diff --git "a/M2/optimization \342\200\223 Cours.html" "b/M2/optimization \342\200\223 Cours.html"
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-<p><span class="math display"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="CHTML" display="true" tabindex="0" ctxtmenu_counter="0" style="font-size: 113.1%; position: relative;"><mjx-math display="true" class="MJX-TEX" aria-hidden="true" style="margin-left: 0px; margin-right: 0px;"></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"></math></mjx-assistive-mml></mjx-container></span></p>
-</div>
-<section id="optimization" class="level1">
-<h1>Optimization</h1>
-<section id="introduction" class="level2">
-<h2 class="anchored" data-anchor-id="introduction">Introduction<a class="anchorjs-link " aria-label="Anchor" data-anchorjs-icon="" href="file:///Users/gergaud/git-ENS/Julia-TSE/enseignants/M2/Cours/_site/optimization.html#introduction" style="font: 1em / 1 anchorjs-icons; margin-left: 0.1875em; padding-right: 0.1875em; padding-left: 0.1875em;"></a></h2>
-<div>
-
-</div>
-<div class="quarto-layout-panel" data-layout-ncol="2">
-<div class="quarto-layout-row">
-<div class="quarto-layout-cell" style="flex-basis: 50.0%;justify-content: flex-start;">
-<div class="quarto-figure quarto-figure-left">
-<figure class="figure">
-<div class="quarto-figure quarto-figure-left">
-<figure class="figure">
-<p><a href="https://en.wikipedia.org/wiki/Pierre_de_Fermat"><img src="./optimization – Cours_files/Pierre_de_Fermat.jpg" class="img-fluid quarto-figure quarto-figure-left figure-img" width="300" alt="Pierre de Fermat"></a></p>
-</figure>
-</div>
-<figcaption>Pierre de Fermat</figcaption>
-</figure>
-</div>
-</div>
-<div class="quarto-layout-cell" style="flex-basis: 50.0%;justify-content: flex-start;">
-<ul>
-<li>Pierre de Fermat, 1601 (Beaumont-de-Lomagne, near Montauban) – 1665 (Castres)</li>
-<li>method <em>maximis et minimis</em></li>
-<li>Early developments that led to infinitesimal calculus</li>
-</ul>
-</div>
-</div>
-</div>
-<p>The problem is to solve</p>
-<p><span class="math display"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="CHTML" display="true" tabindex="0" ctxtmenu_counter="1" style="font-size: 110%; position: relative;"><mjx-math display="true" class="MJX-TEX" aria-hidden="true" style="margin-left: 0px; margin-right: 0px;"><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D443 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mrow space="2"><mjx-mo class="mjx-s3"><mjx-c class="mjx-c7B TEX-S3"></mjx-c></mjx-mo><mjx-mtable style="min-width: 4.174em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="text-align: left; padding-bottom: 0.2em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D440 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi><mjx-mstyle><mjx-mspace style="width: 0.278em;"></mjx-mspace></mjx-mstyle><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D453 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align: left; padding-top: 0.2em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2208"></mjx-c></mjx-mo><mjx-msup space="4"><mjx-texatom texclass="ORD"><mjx-texatom texclass="ORD"><mjx-mi class="mjx-b"><mjx-c class="mjx-c1D411 TEX-B"></mjx-c></mjx-mi></mjx-texatom></mjx-texatom><mjx-script style="vertical-align: 0.363em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msup><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable><mjx-mo class="mjx-n" style="vertical-align: 0.25em;"></mjx-mo></mjx-mrow></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mo stretchy="false">(</mo><mi>P</mi><mo stretchy="false">)</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">{</mo><mtable columnalign="left" columnspacing="1em" rowspacing="4pt"><mtr><mtd><mi>M</mi><mi>i</mi><mi>n</mi><mstyle scriptlevel="0"><mspace width="0.278em"></mspace></mstyle><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mi>x</mi><mo>∈</mo><msup><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">R</mi></mrow></mrow><mi>n</mi></msup></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE" fence="true" stretchy="true" symmetric="true"></mo></mrow></math></mjx-assistive-mml></mjx-container></span></p>
-</section>
-<section id="automatique-differentiation" class="level2">
-<h2 class="anchored" data-anchor-id="automatique-differentiation">Automatique differentiation<a class="anchorjs-link " aria-label="Anchor" data-anchorjs-icon="" href="file:///Users/gergaud/git-ENS/Julia-TSE/enseignants/M2/Cours/_site/optimization.html#automatique-differentiation" style="font: 1em / 1 anchorjs-icons; margin-left: 0.1875em; padding-right: 0.1875em; padding-left: 0.1875em;"></a></h2>
-<div id="7755983b" class="cell" data-execution_count="1">
-<details class="code-fold">
-<summary>Code</summary>
-<div class="sourceCode cell-code" id="cb1"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb1-1"><a href="file:///Users/gergaud/git-ENS/Julia-TSE/enseignants/M2/Cours/_site/optimization.html#cb1-1" aria-hidden="true" tabindex="-1"></a><span class="im">using</span> <span class="bu">LinearAlgebra</span></span>
-<span id="cb1-2"><a href="file:///Users/gergaud/git-ENS/Julia-TSE/enseignants/M2/Cours/_site/optimization.html#cb1-2" aria-hidden="true" tabindex="-1"></a><span class="im">using</span> <span class="bu">ForwardDiff</span></span>
-<span id="cb1-3"><a href="file:///Users/gergaud/git-ENS/Julia-TSE/enseignants/M2/Cours/_site/optimization.html#cb1-3" aria-hidden="true" tabindex="-1"></a>A <span class="op">=</span> [<span class="op">-</span><span class="fl">1</span> <span class="fl">2</span>;<span class="fl">1</span> <span class="fl">4</span>]; b<span class="op">=</span>[<span class="fl">1</span>,<span class="fl">1</span>]; c<span class="op">=</span><span class="fl">1</span></span>
-<span id="cb1-4"><a href="file:///Users/gergaud/git-ENS/Julia-TSE/enseignants/M2/Cours/_site/optimization.html#cb1-4" aria-hidden="true" tabindex="-1"></a><span class="fu">f</span>(x) <span class="op">=</span> <span class="fl">0.5</span><span class="op">*</span>x<span class="op">'*</span>A<span class="op">*</span>x <span class="op">+</span> b<span class="op">'*</span>x <span class="op">+</span>c</span>
