下降gradient-descent">2.5 梯度下降(Gradient Descent)</a></span><span role="listitem" class="md-toc-item md-toc-h2" data-ref="n233"><a class="md-toc-inner" href="#26-梯度下降直观理解gradient-descent-intuition">2.6 梯度下降直观理解(Gradient Descent Intuition)</a></span><span role="listitem" class="md-toc-item md-toc-h2" data-ref="n255"><a class="md-toc-inner" href="#27-线性回归中的梯度下降gradient-descent-for-linear-regression">2.7 线性回归中的梯度下降(Gradient Descent For Linear Regression)</a></span><span role="listitem" class="md-toc-item md-toc-h1" data-ref="n277"><a class="md-toc-inner" href="#3-linear-algebra-review">3 Linear Algebra Review</a></span><span role="listitem" class="md-toc-item md-toc-h2" data-ref="n279"><a class="md-toc-inner" href="#31-matrices-and-vectors">3.1 Matrices and Vectors</a></span><span role="listitem" class="md-toc-item md-toc-h2" data-ref="n284"><a class="md-toc-inner" href="#32-addition-and-scalar-multiplication">3.2 Addition and Scalar Multiplication</a></span><span role="listitem" class="md-toc-item md-toc-h2" data-ref="n289"><a class="md-toc-inner" href="#33-matrix-vector-multiplication">3.3 Matrix Vector Multiplication</a></span><span role="listitem" class="md-toc-item md-toc-h2" data-ref="n294"><a class="md-toc-inner" href="#34-matrix-matrix-multiplication">3.4 Matrix Matrix Multiplication</a></span><span role="listitem" class="md-toc-item md-toc-h2" data-ref="n299"><a class="md-toc-inner" href="#35-matrix-multiplication-properties">3.5 Matrix Multiplication Properties</a></span><span role="listitem" class="md-toc-item md-toc-h2" data-ref="n304"><a class="md-toc-inner" href="#36-inverse-and-transpose">3.6 Inverse and Transpose</a></span></p></div><h1><a name="1-引言introduction" class="md-header-anchor"></a><span>1. 引言(Introduction)</span></h1><h2><a name="11-welcome" class="md-header-anchor"></a><span>1.1 Welcome</span></h2><p><span>随着互联网数据不断累积，硬件不断升级迭代，在这个信息爆炸的时代，机器学习已被应用在各行各业中，可谓无处不在。</span></p><p><span>一些常见的机器学习的应用，例如：</span></p><ul><li><span>手写识别</span></li><li><span>垃圾邮件分类</span></li><li><span>搜索引擎</span></li><li><span>图像处理</span></li><li><span>…</span></li></ul><p><span>使用到机器学习的一些案例：</span></p><ul><li><p><span>数据挖掘</span></p><ul><li><span>网页点击流数据分析</span></li></ul></li><li><p><span>人工无法处理的工作(量大)</span></p><ul><li><span>手写识别</span></li><li><span>计算机视觉</span></li></ul></li><li><p><span>个人定制</span></p><ul><li><span>推荐系统</span></li></ul></li><li><p><span>研究大脑</span></p></li><li><p><span>……</span></p></li></ul><h2><a name="12-什么是机器学习what-is-machine-learning" class="md-header-anchor"></a><span>1.2 什么是机器学习(What is Machine Learning)</span></h2><ol start='' ><li><span>机器学习定义</span>
<span>这里主要有两种定义：</span></li></ol><ul><li><p><span>Arthur Samuel (1959). Machine Learning: Field of study that gives computers the ability to learn without being explicitly programmed.</span></p><p><span>这个定义有点不正式但提出的时间最早，来自于一个懂得计算机编程的下棋菜鸟。他编写了一个程序，但没有显式地编程每一步该怎么走，而是让计算机自己和自己对弈，并不断地计算布局的好坏，来判断什么情况下获胜的概率高，从而积累经验，好似学习，最后，这个计算机程序成为了一个比他自己还厉害的棋手。</span></p></li><li><p><span>Tom Mitchell (1998) Well-posed Learning Problem: A computer program is said to learn from experience E with respect to some </span><strong><span>task T</span></strong><span> and some </span><strong><span>performance measure P</span></strong><span>, if its performance on T, as measured by P, improves with </span><strong><span>experience E</span></strong><span>. </span></p><p><span>Tom Mitchell 的定义更为现代和正式。在过滤垃圾邮件这个例子中，电子邮件系统会根据用户对电子邮件的标记（是/不是垃圾邮件）不断学习，从而提升过滤垃圾邮件的准确率，定义中的三个字母分别代表：</span></p><ul><li><span>T(Task): 过滤垃圾邮件任务。</span></li><li><span>P(Performance): 电子邮件系统过滤垃圾邮件的准确率。</span></li><li><span>E(Experience): 用户对电子邮件的标记。</span></li></ul></li></ul><ol start='2' ><li><p><span>机器学习算法</span></p><p><span>主要有两种机器学习的算法分类</span></p><ol start='' ><li><span>监督学习</span></li><li><span>无监督学习</span></li></ol><p><span>两者的区别为</span><strong><span>是否需要人工参与数据结果的标注</span></strong><span>。这两部分的内容占比很大，并且很重要，掌握好了可以在以后的应用中节省大把大把的时间~</span></p><p><span>还有一些算法也属于机器学习领域，诸如：</span></p><ul><li><span>半监督学习: 介于监督学习于无监督学习之间</span></li><li><span>推荐算法: 没错，就是那些个买完某商品后还推荐同款的某购物网站所用的算法。</span></li><li><span>强化学习: 通过观察来学习如何做出动作，每个动作都会对环境有所影响，而环境的反馈又可以引导该学习算法。</span></li><li><span>迁移学习</span></li></ul></li></ol><h2><a name="13-监督学习supervised-learning" class="md-header-anchor"></a><span>1.3 监督学习(Supervised Learning)</span></h2><p><span>监督学习，即为教计算机如何去完成预测任务（有反馈），预先给一定数据量的输入</span><strong><span>和对应的结果</span></strong><span>即训练集，建模拟合，最后让计算机预测未知数据的结果。</span></p><p><span>监督学习一般有两种：</span></p><ol start='' ><li><p><span>回归问题(Regression)</span></p><p><span>回归问题即为预测一系列的</span><strong><span>连续值</span></strong><span>。</span></p><p><span>在房屋价格预测的例子中，给出了一系列的房屋面基数据，根据这些数据来预测任意面积的房屋价格。给出照片-年龄数据集，预测给定照片的年龄。</span></p><p><img src="images/20180105_194712.png" referrerpolicy="no-referrer"></p></li><li><p><span>分类问题(Classification)</span></p><p><span>分类问题即为预测一系列的</span><strong><span>离散值</span></strong><span>。</span></p><p><span>即根据数据预测被预测对象属于哪个分类。</span></p><p><span>视频中举了癌症肿瘤这个例子，针对诊断结果，分别分类为良性或恶性。还例如垃圾邮件分类问题，也同样属于监督学习中的分类问题。</span></p><p><img src="images/20180105_194839.png" referrerpolicy="no-referrer"></p></li></ol><p><span>视频中提到</span><strong><span>支持向量机</span></strong><span>这个算法，旨在解决当特征量很大的时候(特征即如癌症例子中的肿块大小，颜色，气味等各种特征)，计算机内存一定会不够用的情况。</span><strong><span>支持向量机能让计算机处理无限多个特征。</span></strong></p><h2><a name="14-无监督学习unsupervised-learning" class="md-header-anchor"></a><span>1.4 无监督学习(Unsupervised Learning)</span></h2><p><span>相对于监督学习，训练集不会有人为标注的结果（无反馈），我们</span><strong><span>不会给出</span></strong><span>结果或</span><strong><span>无法得知</span></strong><span>训练集的结果是什么样，而是单纯由计算机通过无监督学习算法自行分析，从而“得出结果”。计算机可能会把特定的数据集归为几个不同的簇，故叫做聚类算法。</span></p><p><span>无监督学习一般分为两种：</span></p><ol start='' ><li><p><span>聚类(Clustering)</span></p><ul><li><span>新闻聚合</span></li><li><span>DNA 个体聚类</span></li><li><span>天文数据分析</span></li><li><span>市场细分</span></li><li><span>社交网络分析</span></li></ul></li><li><p><span>非聚类(Non-clustering)</span></p><ul><li><span>鸡尾酒问题</span></li></ul></li></ol><p><strong><span>新闻聚合</span></strong></p><p><span>在例如谷歌新闻这样的网站中，每天后台都会收集成千上万的新闻，然后将这些新闻分组成一个个的新闻专题，这样一个又一个聚类，就是应用了无监督学习的结果。</span></p><p><strong><span>鸡尾酒问题</span></strong></p><p><img src="images/20180105_201639.png" referrerpolicy="no-referrer"></p><p><span>在鸡尾酒会上，大家说话声音彼此重叠，几乎很难分辨出面前的人说了什么。我们很难对于这个问题进行数据标注，而这里的通过机器学习的无监督学习算法，就可以将说话者的声音同背景音乐分离出来，看视频，效果还不错呢</span><span>~</span><span>~。</span></p><p><span>嗯，这块是打打鸡血的，只需要一行代码就解决了问题，就是这么简单！当然，我没复现过 ^_^……</span></p><p><span>神奇的一行代码：</span>
<code>[W,s,v] = svd((repmat(sum(x.*x,1),size(x,1),1).*x)*x');</code></p><p><strong><span>编程语言建议</span></strong></p><p><span>在机器学习刚开始时，</span><strong><span>推荐使用 Octave 类的工程计算编程软件</span></strong><span>，因为在 C++ 或 Java 等编程语言中，编写对应的代码需要用到复杂的库以及要写大量的冗余代码，比较耗费时间，建议可以在学习过后再考虑使用其他语言来构建系统。</span>
<span>另外，在做</span><strong><span>原型搭建</span></strong><span>的时候也应该先考虑使用类似于 Octave 这种便于计算的编程软件，当其已经可以工作后，才将模型移植到其他的高级编程语言中。</span></p><blockquote><p><span>注：Octave 与 MATLAB 语法相近，由于 MATLAB 为商业软件，课程中使用开源且免费的 Octave。</span></p><p><span>机器学习领域发展迅速，现在也可使用 Tensorflow 等开源机器学习框架编写机器学习代码，这些框架十分友好，易于编写及应用。</span></p></blockquote><h1><a name="2-单变量线性回归linear-regression-with-one-variable" class="md-header-anchor"></a><span>2 单变量线性回归(Linear Regression with One Variable)</span></h1><h2><a name="21-模型表示model-representation" class="md-header-anchor"></a><span>2.1 模型表示(Model Representation)</span></h2><ol start='' ><li><span>房价预测训练集</span></li></ol><figure><table><thead><tr><th><span>Size in </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.334ex" height="2.816ex" viewBox="0 -908.4 2296.6 1212.6" role="img" focusable="false" style="vertical-align: -0.706ex;"><defs><path stroke-width="0" id="E6-MJMATHI-66" 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340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E6-MJMATHI-66" x="0" y="0"></use><use xlink:href="#E6-MJMATHI-65" x="550" y="0"></use><use xlink:href="#E6-MJMATHI-65" x="1016" y="0"></use><g transform="translate(1482,0)"><use xlink:href="#E6-MJMATHI-74" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E6-MJMAIN-32" x="510" y="513"></use></g></g></svg></span><script type="math/tex">feet^2</script><span> (</span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.329ex" height="1.41ex" viewBox="0 -504.6 572 607.1" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E17-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E17-MJMATHI-78" x="0" y="0"></use></g></svg></span><script type="math/tex">x</script><span>)</span></th><th><span>Price (</span><span>$</span><span>) in 1000's(</span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.154ex" height="1.877ex" viewBox="0 -504.6 497 808.1" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="0" id="E32-MJMATHI-79" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E32-MJMATHI-79" x="0" y="0"></use></g></svg></span><script type="math/tex">y</script><span>)</span></th></tr></thead><tbody><tr><td><span>2104</span></td><td><span>460</span></td></tr><tr><td><span>1416</span></td><td><span>232</span></td></tr><tr><td><span>1534</span></td><td><span>315</span></td></tr><tr><td><span>852</span></td><td><span>178</span></td></tr><tr><td><span>...</span></td><td><span>...</span></td></tr></tbody></table></figure><p><span>房价预测训练集中，同时给出了输入 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.329ex" height="1.41ex" viewBox="0 -504.6 572 607.1" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E17-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E17-MJMATHI-78" x="0" y="0"></use></g></svg></span><script type="math/tex">x</script><span> 和输出结果 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.154ex" height="1.877ex" viewBox="0 -504.6 497 808.1" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="0" id="E32-MJMATHI-79" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E32-MJMATHI-79" x="0" y="0"></use></g></svg></span><script type="math/tex">y</script><span>，即给出了人为标注的</span><strong><span>”正确结果“</span></strong><span>，且预测的量是连续的，属于监督学习中的回归问题。</span></p><ol start='2' ><li><strong><span>问题解决模型</span></strong></li></ol><p><img src="images/20180105_212048.png" referrerpolicy="no-referrer"></p><p><span>其中 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.338ex" height="1.994ex" viewBox="0 -755.9 576 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E48-MJMATHI-68" d="M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E48-MJMATHI-68" x="0" y="0"></use></g></svg></span><script type="math/tex">h</script><span> 代表结果函数，也称为</span><strong><span>假设(hypothesis)</span></strong><span> 。假设函数根据输入(房屋的面积)，给出预测结果输出(房屋的价格)，即是一个 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="7.364ex" height="1.994ex" viewBox="0 -755.9 3170.6 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E12-MJMATHI-58" d="M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z"></path><path stroke-width="0" id="E12-MJMAIN-2192" d="M56 237T56 250T70 270H835Q719 357 692 493Q692 494 692 496T691 499Q691 511 708 511H711Q720 511 723 510T729 506T732 497T735 481T743 456Q765 389 816 336T935 261Q944 258 944 250Q944 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634T96 635T83 636T66 637Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E12-MJMATHI-58" x="0" y="0"></use><use xlink:href="#E12-MJMAIN-2192" x="1129" y="0"></use><use xlink:href="#E12-MJMATHI-59" x="2407" y="0"></use></g></svg></span><script type="math/tex">X\to Y</script><span> 的映射。</span></p><p><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="17.026ex" height="2.577ex" viewBox="0 -806.1 7330.7 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="0" id="E77-MJMATHI-68" d="M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z"></path><path stroke-width="0" id="E77-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="0" id="E77-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="0" id="E77-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path stroke-width="0" id="E77-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path><path stroke-width="0" id="E77-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path stroke-width="0" id="E77-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path stroke-width="0" id="E77-MJMAIN-2B" d="M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z"></path><path stroke-width="0" id="E77-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E77-MJMATHI-68" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E77-MJMATHI-3B8" x="814" y="-218"></use><use xlink:href="#E77-MJMAIN-28" x="1007" y="0"></use><use xlink:href="#E77-MJMATHI-78" x="1396" y="0"></use><use xlink:href="#E77-MJMAIN-29" x="1968" y="0"></use><use xlink:href="#E77-MJMAIN-3D" x="2635" y="0"></use><g transform="translate(3691,0)"><use xlink:href="#E77-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E77-MJMAIN-30" x="663" y="-213"></use></g><use xlink:href="#E77-MJMAIN-2B" x="4835" y="0"></use><g transform="translate(5836,0)"><use xlink:href="#E77-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E77-MJMAIN-31" x="663" y="-213"></use></g><use xlink:href="#E77-MJMATHI-78" x="6758" y="0"></use></g></svg></span><script type="math/tex">h_\theta(x)=\theta_0+\theta_1x</script><span>，为解决房价问题的一种可行表达式。</span></p><blockquote><p><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.329ex" height="1.41ex" viewBox="0 -504.6 572 607.1" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E17-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E17-MJMATHI-78" x="0" y="0"></use></g></svg></span><script type="math/tex">x</script><span>: 特征/输入变量。</span></p></blockquote><p><span>上式中，</span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.089ex" height="2.11ex" viewBox="0 -806.1 469 908.7" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E74-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E74-MJMATHI-3B8" x="0" y="0"></use></g></svg></span><script type="math/tex">\theta</script><span> 为参数，</span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.089ex" height="2.11ex" viewBox="0 -806.1 469 908.7" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E74-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E74-MJMATHI-3B8" x="0" y="0"></use></g></svg></span><script type="math/tex">\theta</script><span> 的变化才决定了输出结果，不同以往，这里的 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.329ex" height="1.41ex" viewBox="0 -504.6 572 607.1" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E17-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E17-MJMATHI-78" x="0" y="0"></use></g></svg></span><script type="math/tex">x</script><span> 被我们</span><strong><span>视作已知</span></strong><span>(不论是数据集还是预测时的输入)，所以怎样解得 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.089ex" height="2.11ex" viewBox="0 -806.1 469 908.7" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E74-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E74-MJMATHI-3B8" x="0" y="0"></use></g></svg></span><script type="math/tex">\theta</script><span> 以更好地拟合数据，成了求解该问题的最终问题。</span></p><p><span>单变量，即只有一个特征(如例子中房屋的面积这个特征)。