ブロイデン法

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id="mw-panel" class="vector-main-menu-landmark" aria-label="サイト"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="目次" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">目次</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">サイドバーに移動</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">非表示</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">ページ先頭</div> </a> </li> <li id="toc-手法の詳細" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#手法の詳細"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>手法の詳細</span> </div> </a> <button aria-controls="toc-手法の詳細-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>手法の詳細サブセクションを切り替えます</span> </button> <ul id="toc-手法の詳細-sublist" class="vector-toc-list"> <li id="toc-1変数方程式の求根" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#1変数方程式の求根"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>1変数方程式の求根</span> </div> </a> <ul id="toc-1変数方程式の求根-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-非線形方程式系の求根" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#非線形方程式系の求根"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>非線形方程式系の求根</span> </div> </a> <ul id="toc-非線形方程式系の求根-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Broyden_Classの手法" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Broyden_Classの手法"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Broyden Classの手法</span> </div> </a> <ul id="toc-Broyden_Classの手法-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-関連項目" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#関連項目"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>関連項目</span> </div> </a> <ul id="toc-関連項目-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-出典" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#出典"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>出典</span> </div> </a> <ul id="toc-出典-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-関連文献" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#関連文献"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>関連文献</span> </div> </a> <ul id="toc-関連文献-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-外部リンク" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#外部リンク"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>外部リンク</span> </div> </a> <ul id="toc-外部リンク-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="目次" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="目次の表示・非表示を切り替え" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">目次の表示・非表示を切り替え</span> </label> <div class="vector-dropdown-content"> <div 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id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=%E3%83%96%E3%83%AD%E3%82%A4%E3%83%87%E3%83%B3%E6%B3%95&action=history"><span>履歴表示</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> 全般 </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/%E7%89%B9%E5%88%A5:%E3%83%AA%E3%83%B3%E3%82%AF%E5%85%83/%E3%83%96%E3%83%AD%E3%82%A4%E3%83%87%E3%83%B3%E6%B3%95" title="ここにリンクしている全ウィキページの一覧 [j]" accesskey="j"><span>リンク元</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/%E7%89%B9%E5%88%A5:%E9%96%A2%E9%80%A3%E3%83%9A%E3%83%BC%E3%82%B8%E3%81%AE%E6%9B%B4%E6%96%B0%E7%8A%B6%E6%B3%81/%E3%83%96%E3%83%AD%E3%82%A4%E3%83%87%E3%83%B3%E6%B3%95" rel="nofollow" title="このページからリンクしているページの最近の更新 [k]" accesskey="k"><span>関連ページの更新状況</span></a></li><li id="t-upload" 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data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">表示</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">サイドバーに移動</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">非表示</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">出典: フリー百科事典『ウィキペディア（Wikipedia）』</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="ja" dir="ltr"><style data-mw-deduplicate="TemplateStyles:r101304250">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%;font-size:90%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}html.client-js body.skin-minerva .mw-parser-output .mbox-text-span{margin-left:23px!important}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="plainlinks metadata ambox ambox-content" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><span title="翻訳直後"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Translation_arrow.svg/50px-Translation_arrow.svg.png" decoding="async" width="50" height="17" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Translation_arrow.svg/75px-Translation_arrow.