-<span id="cb1-5"><a href="file:///Users/gergaud/git-ENS/Julia-TSE/enseignants/M2/Cours/_site/optimization.html#cb1-5" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb1-6"><a href="file:///Users/gergaud/git-ENS/Julia-TSE/enseignants/M2/Cours/_site/optimization.html#cb1-6" aria-hidden="true" tabindex="-1"></a><span class="fu">analytic_∇f</span>(x) <span class="op">=</span> <span class="fl">0.5</span><span class="fu">*</span>(A<span class="op">'</span> <span class="op">+</span> A)<span class="op">*</span>x<span class="op">+</span>b</span>
-<span id="cb1-7"><a href="file:///Users/gergaud/git-ENS/Julia-TSE/enseignants/M2/Cours/_site/optimization.html#cb1-7" aria-hidden="true" tabindex="-1"></a><span class="fu">analytic_∇²f</span>(x) <span class="op">=</span> <span class="fl">0.5</span><span class="fu">*</span>(A<span class="op">'</span> <span class="op">+</span> A)</span>
-<span id="cb1-8"><a href="file:///Users/gergaud/git-ENS/Julia-TSE/enseignants/M2/Cours/_site/optimization.html#cb1-8" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb1-9"><a href="file:///Users/gergaud/git-ENS/Julia-TSE/enseignants/M2/Cours/_site/optimization.html#cb1-9" aria-hidden="true" tabindex="-1"></a><span class="fu">∇f</span>(x) <span class="op">=</span> ForwardDiff.<span class="fu">gradient</span>(f, x)</span>
-<span id="cb1-10"><a href="file:///Users/gergaud/git-ENS/Julia-TSE/enseignants/M2/Cours/_site/optimization.html#cb1-10" aria-hidden="true" tabindex="-1"></a><span class="fu">∇²f</span>(x) <span class="op">=</span> ForwardDiff.<span class="fu">hessian</span>(f, x)</span>
-<span id="cb1-11"><a href="file:///Users/gergaud/git-ENS/Julia-TSE/enseignants/M2/Cours/_site/optimization.html#cb1-11" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb1-12"><a href="file:///Users/gergaud/git-ENS/Julia-TSE/enseignants/M2/Cours/_site/optimization.html#cb1-12" aria-hidden="true" tabindex="-1"></a>x0 <span class="op">=</span> [<span class="fl">1</span>,<span class="op">-</span><span class="fl">1</span>]</span>
-<span id="cb1-13"><a href="file:///Users/gergaud/git-ENS/Julia-TSE/enseignants/M2/Cours/_site/optimization.html#cb1-13" aria-hidden="true" tabindex="-1"></a><span class="fu">println</span>(<span class="st">"∇f(x0) = "</span>, <span class="fu">∇f</span>(x0))</span>
-<span id="cb1-14"><a href="file:///Users/gergaud/git-ENS/Julia-TSE/enseignants/M2/Cours/_site/optimization.html#cb1-14" aria-hidden="true" tabindex="-1"></a><span class="fu">println</span>(<span class="st">"analytic_∇f(x0) = "</span>, <span class="fu">analytic_∇f</span>(x0))</span>
-<span id="cb1-15"><a href="file:///Users/gergaud/git-ENS/Julia-TSE/enseignants/M2/Cours/_site/optimization.html#cb1-15" aria-hidden="true" tabindex="-1"></a><span class="fu">println</span>(<span class="st">"analytic_∇f(x0) - ∇f(x0) = "</span>, <span class="fu">analytic_∇f</span>(x0)<span class="op">-</span> <span class="fu">∇f</span>(x0))</span>
-<span id="cb1-16"><a href="file:///Users/gergaud/git-ENS/Julia-TSE/enseignants/M2/Cours/_site/optimization.html#cb1-16" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb1-17"><a href="file:///Users/gergaud/git-ENS/Julia-TSE/enseignants/M2/Cours/_site/optimization.html#cb1-17" aria-hidden="true" tabindex="-1"></a><span class="fu">println</span>(<span class="st">"∇²f(x0) = "</span>, <span class="fu">∇²f</span>(x0))</span>
-<span id="cb1-18"><a href="file:///Users/gergaud/git-ENS/Julia-TSE/enseignants/M2/Cours/_site/optimization.html#cb1-18" aria-hidden="true" tabindex="-1"></a><span class="fu">println</span>(<span class="st">"analytic_∇²f(x0) = "</span>, <span class="fu">analytic_∇²f</span>(x0))</span>
-<span id="cb1-19"><a href="file:///Users/gergaud/git-ENS/Julia-TSE/enseignants/M2/Cours/_site/optimization.html#cb1-19" aria-hidden="true" tabindex="-1"></a><span class="fu">println</span>(<span class="st">"analytic_∇²f(x0) - ∇²f(x0) = "</span>, <span class="fu">analytic_∇²f</span>(x0)<span class="op">-</span> <span class="fu">∇²f</span>(x0))</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
-</details>
-<div class="cell-output cell-output-stdout">
-<pre><code>∇f(x0) = [-1.5, -1.5]
-analytic_∇f(x0) = [-1.5, -1.5]
-analytic_∇f(x0) - ∇f(x0) = [0.0, 0.0]
-∇²f(x0) = [-1.0 1.5; 1.5 4.0]
-analytic_∇²f(x0) = [-1.0 1.5; 1.5 4.0]
-analytic_∇²f(x0) - ∇²f(x0) = [0.0 0.0; 0.0 0.0]</code></pre>
-</div>
-</div>
-</section>
-<section id="newtons-algorithm" class="level2">
-<h2 class="anchored" data-anchor-id="newtons-algorithm">Newton’s Algorithm<a class="anchorjs-link " aria-label="Anchor" data-anchorjs-icon="" href="file:///Users/gergaud/git-ENS/Julia-TSE/enseignants/M2/Cours/_site/optimization.html#newtons-algorithm" style="font: 1em / 1 anchorjs-icons; margin-left: 0.1875em; padding-right: 0.1875em; padding-left: 0.1875em;"></a></h2>
-<p>Solve <span class="math inline"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="CHTML" tabindex="0" ctxtmenu_counter="2" style="font-size: 110%; position: relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-n"><mjx-c class="mjx-c2207"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D453 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c30"></mjx-c></mjx-mn></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">∇</mi><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></math></mjx-assistive-mml></mjx-container></span> by Newton’s method</p>