</span></p><h2><a name="22-代价函数cost-function" class="md-header-anchor"></a><span>2.2 代价函数(Cost Function)</span></h2><blockquote><p><span>李航《统计学习方法》一书中，损失函数与代价函数两者为</span><strong><span>同一概念</span></strong><span>，未作细分区别，全书没有和《深度学习》一书一样混用，而是统一使用</span><strong><span>损失函数</span></strong><span>来指代这类类似概念。</span></p><p><span>吴恩达(Andrew Ng)老师在其公开课中对两者做了细分。</span><strong><span>如果要听他的课做作业，不细分这两个概念是会被打小手扣分的</span></strong><span>！这也可能是因为老师发现了业内混用的乱象，想要治一治吧。</span></p><p><strong><span>损失函数</span></strong><span>(Loss/Error Function): 计算</span><strong><span>单个</span></strong><span>样本的误差。</span><a href='https://www.coursera.org/learn/neural-networks-deep-learning/lecture/yWaRd/logistic-regression-cost-function'><span>link</span></a></p><p><strong><span>代价函数</span></strong><span>(Cost Function): 计算整个训练集</span><strong><span>所有损失函数之和的平均值</span></strong></p><p> </p><p><span>综合考虑，本笔记对两者概念进行细分，若有所谬误，欢迎指正。</span></p><p><a href='https://www.zhihu.com/question/52398145/answer/298003145'><span>机器学习中的目标函数、损失函数、代价函数有什么区别？- 知乎</span></a></p></blockquote><p> </p><p><span>我们的目的在于求解预测结果 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.338ex" height="1.994ex" viewBox="0 -755.9 576 858.4" role="img" focusable="false" style="vertical-align: 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xlink:href="#E32-MJMATHI-79" x="0" y="0"></use></g></svg></span><script type="math/tex">y</script><span> 时 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.089ex" height="2.11ex" viewBox="0 -806.1 469 908.7" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E74-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E74-MJMATHI-3B8" x="0" y="0"></use></g></svg></span><script type="math/tex">\theta</script><span> 的取值，则问题可表达为</span><strong><span>求解 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="18.1ex" height="5.846ex" viewBox="0 -1459.5 7793 2517" role="img" focusable="false" style="vertical-align: -2.456ex;"><defs><path stroke-width="0" id="E22-MJSZ1-2211" d="M61 748Q64 750 489 750H913L954 640Q965 609 976 579T993 533T999 516H979L959 517Q936 579 886 621T777 682Q724 700 655 705T436 710H319Q183 710 183 709Q186 706 348 484T511 259Q517 250 513 244L490 216Q466 188 420 134T330 27L149 -187Q149 -188 362 -188Q388 -188 436 -188T506 -189Q679 -189 778 -162T936 -43Q946 -27 959 6H999L913 -249L489 -250Q65 -250 62 -248Q56 -246 56 -239Q56 -234 118 -161Q186 -81 245 -11L428 206Q428 207 242 462L57 717L56 728Q56 744 61 748Z"></path><path stroke-width="0" id="E22-MJMATHI-69" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 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6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E22-MJSZ1-2211" x="45" y="0"></use><g transform="translate(0,-888)"><use transform="scale(0.707)" xlink:href="#E22-MJMATHI-69" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E22-MJMAIN-3D" x="345" y="0"></use><use transform="scale(0.707)" xlink:href="#E22-MJMAIN-30" x="1123" y="0"></use></g><use transform="scale(0.707)" xlink:href="#E22-MJMATHI-6D" x="372" y="1343"></use><use xlink:href="#E22-MJMAIN-28" x="1147" y="0"></use><g transform="translate(1536,0)"><use xlink:href="#E22-MJMATHI-68" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E22-MJMATHI-3B8" x="814" y="-218"></use></g><use xlink:href="#E22-MJMAIN-28" x="2544" y="0"></use><g transform="translate(2933,0)"><use xlink:href="#E22-MJMATHI-78" x="0" y="0"></use><g transform="translate(572,362)"><use transform="scale(0.707)" xlink:href="#E22-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E22-MJMATHI-69" x="389" y="0"></use><use transform="scale(0.707)" xlink:href="#E22-MJMAIN-29" x="733" y="0"></use></g></g><use xlink:href="#E22-MJMAIN-29" x="4399" y="0"></use><use xlink:href="#E22-MJMAIN-2212" x="5010" y="0"></use><g transform="translate(6010,0)"><use xlink:href="#E22-MJMATHI-79" x="0" y="0"></use><g transform="translate(499,362)"><use transform="scale(0.707)" xlink:href="#E22-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E22-MJMATHI-69" x="389" y="0"></use><use transform="scale(0.707)" xlink:href="#E22-MJMAIN-29" x="733" y="0"></use></g></g><use xlink:href="#E22-MJMAIN-29" x="7404" y="0"></use></g></svg></span><script type="math/tex">\sum\limits_{i=0}^{m}(h_\theta(x^{(i)})-y^{(i)})</script><span> 的最小值</span></strong><span>。</span></p><blockquote><p><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.039ex" height="1.41ex" viewBox="0 -504.6 878 607.1" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E23-MJMATHI-6D" d="M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 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412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E32-MJMATHI-79" x="0" y="0"></use></g></svg></span><script type="math/tex">y</script><span>: 目标变量/输出变量</span></p><p><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.323ex" height="2.577ex" viewBox="0 -806.1 2291.7 1109.7" role="img" focusable="false" style="vertical-align: 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width="9.801ex" height="3.161ex" viewBox="0 -956.9 4220 1361" role="img" focusable="false" style="vertical-align: -0.938ex;"><defs><path stroke-width="0" id="E26-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="0" id="E26-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 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y="0"></use><use transform="scale(0.707)" xlink:href="#E26-MJMAIN-29" x="733" y="0"></use></g></g><use xlink:href="#E26-MJMAIN-2C" x="1924" y="0"></use><g transform="translate(2368,0)"><use xlink:href="#E26-MJMATHI-79" x="0" y="0"></use><g transform="translate(499,362)"><use transform="scale(0.707)" xlink:href="#E26-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E26-MJMATHI-69" x="389" y="0"></use><use transform="scale(0.707)" xlink:href="#E26-MJMAIN-29" x="733" y="0"></use></g></g><use xlink:href="#E26-MJSZ1-29" x="3761" y="-1"></use></g></svg></span><script type="math/tex">\left(x^{\left(i\right)},y^{\left(i\right)}\right)</script><span>: 训练集中的第 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="0.801ex" height="1.994ex" viewBox="0 -755.9 345 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E27-MJMATHI-69" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E27-MJMATHI-69" x="0" y="0"></use></g></svg></span><script type="math/tex">i</script><span> 个样本实例</span></p></blockquote><p><img src="images/20180105_224648.png" referrerpolicy="no-referrer"></p><p><span>上图展示了当 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.089ex" height="2.11ex" viewBox="0 -806.1 469 908.7" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E74-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E74-MJMATHI-3B8" x="0" y="0"></use></g></svg></span><script type="math/tex">\theta</script><span> 取不同值时，</span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.863ex" height="2.577ex" viewBox="0 -806.1 2524.3 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="0" id="E56-MJMATHI-68" d="M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z"></path><path stroke-width="0" id="E56-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 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26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path stroke-width="0" id="E56-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E56-MJMATHI-68" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E56-MJMATHI-3B8" x="814" y="-218"></use><g transform="translate(1174,0)"><use xlink:href="#E56-MJMAIN-28" x="0" y="0"></use><use xlink:href="#E56-MJMATHI-78" x="389" y="0"></use><use xlink:href="#E56-MJMAIN-29" x="961" y="0"></use></g></g></svg></span><script type="math/tex">h_\theta\left(x\right)</script><span> 对数据集的拟合情况，蓝色虚线部分代表</span><strong><span>建模误差</span></strong><span>（预测结果与实际结果之间的误差），我们的目标就是最小化所有误差之和。</span></p><p><span>为了求解最小值，引入代价函数(Cost Function)概念，用于度量建模误差。考虑到要计算最小值，应用二次函数对求和式建模，即应用统计学中的平方损失函数（最小二乘法）：</span></p><p><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="54.767ex" height="6.78ex" viewBox="0 -1660.6 23580.2 2919" role="img" focusable="false" style="vertical-align: -2.923ex;"><defs><path stroke-width="0" id="E30-MJMATHI-4A" d="M447 625Q447 637 354 637H329Q323 642 323 645T325 664Q329 677 335 683H352Q393 681 498 681Q541 681 568 681T605 682T619 682Q633 682 633 672Q633 670 630 658Q626 642 623 640T604 637Q552 637 545 623Q541 610 483 376Q420 128 419 127Q397 64 333 21T195 -22Q137 -22 97 8T57 88Q57 130 80 152T132 174Q177 174 182 130Q182 98 164 80T123 56Q115 54 115 53T122 44Q148 15 197 15Q235 15 271 47T324 130Q328 142 387 380T447 625Z"></path><path stroke-width="0" id="E30-MJMAIN-28" d="M94 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transform="scale(0.707)" xlink:href="#E30-MJMAIN-32" x="7829" y="674"></use></g></g></g></g></svg></span><script type="math/tex">J(\theta_0,\theta_1)=\dfrac{1}{2m}\displaystyle\sum_{i=1}^m\left(\hat{y}_{i}-y_{i} \right)^2=\dfrac{1}{2m}\displaystyle\sum_{i=1}^m\left(h_\theta(x_{i})-y_{i}\right)^2</script><span> </span></p><blockquote><p><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.301ex" height="2.461ex" viewBox="0 -755.9 560.2 1059.4" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="0" id="E31-MJMATHI-79" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="0" id="E31-MJMAIN-5E" d="M112 560L249 694L257 686Q387 562 387 560L361 531Q359 532 303 581L250 627L195 580Q182 569 169 557T148 538L140 532Q138 530 125 546L112 560Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E31-MJMATHI-79" x="1" y="0"></use><use xlink:href="#E31-MJMAIN-5E" x="60" y="-13"></use></g></svg></span><script type="math/tex">\hat{y}</script><span>: </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.154ex" height="1.877ex" viewBox="0 -504.6 497 808.1" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="0" id="E32-MJMATHI-79" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E32-MJMATHI-79" x="0" y="0"></use></g></svg></span><script type="math/tex">y</script><span> 的预测值</span></p><p><span>系数 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.657ex" height="3.278ex" viewBox="0 -956.9 713.6 1411.3" role="img" focusable="false" style="vertical-align: -1.055ex;"><defs><path stroke-width="0" id="E34-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path stroke-width="0" id="E34-MJMAIN-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g transform="translate(120,0)"><rect stroke="none" width="473" height="60" x="0" y="220"></rect><use transform="scale(0.707)" xlink:href="#E34-MJMAIN-31" x="84" y="571"></use><use transform="scale(0.707)" xlink:href="#E34-MJMAIN-32" x="84" y="-531"></use></g></g></svg></span><script type="math/tex">\frac{1}{2}</script><span> 存在与否都不会影响结果，这里是为了在应用梯度下降时便于求解，平方的导数会抵消掉 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.657ex" height="3.278ex" viewBox="0 -956.9 713.6 1411.3" role="img" focusable="false" style="vertical-align: -1.055ex;"><defs><path stroke-width="0" id="E34-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path stroke-width="0" id="E34-MJMAIN-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g 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-194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E35-MJMATHI-4A" x="0" y="0"></use><g transform="translate(799,0)"><use xlink:href="#E35-MJMAIN-28" x="0" y="0"></use><g transform="translate(389,0)"><use xlink:href="#E35-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E35-MJMAIN-30" x="663" y="-213"></use></g><use xlink:href="#E35-MJMAIN-2C" x="1311" y="0"></use><g transform="translate(1756,0)"><use xlink:href="#E35-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E35-MJMAIN-31" x="663" y="-213"></use></g><use xlink:href="#E35-MJMAIN-29" x="2678" y="0"></use></g></g></svg></span><script type="math/tex">J\left( \theta_0, \theta_1 \right)</script><span> 的最小值</span></strong><span>。</span></p><h2><a name="23-代价函数---直观理解1cost-function---intuition-i" class="md-header-anchor"></a><span>2.3 代价函数 - 直观理解1(Cost Function - Intuition I)</span></h2><p><span>根据上节视频，列出如下定义：</span></p><ul><li><span>假设函数(Hypothesis): </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="17.026ex" height="2.577ex" viewBox="0 -806.1 7330.7 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="0" id="E77-MJMATHI-68" d="M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 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299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path stroke-width="0" id="E77-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path><path stroke-width="0" id="E77-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path stroke-width="0" id="E77-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path stroke-width="0" id="E77-MJMAIN-2B" d="M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z"></path><path stroke-width="0" id="E77-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E77-MJMATHI-68" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E77-MJMATHI-3B8" x="814" y="-218"></use><use xlink:href="#E77-MJMAIN-28" x="1007" y="0"></use><use xlink:href="#E77-MJMATHI-78" x="1396" y="0"></use><use xlink:href="#E77-MJMAIN-29" x="1968" y="0"></use><use xlink:href="#E77-MJMAIN-3D" x="2635" y="0"></use><g transform="translate(3691,0)"><use xlink:href="#E77-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E77-MJMAIN-30" x="663" y="-213"></use></g><use xlink:href="#E77-MJMAIN-2B" x="4835" y="0"></use><g transform="translate(5836,0)"><use xlink:href="#E77-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E77-MJMAIN-31" x="663" y="-213"></use></g><use xlink:href="#E77-MJMATHI-78" x="6758" y="0"></use></g></svg></span><script type="math/tex">h_\theta(x)=\theta_0+\theta_1x</script></li><li><span>参数(Parameters): </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.318ex" height="2.461ex" viewBox="0 -806.1 2289.8 1059.4" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="0" id="E52-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="0" id="E52-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 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x="372" y="1343"></use></g><g transform="translate(8016,0)"><use xlink:href="#E78-MJSZ1-28"></use><g transform="translate(458,0)"><use xlink:href="#E78-MJMATHI-68" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E78-MJMATHI-3B8" x="814" y="-218"></use></g><g transform="translate(1632,0)"><use xlink:href="#E78-MJSZ1-28"></use><g transform="translate(458,0)"><use xlink:href="#E78-MJMATHI-78" x="0" y="0"></use><g transform="translate(572,362)"><use transform="scale(0.707)" xlink:href="#E78-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E78-MJMATHI-69" x="389" y="0"></use><use transform="scale(0.707)" xlink:href="#E78-MJMAIN-29" x="733" y="0"></use></g></g><use xlink:href="#E78-MJSZ1-29" x="1924" y="-1"></use></g><use xlink:href="#E78-MJMAIN-2212" x="4236" y="0"></use><g transform="translate(5236,0)"><use xlink:href="#E78-MJMATHI-79" x="0" y="0"></use><g transform="translate(499,362)"><use transform="scale(0.707)" xlink:href="#E78-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E78-MJMATHI-69" x="389" y="0"></use><use transform="scale(0.707)" xlink:href="#E78-MJMAIN-29" x="733" y="0"></use></g></g><use xlink:href="#E78-MJSZ1-29" x="6630" y="-1"></use><use