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Translation_arrow.svg/100px-Translation_arrow.svg.png 2x" data-file-width="60" data-file-height="20" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span">この項目「<b>ブロイデン法</b>」は翻訳されたばかりのものです。不自然あるいは曖昧な表現などが含まれる可能性があり、このままでは読みづらいかもしれません。（原文：<a href="https://en.wikipedia.org/wiki/Special:Permalink/1235927772" class="extiw" title="en:Special:Permalink/1235927772">en: Broyden's method</a>）<br /> 修正、加筆に協力し、現在の表現をより自然な表現にして下さる方を求めています。<a href="/w/index.php?title=%E3%83%8E%E3%83%BC%E3%83%88:%E3%83%96%E3%83%AD%E3%82%A4%E3%83%87%E3%83%B3%E6%B3%95&action=edit&redlink=1" class="new" title="「ノート:ブロイデン法」 (存在しないページ)">ノートページ</a>や<a class="external text" href="https://ja.wikipedia.org/w/index.php?title=%E3%83%96%E3%83%AD%E3%82%A4%E3%83%87%E3%83%B3%E6%B3%95&action=history">履歴</a>も参照してください。<small>（<span title="2024年9月">2024年9月</span>）</small></div></td></tr></tbody></table> <p><a href="/wiki/%E6%95%B0%E5%80%A4%E8%A7%A3%E6%9E%90" title="数値解析">数値解析</a>において、<b>ブロイデン法</b>（ブロイデンほう<a href="/wiki/%E8%8B%B1%E8%AA%9E" title="英語">英</a>: <span lang="en">Broyden's method</span>）とは<a href="/wiki/%E6%BA%96%E3%83%8B%E3%83%A5%E3%83%BC%E3%83%88%E3%83%B3%E6%B3%95" title="準ニュートン法">準ニュートン法</a>の1種であり、<a href="/wiki/%E5%A4%9A%E5%A4%89%E6%95%B0%E9%96%A2%E6%95%B0" class="mw-redirect" title="多変数関数">多変数関数</a>の<a href="/wiki/%E6%B1%82%E6%A0%B9%E3%82%A2%E3%83%AB%E3%82%B4%E3%83%AA%E3%82%BA%E3%83%A0" title="求根アルゴリズム">求根</a>に用いられる<a href="/wiki/%E3%82%A2%E3%83%AB%E3%82%B4%E3%83%AA%E3%82%BA%E3%83%A0" title="アルゴリズム">アルゴリズム</a>である。<a href="/wiki/1965%E5%B9%B4" title="1965年">1965年</a>に<span title="リンク先の項目はまだありません。新規の執筆や他言語版からの翻訳が望まれます。"><a href="/w/index.php?title=%E3%83%81%E3%83%A3%E3%83%BC%E3%83%AB%E3%82%BA%E3%83%BB%E3%82%B8%E3%83%A7%E3%83%BC%E3%82%B8%E3%83%BB%E3%83%96%E3%83%AD%E3%82%A4%E3%83%87%E3%83%B3&action=edit&redlink=1" class="new" title="「チャールズ・ジョージ・ブロイデン」 (存在しないページ)">チャールズ・ジョージ・ブロイデン</a><span style="font-size: 0.77em; font-weight: normal;" class="noprint">（<a href="https://en.wikipedia.org/wiki/Charles_George_Broyden" class="extiw" title="en:Charles George Broyden">英語版</a>）</span></span>が発表した<sup id="cite_ref-Broyden_1965_1-0" class="reference"><a href="#cite_note-Broyden_1965-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>。 </p><p><a href="/wiki/%E3%83%8B%E3%83%A5%E3%83%BC%E3%83%88%E3%83%B3%E6%B3%95" title="ニュートン法">ニュートン法</a>により<span lang="en" class="texhtml"><i><b>f</b></i>(<i><b>x</b></i>) = <b>0</b></span>を解く場合、各イテレーションごとに<a href="/wiki/%E3%83%A4%E3%82%B3%E3%83%93%E8%A1%8C%E5%88%97" title="ヤコビ行列">ヤコビアン</a><span lang="en" class="texhtml mvar" style="font-style:italic;"><b>J</b></span>を用いることになる。しかし、ヤコビアンを計算するには困難で複雑な演算を要する。ブロイデン法では、ヤコビアン全体を最初のイテレーションで1回だけ計算し、以降のイテレーションではランク1更新を用いる。 </p><p><a href="/wiki/1979%E5%B9%B4" title="1979年">1979年</a>、Gayによりブロイデン法はサイズ<span lang="en" class="texhtml"><i>n</i> × <i>n</i></span>の線形システムに適用したとき<span lang="en" class="texhtml">2 <i>n</i></span>ステップで終了することが証明された<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>。しかし、他の準ニュートン法と同様、非線形システムに対しては収束する保証はない。 </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="手法の詳細"><span id=".E6.89.8B.E6.B3.95.E3.81.AE.E8.A9.B3.E7.B4.B0"></span>手法の詳細</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E3%83%96%E3%83%AD%E3%82%A4%E3%83%87%E3%83%B3%E6%B3%95&action=edit&section=1" title="節を編集: 手法の詳細"><span>編集</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="1変数方程式の求根"><span id="1.E5.A4.89.E6.95.B0.E6.96.B9.E7.A8.8B.E5.BC.8F.E3.81.AE.E6.B1.82.E6.A0.B9"></span>1変数方程式の求根</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E3%83%96%E3%83%AD%E3%82%A4%E3%83%87%E3%83%B3%E6%B3%95&action=edit&section=2" title="節を編集: 1変数方程式の求根"><span>編集</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/%E5%89%B2%E7%B7%9A%E6%B3%95" title="割線法">割線法</a>では、<span lang="en" class="texhtml"><i>f</i>′</span>の<span lang="en" class="texhtml"><i>x</i><sub><i>n</i></sub></span>における1階<a href="/wiki/%E5%BE%AE%E5%88%86" title="微分">微分</a>を<a href="/wiki/%E6%9C%89%E9%99%90%E5%B7%AE%E5%88%86" title="有限差分">有限差分</a>近似する。 