-<p><strong>Require</strong> <span class="math inline"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="CHTML" tabindex="0" ctxtmenu_counter="3" style="font-size: 110%; position: relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D453 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c3A"></mjx-c></mjx-mo><mjx-msup space="2"><mjx-texatom texclass="ORD"><mjx-texatom texclass="ORD"><mjx-mi class="mjx-b"><mjx-c class="mjx-c1D411 TEX-B"></mjx-c></mjx-mi></mjx-texatom></mjx-texatom><mjx-script style="vertical-align: 0.363em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msup><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2192"></mjx-c></mjx-mo><mjx-texatom space="4" texclass="ORD"><mjx-texatom texclass="ORD"><mjx-mi class="mjx-b"><mjx-c class="mjx-c1D411 TEX-B"></mjx-c></mjx-mi></mjx-texatom></mjx-texatom></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo data-mjx-texclass="PUNCT">:</mo><msup><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">R</mi></mrow></mrow><mi>n</mi></msup><mo accent="false" stretchy="false">→</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">R</mi></mrow></mrow></math></mjx-assistive-mml></mjx-container></span>, <span class="math inline"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="CHTML" tabindex="0" ctxtmenu_counter="4" style="font-size: 110%; position: relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align: -0.15em;"><mjx-texatom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn></mjx-texatom></mjx-script></mjx-msub><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2208"></mjx-c></mjx-mo><mjx-msup space="4"><mjx-texatom texclass="ORD"><mjx-texatom texclass="ORD"><mjx-mi class="mjx-b"><mjx-c class="mjx-c1D411 TEX-B"></mjx-c></mjx-mi></mjx-texatom></mjx-texatom><mjx-script style="vertical-align: 0.363em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msup></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>x</mi><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msub><mo>∈</mo><msup><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">R</mi></mrow></mrow><mi>n</mi></msup></math></mjx-assistive-mml></mjx-container></span> ( initial point)</p>
-<ul>
-<li><span class="math inline"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="CHTML" tabindex="0" ctxtmenu_counter="5" style="font-size: 110%; position: relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2190"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c30"></mjx-c></mjx-mn></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo stretchy="false">←</mo><mn>0</mn></math></mjx-assistive-mml></mjx-container></span></li>
-<li>continue = true</li>
-<li><strong>While</strong> continue <span class="math inline"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="CHTML" tabindex="0" ctxtmenu_counter="6" style="font-size: 110%; position: relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-n"><mjx-c class="mjx-c23"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">#</mi></math></mjx-assistive-mml></mjx-container></span> See Section <a href="file:///Users/gergaud/git-ENS/Julia-TSE/enseignants/M2/Cours/_site/optimization.html#sec-criteres-arret" class="quarto-xref" aria-expanded="false">Section 1.3.1</a>
-<ul>
-<li>$d_k $ solution of <span class="math inline"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="CHTML" tabindex="0" ctxtmenu_counter="7" style="font-size: 110%; position: relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msup><mjx-mi class="mjx-n"><mjx-c class="mjx-c2207"></mjx-c></mjx-mi><mjx-script style="vertical-align: 0.363em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D453 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align: -0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mstyle><mjx-mspace style="width: 0.167em;"></mjx-mspace></mjx-mstyle><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D451 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mi class="mjx-n"><mjx-c class="mjx-c2207"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D453 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-n"><mjx-c class="mjx-c5F"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi mathvariant="normal">∇</mi><mn>2</mn></msup><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>k</mi></msub><mo stretchy="false">)</mo><mstyle scriptlevel="0"><mspace width="0.167em"></mspace></mstyle><mi>d</mi><mo>=</mo><mo>−</mo><mi mathvariant="normal">∇</mi><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mi mathvariant="normal">_</mi><mi>k</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container></span> <span class="math inline"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="CHTML" tabindex="0" ctxtmenu_counter="8" style="font-size: 110%; position: relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-n"><mjx-c class="mjx-c23"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">#</mi></math></mjx-assistive-mml></mjx-container></span> Newton’s direction</li>
-<li><span class="math inline"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="CHTML" tabindex="0" ctxtmenu_counter="9" style="font-size: 110%; position: relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align: -0.15em;"><mjx-texatom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-texatom></mjx-script></mjx-msub><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2190"></mjx-c></mjx-mo><mjx-msub space="4"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align: -0.15em;"><mjx-texatom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi></mjx-texatom></mjx-script></mjx-msub><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-msub space="3"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D451 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align: -0.15em;"><mjx-texatom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi></mjx-texatom></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>x</mi><mrow data-mjx-texclass="ORD"><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">←</mo><msub><mi>x</mi><mrow data-mjx-texclass="ORD"><mi>k</mi></mrow></msub><mo>+</mo><msub><mi>d</mi><mrow data-mjx-texclass="ORD"><mi>k</mi></mrow></msub></math></mjx-assistive-mml></mjx-container></span> # Mise à jour de l’itéré</li>