transform="scale(0.707)" xlink:href="#E78-MJMAIN-32" x="10024" y="877"></use></g></g></svg></span><script type="math/tex">J\left( \theta_0, \theta_1 \right)=\frac{1}{2m}\sum\limits_{i=1}^{m}{{{\left( {{h}_{\theta }}\left( {{x}^{(i)}} \right)-{{y}^{(i)}} \right)}^{2}}}</script></li><li><span>目标(Goal): </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="18.143ex" height="4.212ex" viewBox="0 -806.1 7811.4 1813.3" role="img" focusable="false" style="vertical-align: -2.339ex;"><defs><path stroke-width="0" id="E39-MJMAIN-6D" d="M41 46H55Q94 46 102 60V68Q102 77 102 91T102 122T103 161T103 203Q103 234 103 269T102 328V351Q99 370 88 376T43 385H25V408Q25 431 27 431L37 432Q47 433 65 434T102 436Q119 437 138 438T167 441T178 442H181V402Q181 364 182 364T187 369T199 384T218 402T247 421T285 437Q305 442 336 442Q351 442 364 440T387 434T406 426T421 417T432 406T441 395T448 384T452 374T455 366L457 361L460 365Q463 369 466 373T475 384T488 397T503 410T523 422T546 432T572 439T603 442Q729 442 740 329Q741 322 741 190V104Q741 66 743 59T754 49Q775 46 803 46H819V0H811L788 1Q764 2 737 2T699 3Q596 3 587 0H579V46H595Q656 46 656 62Q657 64 657 200Q656 335 655 343Q649 371 635 385T611 402T585 404Q540 404 506 370Q479 343 472 315T464 232V168V108Q464 78 465 68T468 55T477 49Q498 46 526 46H542V0H534L510 1Q487 2 460 2T422 3Q319 3 310 0H302V46H318Q379 46 379 62Q380 64 380 200Q379 335 378 343Q372 371 358 385T334 402T308 404Q263 404 229 370Q202 343 195 315T187 232V168V108Q187 78 188 68T191 55T200 49Q221 46 249 46H265V0H257L234 1Q210 2 183 2T145 3Q42 3 33 0H25V46H41Z"></path><path stroke-width="0" id="E39-MJMAIN-69" d="M69 609Q69 637 87 653T131 669Q154 667 171 652T188 609Q188 579 171 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id="E39-MJMAIN-7A" d="M42 263Q44 270 48 345T53 423V431H393Q399 425 399 415Q399 403 398 402L381 378Q364 355 331 309T265 220L134 41L182 40H206Q254 40 283 46T331 77Q352 105 359 185L361 201Q361 202 381 202H401V196Q401 195 393 103T384 6V0H209L34 1L31 3Q28 8 28 17Q28 30 29 31T160 210T294 394H236Q169 393 152 388Q127 382 113 367Q89 344 82 264V255H42V263Z"></path><path stroke-width="0" id="E39-MJMAIN-65" d="M28 218Q28 273 48 318T98 391T163 433T229 448Q282 448 320 430T378 380T406 316T415 245Q415 238 408 231H126V216Q126 68 226 36Q246 30 270 30Q312 30 342 62Q359 79 369 104L379 128Q382 131 395 131H398Q415 131 415 121Q415 117 412 108Q393 53 349 21T250 -11Q155 -11 92 58T28 218ZM333 275Q322 403 238 411H236Q228 411 220 410T195 402T166 381T143 340T127 274V267H333V275Z"></path><path stroke-width="0" id="E39-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="0" id="E39-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path stroke-width="0" id="E39-MJMAIN-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z"></path><path stroke-width="0" id="E39-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path stroke-width="0" id="E39-MJMATHI-4A" d="M447 625Q447 637 354 637H329Q323 642 323 645T325 664Q329 677 335 683H352Q393 681 498 681Q541 681 568 681T605 682T619 682Q633 682 633 672Q633 670 630 658Q626 642 623 640T604 637Q552 637 545 623Q541 610 483 376Q420 128 419 127Q397 64 333 21T195 -22Q137 -22 97 8T57 88Q57 130 80 152T132 174Q177 174 182 130Q182 98 164 80T123 56Q115 54 115 53T122 44Q148 15 197 15Q235 15 271 47T324 130Q328 142 387 380T447 625Z"></path><path stroke-width="0" id="E39-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="0" id="E39-MJMAIN-29" d="M60 749L64 750Q69 750 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xlink:href="#E39-MJMAIN-2C" x="922" y="0"></use><g transform="translate(848,0)"><use transform="scale(0.707)" xlink:href="#E39-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.5)" xlink:href="#E39-MJMAIN-31" x="663" y="-213"></use></g></g><use xlink:href="#E39-MJMATHI-4A" x="3944" y="0"></use><g transform="translate(4743,0)"><use xlink:href="#E39-MJMAIN-28" x="0" y="0"></use><g transform="translate(389,0)"><use xlink:href="#E39-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E39-MJMAIN-30" x="663" y="-213"></use></g><use xlink:href="#E39-MJMAIN-2C" x="1311" y="0"></use><g transform="translate(1756,0)"><use xlink:href="#E39-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E39-MJMAIN-31" x="663" y="-213"></use></g><use xlink:href="#E39-MJMAIN-29" x="2678" y="0"></use></g></g></svg></span><script type="math/tex">\underset{\theta_0, \theta_1}{\text{minimize}} J \left(\theta_0, \theta_1 \right)</script></li></ul><p><span>为了直观理解代价函数到底是在做什么，先假设 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="6.401ex" height="2.344ex" viewBox="0 -806.1 2756.1 1009.2" role="img" focusable="false" style="vertical-align: -0.472ex;"><defs><path stroke-width="0" id="E40-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="0" id="E40-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path stroke-width="0" id="E40-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path stroke-width="0" id="E40-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E40-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E40-MJMAIN-31" x="663" y="-213"></use><use xlink:href="#E40-MJMAIN-3D" x="1200" y="0"></use><use xlink:href="#E40-MJMAIN-30" x="2256" y="0"></use></g></svg></span><script type="math/tex">\theta_1 = 0</script><span>，并假设训练集有三个数据，分别为</span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="18.327ex" height="2.577ex" viewBox="0 -806.1 7890.7 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="0" id="E41-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="0" id="E41-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path stroke-width="0" id="E41-MJMAIN-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z"></path><path stroke-width="0" id="E41-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path><path stroke-width="0" id="E41-MJMAIN-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 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y="0"></use><use xlink:href="#E41-MJMAIN-2C" x="889" y="0"></use><use xlink:href="#E41-MJMAIN-31" x="1333" y="0"></use><use xlink:href="#E41-MJMAIN-29" x="1833" y="0"></use><use xlink:href="#E41-MJMAIN-2C" x="2389" y="0"></use><g transform="translate(2834,0)"><use xlink:href="#E41-MJMAIN-28" x="0" y="0"></use><use xlink:href="#E41-MJMAIN-32" x="389" y="0"></use><use xlink:href="#E41-MJMAIN-2C" x="889" y="0"></use><use xlink:href="#E41-MJMAIN-32" x="1333" y="0"></use><use xlink:href="#E41-MJMAIN-29" x="1833" y="0"></use></g><use xlink:href="#E41-MJMAIN-2C" x="5223" y="0"></use><g transform="translate(5668,0)"><use xlink:href="#E41-MJMAIN-28" x="0" y="0"></use><use xlink:href="#E41-MJMAIN-33" x="389" y="0"></use><use xlink:href="#E41-MJMAIN-2C" x="889" y="0"></use><use xlink:href="#E41-MJMAIN-33" x="1333" y="0"></use><use xlink:href="#E41-MJMAIN-29" x="1833" y="0"></use></g></g></svg></span><script type="math/tex">\left(1, 1\right), \left(2, 2\right), \left(3, 3\right)</script><span>，这样在平面坐标系中绘制出 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.863ex" height="2.577ex" viewBox="0 -806.1 2524.3 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="0" id="E56-MJMATHI-68" d="M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z"></path><path stroke-width="0" id="E56-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="0" id="E56-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="0" id="E56-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path stroke-width="0" id="E56-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E56-MJMATHI-68" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E56-MJMATHI-3B8" x="814" y="-218"></use><g transform="translate(1174,0)"><use xlink:href="#E56-MJMAIN-28" x="0" y="0"></use><use xlink:href="#E56-MJMATHI-78" x="389" y="0"></use><use xlink:href="#E56-MJMAIN-29" x="961" y="0"></use></g></g></svg></span><script type="math/tex">h_\theta\left(x\right)</script><span> ，并分析 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="8.982ex" height="2.577ex" viewBox="0 -806.1 3867.4 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="0" id="E44-MJMATHI-4A" d="M447 625Q447 637 354 637H329Q323 642 323 645T325 664Q329 677 335 683H352Q393 681 498 681Q541 681 568 681T605 682T619 682Q633 682 633 672Q633 670 630 658Q626 642 623 640T604 637Q552 637 545 623Q541 610 483 376Q420 128 419 127Q397 64 333 21T195 -22Q137 -22 97 8T57 88Q57 130 80 152T132 174Q177 174 182 130Q182 98 164 80T123 56Q115 54 115 53T122 44Q148 15 197 15Q235 15 271 47T324 130Q328 142 387 380T447 625Z"></path><path stroke-width="0" id="E44-MJMAIN-28" d="M94 250Q94 319 104 381T127 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transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E44-MJMATHI-4A" x="0" y="0"></use><g transform="translate(799,0)"><use xlink:href="#E44-MJMAIN-28" x="0" y="0"></use><g transform="translate(389,0)"><use xlink:href="#E44-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E44-MJMAIN-30" x="663" y="-213"></use></g><use xlink:href="#E44-MJMAIN-2C" x="1311" y="0"></use><g transform="translate(1756,0)"><use xlink:href="#E44-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E44-MJMAIN-31" x="663" y="-213"></use></g><use xlink:href="#E44-MJMAIN-29" x="2678" y="0"></use></g></g></svg></span><script type="math/tex">J\left(\theta_0, \theta_1\right)</script><span> 的变化。</span></p><p><img src="images/20180106_085915.png" referrerpolicy="no-referrer"></p><p><span>右图 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="8.982ex" height="2.577ex" viewBox="0 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417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="0" id="E44-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path stroke-width="0" id="E44-MJMAIN-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z"></path><path stroke-width="0" id="E44-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path stroke-width="0" id="E44-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E44-MJMATHI-4A" x="0" y="0"></use><g transform="translate(799,0)"><use xlink:href="#E44-MJMAIN-28" x="0" y="0"></use><g transform="translate(389,0)"><use xlink:href="#E44-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E44-MJMAIN-30" x="663" y="-213"></use></g><use xlink:href="#E44-MJMAIN-2C" x="1311" y="0"></use><g transform="translate(1756,0)"><use xlink:href="#E44-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E44-MJMAIN-31" x="663" y="-213"></use></g><use xlink:href="#E44-MJMAIN-29" x="2678" y="0"></use></g></g></svg></span><script type="math/tex">J\left(\theta_0, \theta_1\right)</script><span> 随着 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.143ex" height="2.344ex" viewBox="0 -806.1 922.6 1009.2" role="img" focusable="false" style="vertical-align: -0.472ex;"><defs><path stroke-width="0" id="E73-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="0" id="E73-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E73-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E73-MJMAIN-31" x="663" y="-213"></use></g></svg></span><script type="math/tex">\theta_1</script><span> 的变化而变化，可见</span><strong><span>当 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="6.401ex" height="2.344ex" viewBox="0 -806.1 2756.1 1009.2" role="img" focusable="false" style="vertical-align: -0.472ex;"><defs><path stroke-width="0" id="E46-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="0" id="E46-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path stroke-width="0" id="E46-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E46-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E46-MJMAIN-31" x="663" y="-213"></use><use xlink:href="#E46-MJMAIN-3D" x="1200" y="0"></use><use xlink:href="#E46-MJMAIN-31" x="2256" y="0"></use></g></svg></span><script type="math/tex">\theta_1 = 1</script><span> 时，</span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="13.241ex" height="2.577ex" viewBox="0 -806.1 5701 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="0" id="E47-MJMATHI-4A" d="M447 625Q447 637 354 637H329Q323 642 323 645T325 664Q329 677 335 683H352Q393 681 498 681Q541 681 568 681T605 682T619 682Q633 682 633 672Q633 670 630 658Q626 642 623 640T604 637Q552 637 545 623Q541 610 483 376Q420 128 419 127Q397 64 333 21T195 -22Q137 -22 97 8T57 88Q57 130 80 152T132 174Q177 174 182 130Q182 98 164 80T123 56Q115 54 115 53T122 44Q148 15 197 15Q235 15 271 47T324 130Q328 142 387 380T447 625Z"></path><path stroke-width="0" id="E47-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="0" id="E47-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="0" id="E47-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 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xlink:href="#E47-MJMAIN-29" x="2678" y="0"></use></g><use xlink:href="#E47-MJMAIN-3D" x="4145" y="0"></use><use xlink:href="#E47-MJMAIN-30" x="5200" y="0"></use></g></svg></span><script type="math/tex">J\left(\theta_0, \theta_1 \right) = 0</script><span>，取得最小值，</span></strong><span>对应于左图青色直线，即函数 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.338ex" height="1.994ex" viewBox="0 -755.9 576 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E48-MJMATHI-68" d="M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E48-MJMATHI-68" x="0" y="0"></use></g></svg></span><script type="math/tex">h</script><span> 拟合程度最好的情况。</span></p><h2><a name="24-代价函数---直观理解2cost-function---intuition-ii" class="md-header-anchor"></a><span>2.4 代价函数 - 直观理解2(Cost Function - Intuition II)</span></h2><blockquote><p><span>注：该部分由于涉及到了多变量成像，可能较难理解，要求只需要理解上节内容即可，该节如果不能较好理解可跳过。</span></p></blockquote><p><span>给定数据集：</span></p><p><img src="images/20180106_091307.png" referrerpolicy="no-referrer"></p><p><span>参数在 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.143ex" height="2.461ex" viewBox="0 -806.1 922.6 1059.4" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="0" id="E49-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="0" id="E49-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E49-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E49-MJMAIN-30" x="663" y="-213"></use></g></svg></span><script type="math/tex">\theta_0</script><span> 不恒为 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.161ex" height="1.994ex" viewBox="0 -755.9 500 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E50-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E50-MJMAIN-30" x="0" y="0"></use></g></svg></span><script type="math/tex">0</script><span> 时代价函数 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="4.754ex" height="2.577ex" viewBox="0 -806.1 2046.7 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="0" id="E71-MJMATHI-4A" d="M447 625Q447 637 354 637H329Q323 642 323 645T325 664Q329 677 335 683H352Q393 681 498 681Q541 681 568 681T605 682T619 682Q633 682 633 672Q633 670 630 658Q626 642 623 640T604 637Q552 637 545 623Q541 610 483 376Q420 128 419 127Q397 64 333 21T195 -22Q137 -22 97 8T57 88Q57 130 80 152T132 174Q177 174 182 130Q182 98 164 80T123 56Q115 54 115 53T122 44Q148 15 197 15Q235 15 271 47T324 130Q328 142 387 380T447 625Z"></path><path stroke-width="0" id="E71-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="0" id="E71-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="0" id="E71-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E71-MJMATHI-4A" x="0" y="0"></use><g transform="translate(799,0)"><use xlink:href="#E71-MJMAIN-28" x="0" y="0"></use><use xlink:href="#E71-MJMATHI-3B8" x="389" y="0"></use><use xlink:href="#E71-MJMAIN-29" x="858" y="0"></use></g></g></svg></span><script type="math/tex">J\left(\theta\right)</script><span> 关于 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.318ex" height="2.461ex" viewBox="0 -806.1 2289.8 1059.4" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="0" id="E52-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="0" id="E52-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path stroke-width="0" id="E52-MJMAIN-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z"></path><path stroke-width="0" id="E52-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E52-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E52-MJMAIN-30" x="663" y="-213"></use><use xlink:href="#E52-MJMAIN-2C" x="922" y="0"></use><g transform="translate(1367,0)"><use xlink:href="#E52-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E52-MJMAIN-31" x="663" y="-213"></use></g></g></svg></span><script type="math/tex">\theta_0, \theta_1</script><span> 的3-D图像，图像中的高度为代价函数的值。</span></p><p><img src="images/20180106_090904.png" referrerpolicy="no-referrer"></p><p><span>由于3-D图形不便于标注，所以将3-D图形转换为</span><strong><span>轮廓图(contour plot)</span></strong><span>，下面用轮廓图（下图中的右图）来作直观理解，其中相同颜色的一个圈代表着同一高度（同一 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="4.754ex" height="2.577ex" viewBox="0 -806.1 2046.7 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="0" id="E71-MJMATHI-4A" d="M447 625Q447 637 354 637H329Q323 642 323 645T325 664Q329 677 335 683H352Q393 681 498 681Q541 681 568 681T605 682T619 682Q633 682 633 672Q633 670 630 658Q626 642 623 640T604 637Q552 637 545 623Q541 610 483 376Q420 128 419 127Q397 64 333 21T195 -22Q137 -22 97 8T57 88Q57 130 80 152T132 174Q177 174 182 130Q182 98 164 80T123 56Q115 54 115 53T122 44Q148 15 197 15Q235 15 271 47T324 130Q328 142 387 380T447 625Z"></path><path stroke-width="0" id="E71-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="0" id="E71-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="0" id="E71-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E71-MJMATHI-4A" x="0" y="0"></use><g transform="translate(799,0)"><use xlink:href="#E71-MJMAIN-28" x="0" y="0"></use><use xlink:href="#E71-MJMATHI-3B8" x="389" y="0"></use><use xlink:href="#E71-MJMAIN-29" x="858" y="0"></use></g></g></svg></span><script type="math/tex">J\left(\theta\right)</script><span> 值）。</span></p><p><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="16.158ex" height="2.461ex" viewBox="0 -806.1 6956.9 1059.4" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="0" id="E54-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="0" id="E54-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path stroke-width="0" id="E54-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path stroke-width="0" id="E54-MJMAIN-33" d="M127 463Q100 463 85 480T69 524Q69 579 117 622T233 665Q268 665 277 664Q351 652 390 611T430 522Q430 470 396 421T302 350L299 348Q299 347 308 345T337 336T375 315Q457 262 457 175Q457 96 395 37T238 -22Q158 -22 100 21T42 130Q42 158 60 175T105 193Q133 193 151 175T169 130Q169 119 166 110T159 94T148 82T136 74T126 70T118 67L114 66Q165 21 238 21Q293 21 321 74Q338 107 338 175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z"></path><path stroke-width="0" id="E54-MJMAIN-36" d="M42 313Q42 476 123 571T303 666Q372 666 402 630T432 550Q432 525 418 510T379 495Q356 495 341 509T326 548Q326 592 373 601Q351 623 311 626Q240 626 194 566Q147 500 147 364L148 360Q153 366 156 373Q197 433 263 433H267Q313 433 348 414Q372 400 396 374T435 317Q456 268 456 210V192Q456 169 451 149Q440 90 387 34T253 -22Q225 -22 199 -14T143 16T92 75T56 172T42 313ZM257 397Q227 397 205 380T171 335T154 278T148 216Q148 133 160 97T198 39Q222 21 251 21Q302 21 329 59Q342 77 347 104T352 209Q352 289 347 316T329 361Q302 397 257 397Z"></path><path stroke-width="0" id="E54-MJMAIN-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z"></path><path stroke-width="0" id="E54-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E54-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E54-MJMAIN-30" x="663" y="-213"></use><use xlink:href="#E54-MJMAIN-3D" x="1200" y="0"></use><g transform="translate(2256,0)"><use xlink:href="#E54-MJMAIN-33"></use><use xlink:href="#E54-MJMAIN-36" x="500" y="0"></use><use xlink:href="#E54-MJMAIN-30" x="1000" y="0"></use></g><use xlink:href="#E54-MJMAIN-2C" x="3756" y="0"></use><g transform="translate(4200,0)"><use xlink:href="#E54-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E54-MJMAIN-31" x="663" y="-213"></use></g><use xlink:href="#E54-MJMAIN-3D" x="5401" y="0"></use><use xlink:href="#E54-MJMAIN-30" x="6456" y="0"></use></g></svg></span><script type="math/tex">\theta_0 = 360, \theta_1 =0</script><span> 时：</span></p><p><img src="images/0f38a99c8ceb8aa5b90a5f12136fdf43.png" referrerpolicy="no-referrer"></p><p><span>大概在 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="19.126ex" height="2.461ex" viewBox="0 -806.1 8234.9 1059.4" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="0" id="E55-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="0" id="E55-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path stroke-width="0" id="E55-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path stroke-width="0" id="E55-MJMAIN-2E" d="M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z"></path><path stroke-width="0" id="E55-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path stroke-width="0" id="E55-MJMAIN-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path><path stroke-width="0" id="E55-MJMAIN-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z"></path><path stroke-width="0" id="E55-MJMAIN-35" d="M164 157Q164 133 148 117T109 101H102Q148 22 224 22Q294 22 326 82Q345 115 345 210Q345 313 318 349Q292 382 260 382H254Q176 382 136 314Q132 307 129 306T114 304Q97 304 95 310Q93 314 93 485V614Q93 664 98 664Q100 666 102 666Q103 666 123 658T178 642T253 634Q324 634 389 662Q397 666 402 666Q410 666 410 648V635Q328 538 205 538Q174 538 149 544L139 546V374Q158 388 169 396T205 412T256 420Q337 420 393 355T449 201Q449 109 385 44T229 -22Q148 -22 99 32T50 154Q50 178 61 192T84 210T107 214Q132 214 148 197T164 157Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E55-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E55-MJMAIN-30" x="663" y="-213"></use><use xlink:href="#E55-MJMAIN-3D" x="1200" y="0"></use><g transform="translate(2256,0)"><use xlink:href="#E55-MJMAIN-30"></use><use xlink:href="#E55-MJMAIN-2E" x="500" y="0"></use><use xlink:href="#E55-MJMAIN-31" x="778" y="0"></use><use xlink:href="#E55-MJMAIN-32" x="1278" y="0"></use></g><use xlink:href="#E55-MJMAIN-2C" x="4034" y="0"></use><g transform="translate(4478,0)"><use xlink:href="#E55-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E55-MJMAIN-31" x="663" y="-213"></use></g><use xlink:href="#E55-MJMAIN-3D" x="5679" y="0"></use><g transform="translate(6734,0)"><use xlink:href="#E55-MJMAIN-32"></use><use xlink:href="#E55-MJMAIN-35" x="500" y="0"></use><use xlink:href="#E55-MJMAIN-30" x="1000" y="0"></use></g></g></svg></span><script type="math/tex">\theta_0 = 0.12, \theta_1 =250</script><span> 时：</span></p><p><img src="images/20180106_092119.png" referrerpolicy="no-referrer"></p><p><span>上图中最中心的点（红点），近乎为图像中的最低点，也即代价函数的最小值，此时对应 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.863ex" height="2.577ex" viewBox="0 -806.1 2524.3 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="0" id="E56-MJMATHI-68" d="M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z"></path><path stroke-width="0" id="E56-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="0" id="E56-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 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xlink:href="#E71-MJMAIN-29" x="858" y="0"></use></g></g></svg></span><script type="math/tex">J\left(\theta\right)</script><span> 的最小值，并且大多数情况无法进行可视化，故引入</span><strong><span>梯度下降(Gradient Descent)方法，让计算机自动找出最小化代价函数时对应的 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.089ex" height="2.11ex" viewBox="0 -806.1 469 908.7" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E74-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E74-MJMATHI-3B8" x="0" y="0"></use></g></svg></span><script type="math/tex">\theta</script><span> 值。</span></strong></p><p><span>梯度下降背后的思想是：开始时，我们随机选择一个参数组合</span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="17.694ex" height="2.577ex" viewBox="0 -806.1 7618.4 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="0" id="E59-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="0" id="E59-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 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749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E59-MJMAIN-28" x="0" y="0"></use><g transform="translate(389,0)"><use xlink:href="#E59-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E59-MJMAIN-30" x="663" y="-213"></use></g><use xlink:href="#E59-MJMAIN-2C" x="1311" y="0"></use><g transform="translate(1756,0)"><use xlink:href="#E59-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E59-MJMAIN-31" x="663" y="-213"></use></g><use xlink:href="#E59-MJMAIN-2C" x="2678" y="0"></use><use xlink:href="#E59-MJMAIN-2E" x="3123" y="0"></use><use xlink:href="#E59-MJMAIN-2E" x="3568" y="0"></use><use xlink:href="#E59-MJMAIN-2E" x="4012" y="0"></use><use xlink:href="#E59-MJMAIN-2E" x="4457" y="0"></use><use xlink:href="#E59-MJMAIN-2E" x="4902" y="0"></use><use xlink:href="#E59-MJMAIN-2E" x="5346" y="0"></use><use xlink:href="#E59-MJMAIN-2C" x="5791" y="0"></use><g transform="translate(6236,0)"><use xlink:href="#E59-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E59-MJMATHI-6E" x="663" y="-213"></use></g><use xlink:href="#E59-MJMAIN-29" x="7229" y="0"></use></g></svg></span><script type="math/tex">\left( {\theta_{0}},{\theta_{1}},......,{\theta_{n}} \right)</script><span>即起始点，计算代价函数，然后寻找下一个能使得代价函数下降最多的参数组合。不断迭代，直到找到一个</span><strong><span>局部最小值(local minimum)</span></strong><span>，由于下降的情况只考虑当前参数组合周围的情况，所以无法确定当前的局部最小值是否就是</span><strong><span>全局最小值(global minimum)</span></strong><span>，不同的初始参数组合，可能会产生不同的局部最小值。</span></p><p><span>下图根据不同的起始点，产生了两个不同的局部最小值。</span></p><p><img src="images/db48c81304317847870d486ba5bb2015.jpg" referrerpolicy="no-referrer"></p><p><span>视频中举了下山的例子，即我们在山顶上的某个位置，为了下山，就不断地看一下周围</span><strong><span>下一步往哪走</span></strong><span>下山比较快，然后就</span><strong><span>迈出那一步</span></strong><span>，一直重复，直到我们到达山下的某一处</span><strong><span>陆地</span></strong><span>。</span></p><p><span>梯度下降公式：</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n223" cid="n223" mdtype="math_block"><div class="md-rawblock-container md-math-container" tabindex="-1"><div class="MathJax_SVG_Display" style="text-align: center;"><span class="MathJax_SVG" id="MathJax-Element-477-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="29.62ex" height="12.149ex" viewBox="0 -2866.8 12752.9 5231" role="img" focusable="false" style="vertical-align: -5.491ex; max-width: 100%;"><defs><path stroke-width="0" id="E487-MJMAIN-52" d="M130 622Q123 629 119 631T103 634T60 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& \text{Repeat until convergence:} \; \lbrace \\
&{{\theta }_{j}}:={{\theta }_{j}}-\alpha \frac{\partial }{\partial {{\theta }_{j}}}J\left( {\theta_{0}},{\theta_{1}} \right) \\
\rbrace
\end{align*}</script></div></div><blockquote><p><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.998ex" height="2.694ex" viewBox="0 -806.1 860.3 1160" role="img" focusable="false" style="vertical-align: -0.822ex;"><defs><path stroke-width="0" id="E60-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="0" id="E60-MJMATHI-6A" d="M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405Q240 405 220 393T185 362T161 325T144 293L137 279Q135 278 121 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transform="translate(60,-410)"><use transform="scale(0.707)" xlink:href="#E64-MJMAIN-2202" x="0" y="0"></use><g transform="translate(400,0)"><use transform="scale(0.707)" xlink:href="#E64-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.5)" xlink:href="#E64-MJMATHI-6A" x="663" y="-213"></use></g></g></g><use xlink:href="#E64-MJMATHI-4A" x="1369" y="0"></use><g transform="translate(2168,0)"><use xlink:href="#E64-MJMAIN-28" x="0" y="0"></use><g transform="translate(389,0)"><use xlink:href="#E64-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E64-MJMAIN-30" x="663" y="-213"></use></g><use xlink:href="#E64-MJMAIN-2C" x="1311" y="0"></use><g transform="translate(1756,0)"><use xlink:href="#E64-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E64-MJMAIN-31" x="663" y="-213"></use></g><use xlink:href="#E64-MJMAIN-29" x="2678" y="0"></use></g></g></svg></span><script type="math/tex">\frac{\partial }{\partial {{\theta }_{j}}}J\left( 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xlink:href="#E65-MJMAIN-28" x="0" y="0"></use><g transform="translate(389,0)"><use xlink:href="#E65-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E65-MJMAIN-30" x="663" y="-213"></use></g><use xlink:href="#E65-MJMAIN-2C" x="1311" y="0"></use><g transform="translate(1756,0)"><use xlink:href="#E65-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E65-MJMAIN-31" x="663" y="-213"></use></g><use xlink:href="#E65-MJMAIN-29" x="2678" y="0"></use></g></g></svg></span><script type="math/tex">J\left( \theta_0, \theta_1 \right)</script><span> 的偏导</span></p></blockquote><p><span>公式中，学习速率决定了参数值变化的速率即”</span><strong><span>走多少距离</span></strong><span>“，而偏导这部分决定了下降的方向即”</span><strong><span>下一步往哪里</span></strong><span>“走（当然实际上的走多少距离是由偏导值给出的，学习速率起到调整后决定的作用），收敛处的局部最小值又叫做极小值，即”</span><strong><span>陆地</span></strong><span>“。</span></p><p><img src="images/20180106_101659.png" referrerpolicy="no-referrer"></p><p><span>注意，在计算时要</span><strong><span>批量更新 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.089ex" height="2.11ex" viewBox="0 -806.1 469 908.7" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E74-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E74-MJMATHI-3B8" x="0" y="0"></use></g></svg></span><script type="math/tex">\theta</script><span> 值</span></strong><span>，即如上图中的左图所示，否则结果上会有所出入，原因不做细究。