-<li><span class="math inline"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="CHTML" tabindex="0" ctxtmenu_counter="10" style="font-size: 110%; position: relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2190"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="3"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo stretchy="false">←</mo><mi>k</mi><mo>+</mo><mn>1</mn></math></mjx-assistive-mml></mjx-container></span></li>
-<li>continue = stop_function(<span class="math inline"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="CHTML" tabindex="0" ctxtmenu_counter="11" style="font-size: 110%; position: relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-n"><mjx-c class="mjx-c2207"></mjx-c></mjx-mi><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D453 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align: -0.15em; margin-left: -0.06em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">∇</mi><msub><mi>f</mi><mi>k</mi></msub></math></mjx-assistive-mml></mjx-container></span>,<span class="math inline"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="CHTML" tabindex="0" ctxtmenu_counter="12" style="font-size: 110%; position: relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align: -0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>x</mi><mi>k</mi></msub></math></mjx-assistive-mml></mjx-container></span>,<span class="math inline"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="CHTML" tabindex="0" ctxtmenu_counter="13" style="font-size: 110%; position: relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align: -0.15em;"><mjx-texatom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-texatom></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>x</mi><mrow data-mjx-texclass="ORD"><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></math></mjx-assistive-mml></mjx-container></span>,<span class="math inline"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="CHTML" tabindex="0" ctxtmenu_counter="14" style="font-size: 110%; position: relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D453 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align: -0.15em; margin-left: -0.06em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mi>k</mi></msub></math></mjx-assistive-mml></mjx-container></span>,<span class="math inline"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="CHTML" tabindex="0" ctxtmenu_counter="15" style="font-size: 110%; position: relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D453 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align: -0.15em; margin-left: -0.06em;"><mjx-texatom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-texatom></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mrow data-mjx-texclass="ORD"><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></math></mjx-assistive-mml></mjx-container></span>,AbsTol,RelTol,<span class="math inline"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="CHTML" tabindex="0" ctxtmenu_counter="16" style="font-size: 110%; position: relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D700 TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ε</mi></math></mjx-assistive-mml></mjx-container></span>)</li>
-</ul></li>
-<li><strong>EndWhile</strong></li>
-</ul>
-<section id="sec-criteres-arret" class="level3">
-<h3 class="anchored" data-anchor-id="sec-criteres-arret">Stop criteria<a class="anchorjs-link " aria-label="Anchor" data-anchorjs-icon="" href="file:///Users/gergaud/git-ENS/Julia-TSE/enseignants/M2/Cours/_site/optimization.html#sec-criteres-arret" style="font: 1em / 1 anchorjs-icons; margin-left: 0.1875em; padding-right: 0.1875em; padding-left: 0.1875em;"></a></h3>
-<table class="caption-top table">
-<caption>Stop criteria, Stagnation criteria are more restrictive (<span class="math inline"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="CHTML" tabindex="0" ctxtmenu_counter="17" style="font-size: 109.9%; position: relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D700 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c30"></mjx-c><mjx-c class="mjx-c2E"></mjx-c><mjx-c class="mjx-c30"></mjx-c><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ε</mi><mo>=</mo><mn>0.01</mn></math></mjx-assistive-mml></mjx-container></span>, for example).</caption>
-<colgroup>
-<col style="width: 76%">
-<col style="width: 23%">
-</colgroup>
-<thead>
-<tr class="header">
-<th>Criteria</th>
-<th>Formula</th>
-</tr>
-</thead>
-<tbody>
-<tr class="odd">
-<td><span class="math inline"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="CHTML" tabindex="0" ctxtmenu_counter="18" style="font-size: 110%; position: relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2016"></mjx-c></mjx-mo><mjx-mi class="mjx-n"><mjx-c class="mjx-c2207"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D453 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align: -0.15em;"><mjx-texatom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-texatom></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c2016"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c30"></mjx-c></mjx-mn></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo data-mjx-texclass="OPEN" fence="false" stretchy="false">‖</mo><mi mathvariant="normal">∇</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mrow data-mjx-texclass="ORD"><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mo data-mjx-texclass="CLOSE" fence="false" stretchy="false">‖</mo><mo>=</mo><mn>0</mn></math></mjx-assistive-mml></mjx-container></span></td>