</span></p><h2><a name="26-梯度下降直观理解gradient-descent-intuition" class="md-header-anchor"></a><span>2.6 梯度下降直观理解(Gradient Descent Intuition)</span></h2><p><span>该节探讨 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.143ex" height="2.344ex" viewBox="0 -806.1 922.6 1009.2" role="img" focusable="false" style="vertical-align: -0.472ex;"><defs><path stroke-width="0" id="E73-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="0" id="E73-MJMAIN-31" d="M213 578L200 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type="math/tex">\frac{d}{d\theta_1}J\left(\theta_1\right)</script><span>，如果不熟悉微积分学，就把它视作之前的 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.538ex" height="3.511ex" viewBox="0 -1007.2 1092.6 1511.8" role="img" focusable="false" style="vertical-align: -1.172ex;"><defs><path stroke-width="0" id="E70-MJMAIN-2202" d="M202 508Q179 508 169 520T158 547Q158 557 164 577T185 624T230 675T301 710L333 715H345Q378 715 384 714Q447 703 489 661T549 568T566 457Q566 362 519 240T402 53Q321 -22 223 -22Q123 -22 73 56Q42 102 42 148V159Q42 276 129 370T322 465Q383 465 414 434T455 367L458 378Q478 461 478 515Q478 603 437 639T344 676Q266 676 223 612Q264 606 264 572Q264 547 246 528T202 508ZM430 306Q430 372 401 400T333 428Q270 428 222 382Q197 354 183 323T150 221Q132 149 132 116Q132 21 232 21Q244 21 250 22Q327 35 374 112Q389 137 409 196T430 306Z"></path><path stroke-width="0" id="E70-MJMATHI-3B8" d="M35 200Q35 302 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src="images/20180106_184926.png" referrerpolicy="no-referrer"></p><p><span>把红点定为初始点，切于初始点的红色直线的斜率，表示了函数 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="4.754ex" height="2.577ex" viewBox="0 -806.1 2046.7 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="0" id="E71-MJMATHI-4A" d="M447 625Q447 637 354 637H329Q323 642 323 645T325 664Q329 677 335 683H352Q393 681 498 681Q541 681 568 681T605 682T619 682Q633 682 633 672Q633 670 630 658Q626 642 623 640T604 637Q552 637 545 623Q541 610 483 376Q420 128 419 127Q397 64 333 21T195 -22Q137 -22 97 8T57 88Q57 130 80 152T132 174Q177 174 182 130Q182 98 164 80T123 56Q115 54 115 53T122 44Q148 15 197 15Q235 15 271 47T324 130Q328 142 387 380T447 625Z"></path><path stroke-width="0" id="E71-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 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xlink:href="#E72-MJMATHI-6A" x="663" y="-213"></use></g><use xlink:href="#E72-MJMAIN-2212" x="3554" y="0"></use><use xlink:href="#E72-MJMATHI-3B1" x="4554" y="0"></use><g transform="translate(5194,0)"><g transform="translate(120,0)"><rect stroke="none" width="1129" height="60" x="0" y="220"></rect><use transform="scale(0.707)" xlink:href="#E72-MJMAIN-2202" x="515" y="593"></use><g transform="translate(60,-410)"><use transform="scale(0.707)" xlink:href="#E72-MJMAIN-2202" x="0" y="0"></use><g transform="translate(400,0)"><use transform="scale(0.707)" xlink:href="#E72-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.5)" xlink:href="#E72-MJMATHI-6A" x="663" y="-213"></use></g></g></g></g><use xlink:href="#E72-MJMATHI-4A" x="6563" y="0"></use><g transform="translate(7363,0)"><use xlink:href="#E72-MJMAIN-28" x="0" y="0"></use><g transform="translate(389,0)"><use xlink:href="#E72-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E72-MJMAIN-30" x="663" y="-213"></use></g><use xlink:href="#E72-MJMAIN-2C" x="1311" y="0"></use><g transform="translate(1756,0)"><use xlink:href="#E72-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E72-MJMAIN-31" x="663" y="-213"></use></g><use xlink:href="#E72-MJMAIN-29" x="2678" y="0"></use></g></g></svg></span><script type="math/tex">{{\theta }_{j}}:={{\theta }_{j}}-\alpha \frac{\partial }{\partial {{\theta }_{j}}}J\left( \theta_0, \theta_1 \right)</script><span> 右边的结果是一个正值，即 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.143ex" height="2.344ex" viewBox="0 -806.1 922.6 1009.2" role="img" focusable="false" style="vertical-align: -0.472ex;"><defs><path stroke-width="0" id="E73-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="0" id="E73-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E73-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E73-MJMAIN-31" x="663" y="-213"></use></g></svg></span><script type="math/tex">\theta_1</script><span> 会</span><strong><span>向左边移动</span></strong><span>。这样不断重复，直到收敛（达到局部最小值，即斜率为0）。</span></p><p><span>初始 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.089ex" height="2.11ex" viewBox="0 -806.1 469 908.7" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E74-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E74-MJMATHI-3B8" x="0" y="0"></use></g></svg></span><script type="math/tex">\theta</script><span> 值（初始点）是任意的，若初始点恰好就在极小值点处，梯度下降算法将什么也不做（</span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg 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46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E75-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E75-MJMAIN-31" x="663" y="-213"></use><g transform="translate(1200,0)"><use xlink:href="#E75-MJMAIN-3A"></use><use xlink:href="#E75-MJMAIN-3D" x="278" y="0"></use></g><g transform="translate(2534,0)"><use xlink:href="#E75-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E75-MJMAIN-31" x="663" y="-213"></use></g><use xlink:href="#E75-MJMAIN-2212" x="3678" y="0"></use><use xlink:href="#E75-MJMATHI-3B1" x="4679" y="0"></use><use xlink:href="#E75-MJMAIN-2217" x="5541" y="0"></use><use xlink:href="#E75-MJMAIN-30" x="6263" y="0"></use></g></svg></span><script type="math/tex">\theta_1 := \theta_1 - \alpha*0</script><span>）。</span></p><blockquote><p><span>不熟悉斜率的话，就当斜率的值等于图中三角形的高度除以水平长度好啦，精确地求斜率的方法是求导。</span></p></blockquote><p> </p><p><span>对于学习速率 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.486ex" height="1.41ex" viewBox="0 -504.6 640 607.1" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E76-MJMATHI-3B1" d="M34 156Q34 270 120 356T309 442Q379 442 421 402T478 304Q484 275 485 237V208Q534 282 560 374Q564 388 566 390T582 393Q603 393 603 385Q603 376 594 346T558 261T497 161L486 147L487 123Q489 67 495 47T514 26Q528 28 540 37T557 60Q559 67 562 68T577 70Q597 70 597 62Q597 56 591 43Q579 19 556 5T512 -10H505Q438 -10 414 62L411 69L400 61Q390 53 370 41T325 18T267 -2T203 -11Q124 -11 79 39T34 156ZM208 26Q257 26 306 47T379 90L403 112Q401 255 396 290Q382 405 304 405Q235 405 183 332Q156 292 139 224T121 120Q121 71 146 49T208 26Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E76-MJMATHI-3B1" x="0" y="0"></use></g></svg></span><script type="math/tex">\alpha</script><span>，需要选取一个合适的值才能使得梯度下降算法运行良好。</span></p><ul><li><p><span>学习速率过小图示：</span></p><p><img src="images/20180106_190944.png" referrerpolicy="no-referrer"></p><p><span>收敛的太慢，需要更多次的迭代。</span></p></li><li><p><span>学习速率过大图示：</span></p><p><img src="images/20180106_191023.png" referrerpolicy="no-referrer"></p><p><span>可能越过最低点，甚至导致无法收敛。</span></p></li></ul><p><strong><span>学习速率只需选定即可</span></strong><span>，不需要在运行梯度下降算法的时候进行动态改变，随着斜率越来越接近于0，代价函数的变化幅度会越来越小，直到收敛到局部极小值。</span></p><p><span>如图，品红色点为初始点，代价函数随着迭代的进行，变化的幅度越来越小。</span></p><p><img src="images/20180106_191956.png" referrerpolicy="no-referrer"></p><p><strong><span>最后，梯度下降不止可以用于线性回归中的代价函数，还通用于最小化其他的代价函数。</span></strong></p><h2><a name="27-线性回归中的梯度下降gradient-descent-for-linear-regression" class="md-header-anchor"></a><span>2.7 线性回归中的梯度下降(Gradient Descent For Linear Regression)</span></h2><p><span>线性回归模型</span></p><ul><li><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="17.026ex" height="2.577ex" viewBox="0 -806.1 7330.7 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="0" id="E77-MJMATHI-68" d="M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z"></path><path stroke-width="0" id="E77-MJMATHI-3B8" d="M35 200Q35 302 74 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& \text{Repeat until convergence:} \; \lbrace \\
&{{\theta }_{j}}:={{\theta }_{j}}-\alpha \frac{\partial }{\partial {{\theta }_{j}}}J\left( {\theta_{0}},{\theta_{1}} \right) \\
\rbrace
\end{align*}</script></div></div><p> </p><p><span>直接将线性回归模型公式代入梯度下降公式可得出公式</span></p><p><img src="images/20180106_203726.png" referrerpolicy="no-referrer"></p><p><span>当 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="11.492ex" height="2.461ex" viewBox="-12 -755.9 4947.8 1059.4" role="img" focusable="false" style="vertical-align: -0.705ex; margin-left: -0.028ex;"><defs><path stroke-width="0" id="E79-MJMATHI-6A" d="M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405Q240 405 220 393T185 362T161 325T144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432Q341 418 354 401T367 348V332L318 133Q267 -67 264 -75Q246 -125 194 -164T75 -204Q25 -204 7 -183T-12 -137Q-12 -110 7 -91T53 -71Q70 -71 82 -81T95 -112Q95 -148 63 -167Q69 -168 77 -168Q111 -168 139 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x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E489-MJMAIN-31" x="663" y="-213"></use></g><g transform="translate(5130,0)"><use xlink:href="#E489-MJMATHI-78" x="0" y="0"></use><g transform="translate(572,521)"><use transform="scale(0.707)" xlink:href="#E489-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E489-MJMATHI-69" x="389" y="0"></use><use transform="scale(0.707)" xlink:href="#E489-MJMAIN-29" x="733" y="0"></use></g><use transform="scale(0.707)" xlink:href="#E489-MJMAIN-31" x="808" y="-434"></use></g><use xlink:href="#E489-MJMAIN-2212" x="6818" y="0"></use><g transform="translate(7819,0)"><use xlink:href="#E489-MJMATHI-79" x="0" y="0"></use><g transform="translate(499,412)"><use transform="scale(0.707)" xlink:href="#E489-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E489-MJMATHI-69" x="389" y="0"></use><use transform="scale(0.707)" xlink:href="#E489-MJMAIN-29" x="733" y="0"></use></g></g><use xlink:href="#E489-MJSZ2-29" x="9212" y="-1"></use></g></g></g></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-479">\begin{align*}
\frac{\partial}{\partial\theta_j} J(\theta_1, \theta_2)&=\frac{\partial}{\partial\theta_j} \left(\frac{1}{2m}\sum\limits_{i=1}^{m}{{\left( {{h}_{\theta }}\left( {{x}^{(i)}} \right)-{{y}^{(i)}} \right)}^{2}} \right)\\
&=\left(\frac{1}{2m}*2\sum\limits_{i=1}^{m}{{\left( {{h}_{\theta }}\left( {{x}^{(i)}} \right)-{{y}^{(i)}} \right)}} \right)*\frac{\partial}{\partial\theta_j}{{\left( {{h}_{\theta }}\left( {{x}^{(i)}} \right)-{{y}^{(i)}} \right)}}\\
&=\left(\frac{1}{m}\sum\limits_{i=1}^{m}{{\left( {{h}_{\theta }}\left( {{x}^{(i)}} \right)-{{y}^{(i)}} \right)}} \right)*\frac{\partial}{\partial\theta_j}{{\left(\theta_0{x_0^{(i)}} + \theta_1{x_1^{(i)}}-{{y}^{(i)}} \right)}}
\end{align*}</script></div></div><p><span>所以当 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.243ex" height="2.461ex" viewBox="-12 -755.9 2257.6 1059.4" role="img" focusable="false" style="vertical-align: -0.705ex; margin-left: -0.028ex;"><defs><path stroke-width="0" id="E80-MJMATHI-6A" d="M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405Q240 405 220 393T185 362T161 325T144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432Q341 418 354 401T367 348V332L318 133Q267 -67 264 -75Q246 -125 194 -164T75 -204Q25 -204 7 -183T-12 -137Q-12 -110 7 -91T53 -71Q70 -71 82 -81T95 -112Q95 -148 63 -167Q69 -168 77 -168Q111 -168 139 -140T182 -74L193 -32Q204 11 219 72T251 197T278 308T289 365Q289 372 288 376Z"></path><path stroke-width="0" id="E80-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path stroke-width="0" id="E80-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E80-MJMATHI-6A" x="0" y="0"></use><use xlink:href="#E80-MJMAIN-3D" x="689" y="0"></use><use xlink:href="#E80-MJMAIN-30" x="1745" y="0"></use></g></svg></span><script type="math/tex">j = 0</script><span> 时：</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" 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xlink:href="#E491-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E491-MJMATHI-69" x="389" y="0"></use><use transform="scale(0.707)" xlink:href="#E491-MJMAIN-29" x="733" y="0"></use></g><use transform="scale(0.707)" xlink:href="#E491-MJMAIN-31" x="808" y="-434"></use></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-481">\frac{\partial}{\partial\theta_1} J(\theta)=\frac{1}{m}\sum\limits_{i=1}^{m}{{\left( {{h}_{\theta }}\left( {{x}^{(i)}} \right)-{{y}^{(i)}} \right)}} *x_1^{(i)}</script></div></div><p><span>上文中所提到的梯度下降，都为批量梯度下降(Batch Gradient Descent)，即每次计算都使用</span><strong><span>所有</span></strong><span>的数据集 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="6.084ex" height="5.963ex" viewBox="0 -1509.8 2619.6 2567.2" role="img" focusable="false" style="vertical-align: -2.456ex;"><defs><path stroke-width="0" id="E82-MJMAIN-28" 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transform="translate(736,0)"><use xlink:href="#E82-MJSZ1-2211" x="45" y="0"></use><g transform="translate(0,-888)"><use transform="scale(0.707)" xlink:href="#E82-MJMATHI-69" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E82-MJMAIN-3D" x="345" y="0"></use><use transform="scale(0.707)" xlink:href="#E82-MJMAIN-31" x="1123" y="0"></use></g><use transform="scale(0.707)" xlink:href="#E82-MJMATHI-6D" x="372" y="1343"></use></g><use xlink:href="#E82-MJSZ3-29" x="1883" y="-1"></use></g></svg></span><script type="math/tex">\left(\sum\limits_{i=1}^{m}\right)</script><span> 更新。</span></p><p><span>由于线性回归函数呈现</span><strong><span>碗状</span></strong><span>，且</span><strong><span>只有一个</span></strong><span>全局的最优值，所以函数</span><strong><span>一定总会</span></strong><span>收敛到全局最小值（学习速率不可过大）。同时，函数 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.47ex" height="1.994ex" viewBox="0 -755.9 633 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E83-MJMATHI-4A" d="M447 625Q447 637 354 637H329Q323 642 323 645T325 664Q329 677 335 683H352Q393 681 498 681Q541 681 568 681T605 682T619 682Q633 682 633 672Q633 670 630 658Q626 642 623 640T604 637Q552 637 545 623Q541 610 483 376Q420 128 419 127Q397 64 333 21T195 -22Q137 -22 97 8T57 88Q57 130 80 152T132 174Q177 174 182 130Q182 98 164 80T123 56Q115 54 115 53T122 44Q148 15 197 15Q235 15 271 47T324 130Q328 142 387 380T447 625Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E83-MJMATHI-4A" x="0" y="0"></use></g></svg></span><script type="math/tex">J</script><span> 被称为</span><strong><span>凸二次函数</span></strong><span>，而线性回归函数求解最小值问题属于</span><strong><span>凸函数优化问题</span></strong><span>。</span></p><p><img src="images/24e9420f16fdd758ccb7097788f879e7.png" referrerpolicy="no-referrer"></p><p><span>另外，使用循环求解，代码较为冗余，后面会讲到如何使用</span><strong><span>向量化(Vectorization)</span></strong><span>来简化代码并优化计算，使梯度下降运行的更快更好。</span></p><h1><a name="3-linear-algebra-review" class="md-header-anchor"></a><span>3 Linear Algebra Review</span></h1><p><span>这部分，学过线性代数的可以复习一下，比较基础。笔记整理暂留。</span></p><h2><a name="31-matrices-and-vectors" class="md-header-anchor"></a><span>3.1 Matrices and Vectors</span></h2><p><span>Octave/Matlab 代码:</span></p><pre spellcheck="false" class="md-fences md-end-block md-fences-with-lineno ty-contain-cm modeLoaded" lang="matlab" style="break-inside: unset;"><div class="CodeMirror cm-s-inner CodeMirror-wrap" lang="matlab"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 0px; left: 44px;"><textarea autocorrect="off" autocapitalize="off" spellcheck="false" tabindex="0" style="position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;"></textarea></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 36px; margin-bottom: 0px; border-right-width: 0px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre><span>xxxxxxxxxx</span></pre><div class="CodeMirror-linenumber CodeMirror-gutter-elt"><div>17</div></div></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation" style=""><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: -36px; width: 36px;"></div><div class="CodeMirror-gutter-wrapper CodeMirror-activeline-gutter" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">1</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">% The ; denotes we are going back to a new row.</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">2</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">A</span> = [<span class="cm-number">1</span>, <span class="cm-number">2</span>, <span class="cm-number">3</span>; <span class="cm-number">4</span>, <span class="cm-number">5</span>, <span class="cm-number">6</span>; <span class="cm-number">7</span>, <span class="cm-number">8</span>, <span class="cm-number">9</span>; <span class="cm-number">10</span>, <span class="cm-number">11</span>, <span class="cm-number">12</span>]</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">3</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">4</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">% Initialize a vector </span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">5</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">v</span> = [<span class="cm-number">1</span>;<span class="cm-number">2</span>;<span class="cm-number">3</span>] </span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">6</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">7</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">% Get the dimension of the matrix A where m = rows and n = columns</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">8</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">[<span class="cm-variable">m</span>,<span class="cm-variable">n</span>] = <span class="cm-builtin">size</span>(<span class="cm-variable">A</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">9</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">10</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">% You could also store it this way</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">11</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">dim_A</span> = <span class="cm-builtin">size</span>(<span class="cm-variable">A</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">12</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">13</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">% Get the dimension of the vector v </span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">14</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">dim_v</span> = <span class="cm-builtin">size</span>(<span class="cm-variable">v</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">15</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">16</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">% Now let's index into the 2nd row 3rd column of matrix A</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">17</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">A_23</span> = <span class="cm-variable">A</span>(<span class="cm-number">2</span>,<span class="cm-number">3</span>)</span></pre></div></div></div></div></div></div><div style="position: absolute; height: 0px; width: 1px; border-bottom: 0px solid transparent; top: 391px;"></div><div class="CodeMirror-gutters" style="height: 391px;"><div class="CodeMirror-gutter CodeMirror-linenumbers" style="width: 35px;"></div></div></div></div></pre><p><span>执行结果:</span></p><pre spellcheck="false" class="md-fences md-end-block md-fences-with-lineno ty-contain-cm modeLoaded" lang="" style="break-inside: unset;"><div class="CodeMirror cm-s-inner CodeMirror-wrap" lang=""><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 0px; left: 44px;"><textarea autocorrect="off" autocapitalize="off" spellcheck="false" tabindex="0" style="position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;"></textarea></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 36px; margin-bottom: 0px; border-right-width: 0px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre><span>xxxxxxxxxx</span></pre><div class="CodeMirror-linenumber CodeMirror-gutter-elt"><div>24</div></div></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation" style=""><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: -36px; width: 36px;"></div><div class="CodeMirror-gutter-wrapper CodeMirror-activeline-gutter" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">1</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">A =</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">2</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">3</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> 1 2 3</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">4</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> 4 5 6</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">5</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> 7 8 9</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">6</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> 10 11 12</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">7</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">8</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">v =</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">9</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">10</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> 1</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">11</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> 2</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">12</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> 3</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">13</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">14</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">m = 4</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">15</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">n = 3</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">16</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">dim_A =</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">17</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">18</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> 4 3</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">19</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">20</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">dim_v =</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">21</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">22</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> 3 1</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">23</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">24</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">A_23 = 6</span></pre></div></div></div></div></div></div><div style="position: absolute; height: 0px; width: 1px; border-bottom: 0px solid transparent; top: 552px;"></div><div class="CodeMirror-gutters" style="height: 552px;"><div class="CodeMirror-gutter CodeMirror-linenumbers" style="width: 35px;"></div></div></div></div></pre><h2><a name="32-addition-and-scalar-multiplication" class="md-header-anchor"></a><span>3.2 Addition and Scalar Multiplication</span></h2><p><span>Octave/Matlab 代码:</span></p><pre spellcheck="false" class="md-fences md-end-block md-fences-with-lineno ty-contain-cm modeLoaded" lang="matlab" style="break-inside: unset;"><div class="CodeMirror cm-s-inner CodeMirror-wrap" lang="matlab"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 0px; left: 44px;"><textarea autocorrect="off" autocapitalize="off" spellcheck="false" tabindex="0" style="position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;"></textarea></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 36px; margin-bottom: 0px; border-right-width: 0px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre><span>xxxxxxxxxx</span></pre><div class="CodeMirror-linenumber CodeMirror-gutter-elt"><div>21</div></div></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation" style=""><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: -36px; width: 36px;"></div><div class="CodeMirror-gutter-wrapper CodeMirror-activeline-gutter" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">1</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">% Initialize matrix A and B </span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">2</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">A</span> = [<span class="cm-number">1</span>, <span class="cm-number">2</span>, <span class="cm-number">4</span>; <span class="cm-number">5</span>, <span class="cm-number">3</span>, <span class="cm-number">2</span>]</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">3</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">B</span> = [<span class="cm-number">1</span>, <span class="cm-number">3</span>, <span class="cm-number">4</span>; <span class="cm-number">1</span>, <span class="cm-number">1</span>, <span class="cm-number">1</span>]</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">4</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">5</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">% Initialize constant s </span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">6</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">s</span> = <span class="cm-number">2</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">7</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">8</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">% See how element-wise addition works</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">9</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">add_AB</span> = <span class="cm-variable">A</span> <span class="cm-operator">+</span> <span class="cm-variable">B</span> </span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">10</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">11</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">% See how element-wise subtraction works</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">12</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">sub_AB</span> = <span class="cm-variable">A</span> <span class="cm-operator">-</span> <span class="cm-variable">B</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">13</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">14</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">% See how scalar multiplication works</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">15</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">mult_As</span> = <span class="cm-variable">A</span> <span class="cm-operator">*</span> <span class="cm-variable">s</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">16</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">17</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">% Divide A by s</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">18</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">div_As</span> = <span class="cm-variable">A</span> <span class="cm-operator">/</span> <span class="cm-variable">s</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">19</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">20</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">% What happens if we have a Matrix + scalar?</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">21</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">add_As</span> = <span class="cm-variable">A</span> <span class="cm-operator">+</span> <span class="cm-variable">s</span></span></pre></div></div></div></div></div></div><div style="position: absolute; height: 0px; width: 1px; border-bottom: 0px solid transparent; top: 483px;"></div><div class="CodeMirror-gutters" style="height: 483px;"><div class="CodeMirror-gutter CodeMirror-linenumbers" style="width: 35px;"></div></div></div></div></pre><p><span>执行结果:</span></p><pre spellcheck="false" class="md-fences md-end-block md-fences-with-lineno ty-contain-cm modeLoaded" lang="" style="break-inside: unset;"><div class="CodeMirror cm-s-inner CodeMirror-wrap" lang=""><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 0px; left: 44px;"><textarea autocorrect="off" autocapitalize="off" spellcheck="false" tabindex="0" style="position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;"></textarea></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 36px; margin-bottom: 0px; border-right-width: 0px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre><span>xxxxxxxxxx</span></pre><div class="CodeMirror-linenumber CodeMirror-gutter-elt"><div>35</div></div></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation" style=""><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: -36px; width: 36px;"></div><div class="CodeMirror-gutter-wrapper CodeMirror-activeline-gutter" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">1</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">A =</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">2</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">3</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> 1 2 4</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">4</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> 5 3 2</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">5</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">6</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">B =</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">7</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">8</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> 1 3 4</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">9</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> 1 1 1</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">10</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">11</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">s = 2</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">12</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">add_AB =</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">13</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">14</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> 2 5 8</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">15</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> 6 4 3</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">16</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">17</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">sub_AB =</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">18</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">19</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> 0 -1 0</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">20</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> 4 2 1</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">21</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">22</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">mult_As =</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">23</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">24</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> 2 4 