-<td><span class="math inline"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="CHTML" tabindex="0" ctxtmenu_counter="19" style="font-size: 110%; position: relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2016"></mjx-c></mjx-mo><mjx-mi class="mjx-n"><mjx-c class="mjx-c2207"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D453 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align: -0.15em;"><mjx-texatom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-texatom></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c2016"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3C"></mjx-c></mjx-mo><mjx-texatom space="4" texclass="ORD"><mjx-mi class="mjx-n"><mjx-c class="mjx-c6D"></mjx-c><mjx-c class="mjx-c61"></mjx-c><mjx-c class="mjx-c78"></mjx-c></mjx-mi></mjx-texatom><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-texatom texclass="ORD"><mjx-mi class="mjx-n"><mjx-c class="mjx-c52"></mjx-c><mjx-c class="mjx-c65"></mjx-c><mjx-c class="mjx-c6C"></mjx-c><mjx-c class="mjx-c54"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c6C"></mjx-c></mjx-mi></mjx-texatom><mjx-mo class="mjx-n"><mjx-c class="mjx-c2016"></mjx-c></mjx-mo><mjx-mi class="mjx-n"><mjx-c class="mjx-c2207"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D453 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align: -0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c30"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c2016"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-texatom space="2" texclass="ORD"><mjx-mi class="mjx-n"><mjx-c class="mjx-c41"></mjx-c><mjx-c class="mjx-c62"></mjx-c><mjx-c class="mjx-c73"></mjx-c><mjx-c class="mjx-c54"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c6C"></mjx-c></mjx-mi></mjx-texatom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo data-mjx-texclass="OPEN" fence="false" stretchy="false">‖</mo><mi mathvariant="normal">∇</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mrow data-mjx-texclass="ORD"><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mo data-mjx-texclass="CLOSE" fence="false" stretchy="false">‖</mo><mo><</mo><mrow data-mjx-texclass="ORD"><mi data-mjx-auto-op="false">max</mi></mrow><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mi data-mjx-auto-op="false">RelTol</mi></mrow><mo data-mjx-texclass="OPEN" fence="false" stretchy="false">‖</mo><mi mathvariant="normal">∇</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>0</mn></msub><mo stretchy="false">)</mo><mo data-mjx-texclass="CLOSE" fence="false" stretchy="false">‖</mo><mo>,</mo><mrow data-mjx-texclass="ORD"><mi data-mjx-auto-op="false">AbsTol</mi></mrow><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container></span></td>
-</tr>
-<tr class="even">
-<td>Stagnation of the iterate</td>
-<td><span class="math inline"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="CHTML" tabindex="0" ctxtmenu_counter="20" style="font-size: 110%; position: relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2016"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align: -0.15em;"><mjx-texatom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-texatom></mjx-script></mjx-msub><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-msub space="3"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align: -0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c2016"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3C"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D700 TEX-I"></mjx-c></mjx-mi><mjx-texatom texclass="ORD"><mjx-mi class="mjx-n"><mjx-c class="mjx-c6D"></mjx-c><mjx-c class="mjx-c61"></mjx-c><mjx-c class="mjx-c78"></mjx-c></mjx-mi></mjx-texatom><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-texatom texclass="ORD"><mjx-mi class="mjx-n"><mjx-c class="mjx-c52"></mjx-c><mjx-c class="mjx-c65"></mjx-c><mjx-c class="mjx-c6C"></mjx-c><mjx-c class="mjx-c54"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c6C"></mjx-c></mjx-mi></mjx-texatom><mjx-mo class="mjx-n"><mjx-c class="mjx-c2016"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align: -0.15em;"><mjx-texatom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-texatom></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c2016"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-texatom space="2" texclass="ORD"><mjx-mi class="mjx-n"><mjx-c class="mjx-c41"></mjx-c><mjx-c class="mjx-c62"></mjx-c><mjx-c class="mjx-c73"></mjx-c><mjx-c class="mjx-c54"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c6C"></mjx-c></mjx-mi></mjx-texatom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo data-mjx-texclass="OPEN" fence="false" stretchy="false">‖</mo><msub><mi>x</mi><mrow data-mjx-texclass="ORD"><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>−</mo><msub><mi>x</mi><mi>k</mi></msub><mo data-mjx-texclass="CLOSE" fence="false" stretchy="false">‖</mo><mo><</mo><mi>ε</mi><mrow data-mjx-texclass="ORD"><mi data-mjx-auto-op="false">max</mi></mrow><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mi data-mjx-auto-op="false">RelTol</mi></mrow><mo data-mjx-texclass="OPEN" fence="false" stretchy="false">‖</mo><msub><mi>x</mi><mrow data-mjx-texclass="ORD"><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo data-mjx-texclass="CLOSE" fence="false" stretchy="false">‖</mo><mo>,</mo><mrow data-mjx-texclass="ORD"><mi data-mjx-auto-op="false">AbsTol</mi></mrow><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container></span></td>
-</tr>
-<tr class="odd">
-<td>Stagnation of the function</td>
-<td><span class="math inline"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="CHTML" tabindex="0" ctxtmenu_counter="21" style="font-size: 110%; position: relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mo class="mjx-n"><mjx-c class="mjx-c7C"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D453 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align: -0.15em;"><mjx-texatom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-texatom></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D453 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align: -0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c7C"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3C"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D700 TEX-I"></mjx-c></mjx-mi><mjx-texatom texclass="ORD"><mjx-mi class="mjx-n"><mjx-c class="mjx-c6D"></mjx-c><mjx-c class="mjx-c61"></mjx-c><mjx-c class="mjx-c78"></mjx-c></mjx-mi></mjx-texatom><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-texatom texclass="ORD"><mjx-mi class="mjx-n"><mjx-c class="mjx-c52"></mjx-c><mjx-c class="mjx-c65"></mjx-c><mjx-c class="mjx-c6C"></mjx-c><mjx-c class="mjx-c54"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c6C"></mjx-c></mjx-mi></mjx-texatom><mjx-mo class="mjx-n"><mjx-c class="mjx-c7C"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D453 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align: -0.15em;"><mjx-texatom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-texatom></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c7C"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-texatom space="2" texclass="ORD"><mjx-mi class="mjx-n"><mjx-c class="mjx-c41"></mjx-c><mjx-c class="mjx-c62"></mjx-c><mjx-c class="mjx-c73"></mjx-c><mjx-c class="mjx-c54"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c6C"></mjx-c></mjx-mi></mjx-texatom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo data-mjx-texclass="OPEN" fence="false" stretchy="false">|</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mrow data-mjx-texclass="ORD"><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mo>−</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>k</mi></msub><mo stretchy="false">)</mo><mo data-mjx-texclass="CLOSE" fence="false" stretchy="false">|</mo><mo><</mo><mi>ε</mi><mrow data-mjx-texclass="ORD"><mi data-mjx-auto-op="false">max</mi></mrow><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mi data-mjx-auto-op="false">RelTol</mi></mrow><mo data-mjx-texclass="OPEN" fence="false" stretchy="false">|</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mrow data-mjx-texclass="ORD"><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mo data-mjx-texclass="CLOSE" fence="false" stretchy="false">|</mo><mo>,</mo><mrow data-mjx-texclass="ORD"><mi data-mjx-auto-op="false">AbsTol</mi></mrow><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container></span></td>
-</tr>
-<tr class="even">
-<td>Maximum number of iteration</td>
-<td><span class="math inline"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="CHTML" tabindex="0" ctxtmenu_counter="22" style="font-size: 110%; position: relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="3"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-texatom space="4" texclass="ORD"><mjx-mi class="mjx-n"><mjx-c class="mjx-c6D"></mjx-c><mjx-c class="mjx-c61"></mjx-c><mjx-c class="mjx-c78"></mjx-c></mjx-mi><mjx-mi class="mjx-n"><mjx-c class="mjx-c5F"></mjx-c></mjx-mi><mjx-mi class="mjx-n"><mjx-c class="mjx-c69"></mjx-c><mjx-c class="mjx-c74"></mjx-c><mjx-c class="mjx-c65"></mjx-c><mjx-c class="mjx-c72"></mjx-c></mjx-mi></mjx-texatom></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>+</mo><mn>1</mn><mo>=</mo><mrow data-mjx-texclass="ORD"><mi data-mjx-auto-op="false">max</mi><mi mathvariant="normal">_</mi><mi data-mjx-auto-op="false">iter</mi></mrow></math></mjx-assistive-mml></mjx-container></span></td>
-</tr>
-</tbody>
-</table>
-</section>
-<section id="exercice" class="level3">
-<h3 class="anchored" data-anchor-id="exercice">Exercice<a class="anchorjs-link " aria-label="Anchor" data-anchorjs-icon="" href="file:///Users/gergaud/git-ENS/Julia-TSE/enseignants/M2/Cours/_site/optimization.html#exercice" style="font: 1em / 1 anchorjs-icons; margin-left: 0.1875em; padding-right: 0.1875em; padding-left: 0.1875em;"></a></h3>
-<div id="50e63fb5" class="cell" data-execution_count="2">
-<details class="code-fold">
-<summary>Code</summary>
-<div class="sourceCode cell-code" id="cb3"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb3-1"><a href="file:///Users/gergaud/git-ENS/Julia-TSE/enseignants/M2/Cours/_site/optimization.html#cb3-1" aria-hidden="true" tabindex="-1"></a><span class="im">using</span> <span class="bu">LinearAlgebra</span></span>
-<span id="cb3-2"><a href="file:///Users/gergaud/git-ENS/Julia-TSE/enseignants/M2/Cours/_site/optimization.html#cb3-2" aria-hidden="true" tabindex="-1"></a><span class="im">using</span> <span class="bu">ForwardDiff</span></span>
-<span id="cb3-3"><a href="file:///Users/gergaud/git-ENS/Julia-TSE/enseignants/M2/Cours/_site/optimization.html#cb3-3" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb3-4"><a href="file:///Users/gergaud/git-ENS/Julia-TSE/enseignants/M2/Cours/_site/optimization.html#cb3-4" aria-hidden="true" tabindex="-1"></a><span class="st">"""</span></span>
-<span id="cb3-5"><a href="file:///Users/gergaud/git-ENS/Julia-TSE/enseignants/M2/Cours/_site/optimization.html#cb3-5" aria-hidden="true" tabindex="-1"></a><span class="st"> Solve by Newton's algorithm the optimization problem Min f(x)</span></span>
-<span id="cb3-6"><a href="file:///Users/gergaud/git-ENS/Julia-TSE/enseignants/M2/Cours/_site/optimization.html#cb3-6" aria-hidden="true" tabindex="-1"></a><span class="st"> Case where f is a function from R^n to R</span></span>
-<span id="cb3-7"><a href="file:///Users/gergaud/git-ENS/Julia-TSE/enseignants/M2/Cours/_site/optimization.html#cb3-7" aria-hidden="true" tabindex="-1"></a><span class="st">"""</span></span>
-<span id="cb3-8"><a href="file:///Users/gergaud/git-ENS/Julia-TSE/enseignants/M2/Cours/_site/optimization.html#cb3-8" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb3-9"><a href="file:///Users/gergaud/git-ENS/Julia-TSE/enseignants/M2/Cours/_site/optimization.html#cb3-9" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb3-10"><a href="file:///Users/gergaud/git-ENS/Julia-TSE/enseignants/M2/Cours/_site/optimization.html#cb3-10" aria-hidden="true" tabindex="-1"></a><span class="kw">function</span> <span class="fu">algo_Newton</span>(f,x0<span class="op">::</span><span class="dt">Vector{<:Real}</span>;AbsTol<span class="op">=</span> <span class="fu">abs</span>(<span class="fu">eps</span>()), RelTol <span class="op">=</span> <span class="fu">abs</span>(<span class="fu">eps</span>()), ε<span class="op">=</span><span class="fl">0.01</span>, nbit_max <span class="op">=</span> <span class="fl">0</span>)</span>