8</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">25</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> 10 6 4</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">26</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">27</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">div_As =</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">28</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">29</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> 0.50000 1.00000 2.00000</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">30</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> 2.50000 1.50000 1.00000</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">31</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">32</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">add_As =</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">33</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">34</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> 3 4 6</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">35</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> 7 5 4</span></pre></div></div></div></div></div></div><div style="position: absolute; height: 0px; width: 1px; border-bottom: 0px solid transparent; top: 805px;"></div><div class="CodeMirror-gutters" style="height: 805px;"><div class="CodeMirror-gutter CodeMirror-linenumbers" style="width: 35px;"></div></div></div></div></pre><h2><a name="33-matrix-vector-multiplication" class="md-header-anchor"></a><span>3.3 Matrix Vector Multiplication</span></h2><p><span>Octave/Matlab 代码:</span></p><pre spellcheck="false" class="md-fences md-end-block md-fences-with-lineno ty-contain-cm modeLoaded" lang="matlab"><div class="CodeMirror cm-s-inner CodeMirror-wrap" lang="matlab"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 0px; left: 44px;"><textarea autocorrect="off" autocapitalize="off" spellcheck="false" tabindex="0" style="position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;"></textarea></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 36px; margin-bottom: 0px; border-right-width: 0px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre><span>xxxxxxxxxx</span></pre><div class="CodeMirror-linenumber CodeMirror-gutter-elt"><div>10</div></div></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation" style=""><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: -36px; width: 36px;"></div><div class="CodeMirror-gutter-wrapper CodeMirror-activeline-gutter" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">1</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">% Initialize matrix A </span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">2</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">A</span> = [<span class="cm-number">1</span>, <span class="cm-number">2</span>, <span class="cm-number">3</span>; <span class="cm-number">4</span>, <span class="cm-number">5</span>, <span class="cm-number">6</span>;<span class="cm-number">7</span>, <span class="cm-number">8</span>, <span class="cm-number">9</span>] </span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">3</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">4</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">% Initialize vector v </span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">5</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">v</span> = [<span class="cm-number">1</span>; <span class="cm-number">1</span>; <span class="cm-number">1</span>] </span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">6</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">7</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">% Multiply A * v</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">8</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">Av</span> = <span class="cm-variable">A</span> <span class="cm-operator">*</span> <span class="cm-variable">v</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">9</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">10</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div></div></div></div></div></div><div style="position: absolute; height: 0px; width: 1px; border-bottom: 0px solid transparent; top: 230px;"></div><div class="CodeMirror-gutters" style="height: 230px;"><div class="CodeMirror-gutter CodeMirror-linenumbers" style="width: 35px;"></div></div></div></div></pre><p><span>执行结果:</span></p><pre spellcheck="false" class="md-fences md-end-block md-fences-with-lineno ty-contain-cm modeLoaded" lang="" style="break-inside: unset;"><div class="CodeMirror cm-s-inner CodeMirror-wrap" lang=""><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 0px; left: 44px;"><textarea autocorrect="off" autocapitalize="off" spellcheck="false" tabindex="0" style="position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;"></textarea></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 36px; margin-bottom: 0px; border-right-width: 0px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre><span>xxxxxxxxxx</span></pre><div class="CodeMirror-linenumber CodeMirror-gutter-elt"><div>18</div></div></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation" style=""><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: -36px; width: 36px;"></div><div class="CodeMirror-gutter-wrapper CodeMirror-activeline-gutter" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">1</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">A =</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">2</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">3</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> 1 2 3</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">4</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> 4 5 6</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">5</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> 7 8 9</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">6</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">7</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">v =</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">8</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">9</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> 1</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">10</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> 1</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">11</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> 1</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">12</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">13</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">Av =</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">14</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">15</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> 6</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">16</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> 15</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">17</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> 24</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">18</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div></div></div></div></div></div><div style="position: absolute; height: 0px; width: 1px; border-bottom: 0px solid transparent; top: 414px;"></div><div class="CodeMirror-gutters" style="height: 414px;"><div class="CodeMirror-gutter CodeMirror-linenumbers" style="width: 35px;"></div></div></div></div></pre><h2><a name="34-matrix-matrix-multiplication" class="md-header-anchor"></a><span>3.4 Matrix Matrix Multiplication</span></h2><p><span>Octave/Matlab 代码:</span></p><pre spellcheck="false" class="md-fences md-end-block md-fences-with-lineno ty-contain-cm modeLoaded" lang="matlab"><div class="CodeMirror cm-s-inner CodeMirror-wrap" lang="matlab"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 0px; left: 44px;"><textarea autocorrect="off" autocapitalize="off" spellcheck="false" tabindex="0" style="position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;"></textarea></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 36px; margin-bottom: 0px; border-right-width: 0px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre><span>xxxxxxxxxx</span></pre><div class="CodeMirror-linenumber CodeMirror-gutter-elt"><div>10</div></div></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation" style=""><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: -36px; width: 36px;"></div><div class="CodeMirror-gutter-wrapper CodeMirror-activeline-gutter" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">1</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">% Initialize a 3 by 2 matrix </span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">2</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">A</span> = [<span class="cm-number">1</span>, <span class="cm-number">2</span>; <span class="cm-number">3</span>, <span class="cm-number">4</span>;<span class="cm-number">5</span>, <span class="cm-number">6</span>]</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">3</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">4</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">% Initialize a 2 by 1 matrix </span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">5</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">B</span> = [<span class="cm-number">1</span>; <span class="cm-number">2</span>] </span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">6</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">7</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">% We expect a resulting matrix of (3 by 2)*(2 by 1) = (3 by 1) </span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">8</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">mult_AB</span> = <span class="cm-variable">A</span><span class="cm-operator">*</span><span class="cm-variable">B</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">9</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">10</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">% Make sure you understand why we got that result</span></span></pre></div></div></div></div></div></div><div style="position: absolute; height: 0px; width: 1px; border-bottom: 0px solid transparent; top: 230px;"></div><div class="CodeMirror-gutters" style="height: 230px;"><div class="CodeMirror-gutter CodeMirror-linenumbers" style="width: 35px;"></div></div></div></div></pre><p><span>执行结果:</span></p><pre spellcheck="false" class="md-fences md-end-block md-fences-with-lineno ty-contain-cm modeLoaded" lang="" style="break-inside: unset;"><div class="CodeMirror cm-s-inner CodeMirror-wrap" lang=""><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 0px; left: 44px;"><textarea autocorrect="off" autocapitalize="off" spellcheck="false" tabindex="0" style="position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;"></textarea></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 36px; margin-bottom: 0px; border-right-width: 0px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre><span>xxxxxxxxxx</span></pre><div class="CodeMirror-linenumber CodeMirror-gutter-elt"><div>17</div></div></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation" style=""><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: -36px; width: 36px;"></div><div class="CodeMirror-gutter-wrapper CodeMirror-activeline-gutter" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">1</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">A =</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">2</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">3</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> 1 2</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">4</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> 3 4</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">5</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> 5 6</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">6</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">7</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">B =</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">8</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">9</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> 1</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">10</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> 2</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">11</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">12</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">mult_AB =</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">13</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">14</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> 5</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">15</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> 11</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">16</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> 17</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">17</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div></div></div></div></div></div><div style="position: absolute; height: 0px; width: 1px; border-bottom: 0px solid transparent; top: 391px;"></div><div class="CodeMirror-gutters" style="height: 391px;"><div class="CodeMirror-gutter CodeMirror-linenumbers" style="width: 35px;"></div></div></div></div></pre><h2><a name="35-matrix-multiplication-properties" class="md-header-anchor"></a><span>3.5 Matrix Multiplication Properties</span></h2><p><span>Octave/Matlab 代码:</span></p><pre spellcheck="false" class="md-fences md-end-block md-fences-with-lineno ty-contain-cm modeLoaded" lang="matlab" style="break-inside: unset;"><div class="CodeMirror cm-s-inner CodeMirror-wrap" lang="matlab"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 0px; left: 44px;"><textarea autocorrect="off" autocapitalize="off" spellcheck="false" tabindex="0" style="position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;"></textarea></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 36px; margin-bottom: 0px; border-right-width: 0px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre><span>xxxxxxxxxx</span></pre><div class="CodeMirror-linenumber CodeMirror-gutter-elt"><div>22</div></div></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation" style=""><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: -36px; width: 36px;"></div><div class="CodeMirror-gutter-wrapper CodeMirror-activeline-gutter" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">1</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">% Initialize random matrices A and B </span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">2</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">A</span> = [<span class="cm-number">1</span>,<span class="cm-number">2</span>;<span class="cm-number">4</span>,<span class="cm-number">5</span>]</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">3</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">B</span> = [<span class="cm-number">1</span>,<span class="cm-number">1</span>;<span class="cm-number">0</span>,<span class="cm-number">2</span>]</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">4</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">5</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">% Initialize a 2 by 2 identity matrix</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">6</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">I</span> = <span class="cm-builtin">eye</span>(<span class="cm-number">2</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">7</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">8</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">% The above notation is the same as I = [1,0;0,1]</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">9</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">10</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">% What happens when we multiply I*A ? </span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">11</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">IA</span> = <span class="cm-variable">I</span><span class="cm-operator">*</span><span class="cm-variable">A</span> </span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">12</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">13</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">% How about A*I ? </span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">14</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">AI</span> = <span class="cm-variable">A</span><span class="cm-operator">*</span><span class="cm-variable">I</span> </span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">15</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">16</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">% Compute A*B </span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">17</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">AB</span> = <span class="cm-variable">A</span><span class="cm-operator">*</span><span class="cm-variable">B</span> </span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">18</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">19</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">% Is it equal to B*A? </span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">20</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">BA</span> = <span class="cm-variable">B</span><span class="cm-operator">*</span><span class="cm-variable">A</span> </span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">21</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">22</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">% Note that IA = AI but AB != BA</span></span></pre></div></div></div></div></div></div><div style="position: absolute; height: 0px; width: 1px; border-bottom: 0px solid transparent; top: 506px;"></div><div class="CodeMirror-gutters" style="height: 506px;"><div class="CodeMirror-gutter CodeMirror-linenumbers" style="width: 35px;"></div></div></div></div></pre><p><span>执行结果:</span></p><pre spellcheck="false" class="md-fences md-end-block md-fences-with-lineno ty-contain-cm modeLoaded" lang="" style="break-inside: unset;"><div class="CodeMirror cm-s-inner CodeMirror-wrap" lang=""><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 0px; left: 44px;"><textarea autocorrect="off" autocapitalize="off" spellcheck="false" tabindex="0" style="position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;"></textarea></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 36px; margin-bottom: 0px; border-right-width: 0px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre><span>xxxxxxxxxx</span></pre><div class="CodeMirror-linenumber CodeMirror-gutter-elt"><div>36</div></div></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation" style=""><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: -36px; width: 36px;"></div><div class="CodeMirror-gutter-wrapper CodeMirror-activeline-gutter" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">1</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">A =</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">2</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">3</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> 1 2</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">4</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> 4 5</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">5</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">6</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">B =</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">7</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">8</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> 1 1</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">9</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> 0 2</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">10</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">11</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">I =</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">12</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">13</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">Diagonal Matrix</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">14</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">15</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> 1 0</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">16</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> 0 1</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">17</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">18</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">IA =</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">19</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">20</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> 1 2</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">21</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> 4 5</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">22</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">23</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">AI =</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">24</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">25</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> 1 2</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">26</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> 4 5</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">27</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">28</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">AB =</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">29</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">30</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> 1 5</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">31</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> 4 14</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">32</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">33</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">BA =</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">34</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">35</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> 5 7</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">36</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> 8 10</span></pre></div></div></div></div></div></div><div style="position: absolute; height: 0px; width: 1px; border-bottom: 0px solid transparent; top: 828px;"></div><div class="CodeMirror-gutters" style="height: 828px;"><div class="CodeMirror-gutter CodeMirror-linenumbers" style="width: 35px;"></div></div></div></div></pre><h2><a name="36-inverse-and-transpose" class="md-header-anchor"></a><span>3.6 Inverse and Transpose</span></h2><p><span>Octave/Matlab 代码:</span></p><pre spellcheck="false" class="md-fences md-end-block md-fences-with-lineno ty-contain-cm modeLoaded" lang="matlab"><div class="CodeMirror cm-s-inner CodeMirror-wrap" lang="matlab"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 0px; left: 44px;"><textarea autocorrect="off" autocapitalize="off" spellcheck="false" tabindex="0" style="position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;"></textarea></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 36px; margin-bottom: 0px; border-right-width: 0px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre><span>xxxxxxxxxx</span></pre><div class="CodeMirror-linenumber CodeMirror-gutter-elt"><div>13</div></div></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation" style=""><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: -36px; width: 36px;"></div><div class="CodeMirror-gutter-wrapper CodeMirror-activeline-gutter" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">1</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">% Initialize matrix A </span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">2</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">A</span> = [<span class="cm-number">1</span>,<span class="cm-number">2</span>,<span class="cm-number">0</span>;<span class="cm-number">0</span>,<span class="cm-number">5</span>,<span class="cm-number">6</span>;<span class="cm-number">7</span>,<span class="cm-number">0</span>,<span class="cm-number">9</span>]</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">3</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">4</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">% Transpose A </span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">5</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">A_trans</span> = <span class="cm-variable">A</span><span class="cm-operator">'</span> </span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">6</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">7</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">% Take the inverse of A </span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">8</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">A_inv</span> = <span class="cm-builtin">inv</span>(<span class="cm-variable">A</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">9</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">10</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">% What is A^(-1)*A? </span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">11</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">A_invA</span> = <span class="cm-builtin">inv</span>(<span class="cm-variable">A</span>)<span class="cm-operator">*</span><span class="cm-variable">A</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">12</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">13</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div></div></div></div></div></div><div style="position: absolute; height: 0px; width: 1px; border-bottom: 0px solid transparent; top: 299px;"></div><div class="CodeMirror-gutters" style="height: 299px;"><div class="CodeMirror-gutter CodeMirror-linenumbers" style="width: 35px;"></div></div></div></div></pre><p><span>执行结果:</span></p><pre spellcheck="false" class="md-fences md-end-block md-fences-with-lineno ty-contain-cm modeLoaded" lang="" style="break-inside: unset;"><div class="CodeMirror cm-s-inner CodeMirror-wrap" lang=""><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 0px; left: 44px;"><textarea autocorrect="off" autocapitalize="off" spellcheck="false" tabindex="0" style="position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;"></textarea></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 36px; margin-bottom: 0px; border-right-width: 0px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre><span>xxxxxxxxxx</span></pre><div class="CodeMirror-linenumber CodeMirror-gutter-elt"><div>24</div></div></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation" style=""><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: -36px; width: 36px;"></div><div class="CodeMirror-gutter-wrapper CodeMirror-activeline-gutter" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">1</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">A =</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">2</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">3</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> 1 2 0</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">4</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> 0 5 6</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">5</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> 7 0 9</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">6</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">7</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">A_trans =</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">8</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">9</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> 1 0 7</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">10</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> 2 5 0</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">11</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> 0 6 9</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">12</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">13</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">A_inv =</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">14</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">15</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> 0.348837 -0.139535 0.093023</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">16</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> 0.325581 0.069767 -0.046512</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">17</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> -0.271318 0.108527 0.038760</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">18</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">19</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">A_invA =</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">20</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">21</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> 1.00000 -0.00000 0.00000</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">22</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> 0.00000 1.00000 -0.00000</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">23</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> -0.00000 0.00000 1.00000</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">24</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div></div></div></div></div></div><div style="position: absolute; height: 0px; width: 1px; border-bottom: 0px solid transparent; top: 552px;"></div><div class="CodeMirror-gutters" style="height: 552px;"><div class="CodeMirror-gutter CodeMirror-linenumbers" style="width: 35px;"></div></div></div></div></pre></div>
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