-<span id="cb3-11"><a href="file:///Users/gergaud/git-ENS/Julia-TSE/enseignants/M2/Cours/_site/optimization.html#cb3-11" aria-hidden="true" tabindex="-1"></a><span class="co"># to complete</span></span>
-<span id="cb3-12"><a href="file:///Users/gergaud/git-ENS/Julia-TSE/enseignants/M2/Cours/_site/optimization.html#cb3-12" aria-hidden="true" tabindex="-1"></a> </span>
-<span id="cb3-13"><a href="file:///Users/gergaud/git-ENS/Julia-TSE/enseignants/M2/Cours/_site/optimization.html#cb3-13" aria-hidden="true" tabindex="-1"></a> <span class="co"># flag = 0 if the program stop on the first criteria</span></span>
-<span id="cb3-14"><a href="file:///Users/gergaud/git-ENS/Julia-TSE/enseignants/M2/Cours/_site/optimization.html#cb3-14" aria-hidden="true" tabindex="-1"></a> <span class="co"># flag = 2 if the program stop on the second criteria</span></span>
-<span id="cb3-15"><a href="file:///Users/gergaud/git-ENS/Julia-TSE/enseignants/M2/Cours/_site/optimization.html#cb3-15" aria-hidden="true" tabindex="-1"></a> <span class="co"># ...</span></span>
-<span id="cb3-16"><a href="file:///Users/gergaud/git-ENS/Julia-TSE/enseignants/M2/Cours/_site/optimization.html#cb3-16" aria-hidden="true" tabindex="-1"></a> <span class="cf">return</span> xₖ, flag, fₖ, ∇fₖ , k </span>
-<span id="cb3-17"><a href="file:///Users/gergaud/git-ENS/Julia-TSE/enseignants/M2/Cours/_site/optimization.html#cb3-17" aria-hidden="true" tabindex="-1"></a><span class="kw">end</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
-</details>
-<div class="cell-output cell-output-display" data-execution_count="53">
-<pre><code>algo_Newton (generic function with 1 method)</code></pre>
-</div>
-</div>
-<div id="fdaa1a09" class="cell" data-execution_count="3">
-<details class="code-fold">
-<summary>Code</summary>
-<div class="sourceCode cell-code" id="cb5"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb5-1"><a href="file:///Users/gergaud/git-ENS/Julia-TSE/enseignants/M2/Cours/_site/optimization.html#cb5-1" aria-hidden="true" tabindex="-1"></a><span class="fu">include</span>(<span class="st">"src/MyOptims.jl"</span>)</span>
-<span id="cb5-2"><a href="file:///Users/gergaud/git-ENS/Julia-TSE/enseignants/M2/Cours/_site/optimization.html#cb5-2" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb5-3"><a href="file:///Users/gergaud/git-ENS/Julia-TSE/enseignants/M2/Cours/_site/optimization.html#cb5-3" aria-hidden="true" tabindex="-1"></a>A <span class="op">=</span> [<span class="fl">1</span> <span class="fl">0</span> ; <span class="fl">0</span> <span class="fl">9</span><span class="op">/</span><span class="fl">2</span>]</span>
-<span id="cb5-4"><a href="file:///Users/gergaud/git-ENS/Julia-TSE/enseignants/M2/Cours/_site/optimization.html#cb5-4" aria-hidden="true" tabindex="-1"></a>b <span class="op">=</span> [<span class="fl">0</span>,<span class="fl">0</span>]; c<span class="op">=</span><span class="fl">0</span>.</span>
-<span id="cb5-5"><a href="file:///Users/gergaud/git-ENS/Julia-TSE/enseignants/M2/Cours/_site/optimization.html#cb5-5" aria-hidden="true" tabindex="-1"></a><span class="fu">f1</span>(x) <span class="op">=</span> <span class="fl">0.5</span><span class="op">*</span>x<span class="op">'*</span>A<span class="op">*</span>x <span class="op">+</span> b<span class="op">'*</span>x <span class="op">+</span>c</span>
-<span id="cb5-6"><a href="file:///Users/gergaud/git-ENS/Julia-TSE/enseignants/M2/Cours/_site/optimization.html#cb5-6" aria-hidden="true" tabindex="-1"></a>x0 <span class="op">=</span> [<span class="fl">1000</span>,<span class="op">-</span><span class="fl">20</span>]</span>
-<span id="cb5-7"><a href="file:///Users/gergaud/git-ENS/Julia-TSE/enseignants/M2/Cours/_site/optimization.html#cb5-7" aria-hidden="true" tabindex="-1"></a><span class="fu">println</span>(<span class="st">"Results for Newton on f1 : "</span>, <span class="fu">algo_Newton</span>(f1,x0))</span>
-<span id="cb5-8"><a href="file:///Users/gergaud/git-ENS/Julia-TSE/enseignants/M2/Cours/_site/optimization.html#cb5-8" aria-hidden="true" tabindex="-1"></a><span class="fu">println</span>(<span class="st">"eigen value of 0.5(A^T+A) = "</span>, <span class="fl">0.5</span><span class="fu">*eigen</span>(A<span class="op">'+</span>A).values)</span>
-<span id="cb5-9"><a href="file:///Users/gergaud/git-ENS/Julia-TSE/enseignants/M2/Cours/_site/optimization.html#cb5-9" aria-hidden="true" tabindex="-1"></a><span class="im">using</span> <span class="bu">Plots;</span></span>
-<span id="cb5-10"><a href="file:///Users/gergaud/git-ENS/Julia-TSE/enseignants/M2/Cours/_site/optimization.html#cb5-10" aria-hidden="true" tabindex="-1"></a>x<span class="op">=</span><span class="fu">range</span>(<span class="op">-</span><span class="fl">10</span>,stop<span class="op">=</span><span class="fl">10</span>,length<span class="op">=</span><span class="fl">100</span>)</span>
-<span id="cb5-11"><a href="file:///Users/gergaud/git-ENS/Julia-TSE/enseignants/M2/Cours/_site/optimization.html#cb5-11" aria-hidden="true" tabindex="-1"></a>y<span class="op">=</span><span class="fu">range</span>(<span class="op">-</span><span class="fl">10</span>,stop<span class="op">=</span><span class="fl">10</span>,length<span class="op">=</span><span class="fl">100</span>)</span>
-<span id="cb5-12"><a href="file:///Users/gergaud/git-ENS/Julia-TSE/enseignants/M2/Cours/_site/optimization.html#cb5-12" aria-hidden="true" tabindex="-1"></a><span class="fu">f11</span>(x,y) <span class="op">=</span> <span class="fu">f1</span>([x,y])</span>
-<span id="cb5-13"><a href="file:///Users/gergaud/git-ENS/Julia-TSE/enseignants/M2/Cours/_site/optimization.html#cb5-13" aria-hidden="true" tabindex="-1"></a>p1 <span class="op">=</span> <span class="fu">plot</span>(x,y,f11,st<span class="op">=:</span>contourf)</span>
-<span id="cb5-14"><a href="file:///Users/gergaud/git-ENS/Julia-TSE/enseignants/M2/Cours/_site/optimization.html#cb5-14" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb5-15"><a href="file:///Users/gergaud/git-ENS/Julia-TSE/enseignants/M2/Cours/_site/optimization.html#cb5-15" aria-hidden="true" tabindex="-1"></a>A <span class="op">=</span> [<span class="op">-</span><span class="fl">1</span> <span class="fl">0</span> ; <span class="fl">0</span> <span class="fl">3</span>]</span>
-<span id="cb5-16"><a href="file:///Users/gergaud/git-ENS/Julia-TSE/enseignants/M2/Cours/_site/optimization.html#cb5-16" aria-hidden="true" tabindex="-1"></a><span class="fu">f2</span>(x) <span class="op">=</span> <span class="fl">0.5</span><span class="op">*</span>x<span class="op">'*</span>A<span class="op">*</span>x <span class="op">+</span> b<span class="op">'*</span>x <span class="op">+</span>c</span>
-<span id="cb5-17"><a href="file:///Users/gergaud/git-ENS/Julia-TSE/enseignants/M2/Cours/_site/optimization.html#cb5-17" aria-hidden="true" tabindex="-1"></a><span class="fu">println</span>(<span class="st">"Results for Newton on f2 : "</span>, <span class="fu">algo_Newton</span>(f2,x0))</span>
-<span id="cb5-18"><a href="file:///Users/gergaud/git-ENS/Julia-TSE/enseignants/M2/Cours/_site/optimization.html#cb5-18" aria-hidden="true" tabindex="-1"></a><span class="fu">println</span>(<span class="st">"eigen value of 0.5(A^T+A) = "</span>, <span class="fl">0.5</span><span class="fu">*eigen</span>(A<span class="op">'+</span>A).values)</span>
-<span id="cb5-19"><a href="file:///Users/gergaud/git-ENS/Julia-TSE/enseignants/M2/Cours/_site/optimization.html#cb5-19" aria-hidden="true" tabindex="-1"></a><span class="fu">f21</span>(x,y) <span class="op">=</span> <span class="fu">f1</span>([x,y])</span>
-<span id="cb5-20"><a href="file:///Users/gergaud/git-ENS/Julia-TSE/enseignants/M2/Cours/_site/optimization.html#cb5-20" aria-hidden="true" tabindex="-1"></a>p2 <span class="op">=</span> <span class="fu">plot</span>(x,y,f21,st<span class="op">=:</span>contourf)</span>
-<span id="cb5-21"><a href="file:///Users/gergaud/git-ENS/Julia-TSE/enseignants/M2/Cours/_site/optimization.html#cb5-21" aria-hidden="true" tabindex="-1"></a><span class="co"># Rosenbrock function</span></span>
-<span id="cb5-22"><a href="file:///Users/gergaud/git-ENS/Julia-TSE/enseignants/M2/Cours/_site/optimization.html#cb5-22" aria-hidden="true" tabindex="-1"></a>x<span class="op">=</span><span class="fu">range</span>(<span class="op">-</span><span class="fl">1.5</span>,stop<span class="op">=</span><span class="fl">2</span>,length<span class="op">=</span><span class="fl">100</span>)</span>
-<span id="cb5-23"><a href="file:///Users/gergaud/git-ENS/Julia-TSE/enseignants/M2/Cours/_site/optimization.html#cb5-23" aria-hidden="true" tabindex="-1"></a>y<span class="op">=</span><span class="fu">range</span>(<span class="op">-</span><span class="fl">0.5</span>,stop<span class="op">=</span><span class="fl">3</span>.,length<span class="op">=</span><span class="fl">100</span>)</span>
-<span id="cb5-24"><a href="file:///Users/gergaud/git-ENS/Julia-TSE/enseignants/M2/Cours/_site/optimization.html#cb5-24" aria-hidden="true" tabindex="-1"></a><span class="fu">f3</span>(x) <span class="op">=</span> (<span class="fl">1</span><span class="op">-</span>x[<span class="fl">1</span>])<span class="op">^</span><span class="fl">2</span> <span class="op">+</span> <span class="fl">100</span><span class="fu">*</span>(x[<span class="fl">2</span>]<span class="op">-</span>x[<span class="fl">1</span>]<span class="op">^</span><span class="fl">2</span>)<span class="op">^</span><span class="fl">2</span></span>
-<span id="cb5-25"><a href="file:///Users/gergaud/git-ENS/Julia-TSE/enseignants/M2/Cours/_site/optimization.html#cb5-25" aria-hidden="true" tabindex="-1"></a><span class="fu">f31</span>(x,y) <span class="op">=</span> <span class="fu">f3</span>([x,y])</span>
-<span id="cb5-26"><a href="file:///Users/gergaud/git-ENS/Julia-TSE/enseignants/M2/Cours/_site/optimization.html#cb5-26" aria-hidden="true" tabindex="-1"></a>p3 <span class="op">=</span> <span class="fu">plot</span>(x,y,f31,st<span class="op">=:</span>contourf)</span>
-<span id="cb5-27"><a href="file:///Users/gergaud/git-ENS/Julia-TSE/enseignants/M2/Cours/_site/optimization.html#cb5-27" aria-hidden="true" tabindex="-1"></a>x0 <span class="op">=</span> [<span class="fl">1.1</span>,<span class="fl">1.2</span>]</span>
-<span id="cb5-28"><a href="file:///Users/gergaud/git-ENS/Julia-TSE/enseignants/M2/Cours/_site/optimization.html#cb5-28" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb5-29"><a href="file:///Users/gergaud/git-ENS/Julia-TSE/enseignants/M2/Cours/_site/optimization.html#cb5-29" aria-hidden="true" tabindex="-1"></a><span class="fu">println</span>(<span class="st">"Results for Newton on f3 : "</span>, <span class="fu">algo_Newton</span>(f3,[<span class="fl">1.1</span>,<span class="fl">1.2</span>]))</span>
-<span id="cb5-30"><a href="file:///Users/gergaud/git-ENS/Julia-TSE/enseignants/M2/Cours/_site/optimization.html#cb5-30" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb5-31"><a href="file:///Users/gergaud/git-ENS/Julia-TSE/enseignants/M2/Cours/_site/optimization.html#cb5-31" aria-hidden="true" tabindex="-1"></a><span class="fu">println</span>(<span class="st">"Results for Newton on f3 : "</span>, <span class="fu">algo_Newton</span>(f3,[<span class="fl">3</span>.,<span class="fl">0</span>.]))</span>
-<span id="cb5-32"><a href="file:///Users/gergaud/git-ENS/Julia-TSE/enseignants/M2/Cours/_site/optimization.html#cb5-32" aria-hidden="true" tabindex="-1"></a><span class="fu">plot</span>(p1,p2,p3)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
-</details>
-<div class="cell-output cell-output-stdout">
-<pre><code>Results for Newton on f1 : ([0.0, 0.0], 0, 0.0, [0.0, 0.0], 1)
-eigen value of 0.5(A^T+A) = [1.0, 4.5]
-Results for Newton on f2 : ([0.0, 0.0], 0, 0.0, [0.0, 0.0], 1)
-eigen value of 0.5(A^T+A) = [-1.0, 3.0]
-Results for Newton on f3 : ([1.0, 1.0], 0, 0.0, [-0.0, 0.0], 7)
-Results for Newton on f3 : ([1.0, 1.0], 0, 0.0, [-0.0, 0.0], 5)</code></pre>
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