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f'(x_{n})\simeq {\frac {f(x_{n})-f(x_{n-1})}{x_{n}-x_{n-1}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>≃</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f'(x_{n})\simeq {\frac {f(x_{n})-f(x_{n-1})}{x_{n}-x_{n-1}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7267b0cc36895f5e29ee435bb9bfc8010e130e75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:26.51ex; height:6.176ex;" alt="{\displaystyle f'(x_{n})\simeq {\frac {f(x_{n})-f(x_{n-1})}{x_{n}-x_{n-1}}}}"></span></dd></dl> <p>その上で、<a href="/wiki/%E3%83%8B%E3%83%A5%E3%83%BC%E3%83%88%E3%83%B3%E6%B3%95" title="ニュートン法">ニュートン法</a>と同様の操作を繰り返す </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{n+1}=x_{n}-{\frac {f(x_{n})}{f^{\prime }(x_{n})}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{n+1}=x_{n}-{\frac {f(x_{n})}{f^{\prime }(x_{n})}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/141fdb46e7bc9c5dc27e7c1fb54f14550f638c62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:20.335ex; height:6.509ex;" alt="{\displaystyle x_{n+1}=x_{n}-{\frac {f(x_{n})}{f^{\prime }(x_{n})}}}"></span></dd></dl> <p>ここで<span lang="en" class="texhtml"><i>n</i></span>はイテレーション指数である。 </p> <div class="mw-heading mw-heading3"><h3 id="非線形方程式系の求根"><span id=".E9.9D.9E.E7.B7.9A.E5.BD.A2.E6.96.B9.E7.A8.8B.E5.BC.8F.E7.B3.BB.E3.81.AE.E6.B1.82.E6.A0.B9"></span>非線形方程式系の求根</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E3%83%96%E3%83%AD%E3%82%A4%E3%83%87%E3%83%B3%E6%B3%95&action=edit&section=3" title="節を編集: 非線形方程式系の求根"><span>編集</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><span lang="en" class="texhtml"><i>k</i></span>本の非線形方程式の系 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {f}}({\boldsymbol {x}})=\mathbf {0} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">f</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">x</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {f}}({\boldsymbol {x}})=\mathbf {0} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1f4b82eb33a8698e2b6dfda4f8e65577d3ab378" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.227ex; height:2.843ex;" alt="{\displaystyle {\boldsymbol {f}}({\boldsymbol {x}})=\mathbf {0} }"></span></dd></dl> <p>を考える。ここで<span lang="en" class="texhtml mvar" style="font-style:italic;"><b>f</b></span>は<a href="/wiki/%E3%83%99%E3%82%AF%E3%83%88%E3%83%AB%E7%A9%BA%E9%96%93" title="ベクトル空間">ベクトル</a><span lang="en" class="texhtml mvar" style="font-style:italic;"><b>x</b></span>の<a href="/wiki/%E3%83%99%E3%82%AF%E3%83%88%E3%83%AB%E5%80%A4%E9%96%A2%E6%95%B0" class="mw-redirect" title="ベクトル値関数">ベクトル値関数</a>である。 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {x}}=(x_{1},x_{2},x_{3},\dotsc ,x_{k})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">x</mi> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {x}}=(x_{1},x_{2},x_{3},\dotsc ,x_{k})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b7e338f9c386a0d34d7c3c40bdf4e72fb3e4fe2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.256ex; height:2.843ex;" alt="{\displaystyle {\boldsymbol {x}}=(x_{1},x_{2},x_{3},\dotsc ,x_{k})}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {f}}({\boldsymbol {x}})={\big (}f_{1}(x_{1},x_{2},\dotsc ,x_{k}),f_{2}(x_{1},x_{2},\dotsc ,x_{k}),\dotsc ,f_{k}(x_{1},x_{2},\dotsc ,x_{k}){\big )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">f</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">x</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {f}}({\boldsymbol {x}})={\big (}f_{1}(x_{1},x_{2},\dotsc ,x_{k}),f_{2}(x_{1},x_{2},\dotsc ,x_{k}),\dotsc ,f_{k}(x_{1},x_{2},\dotsc ,x_{k}){\big )}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ed94f5e8ef95aa658a0ee9877a0c9a38ca5115d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:68.47ex; height:3.176ex;" alt="{\displaystyle {\boldsymbol {f}}({\boldsymbol {x}})={\big (}f_{1}(x_{1},x_{2},\dotsc ,x_{k}),f_{2}(x_{1},x_{2},\dotsc ,x_{k}),\dotsc ,f_{k}(x_{1},x_{2},\dotsc ,x_{k}){\big )}}"></span></dd> <dd></dd> <dd></dd></dl> <p>このような問題に対して、ブロイデンは1次元ニュートン法の微分をヤコビアン<span lang="en" class="texhtml mvar" style="font-style:italic;"><b>J</b></span>で置き換えて一般化した手法を考案した。ヤコビアンは、次のように割線法における有限差分近似にもとづいて反復的に決定される。 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {J}}_{n}({\boldsymbol {x}}_{n}-{\boldsymbol {x}}_{n-1})\simeq {\boldsymbol {f}}({\boldsymbol {x}}_{n})-{\boldsymbol {f}}({\boldsymbol {x}}_{n-1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">J</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>≃</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">f</mi> </mrow> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">f</mi> </mrow> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {J}}_{n}({\boldsymbol {x}}_{n}-{\boldsymbol {x}}_{n-1})\simeq {\boldsymbol {f}}({\boldsymbol {x}}_{n})-{\boldsymbol {f}}({\boldsymbol {x}}_{n-1})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a6b9428c33b467ab99bf7eac6c73c3d5991af5f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.995ex; height:2.843ex;" alt="{\displaystyle {\boldsymbol {J}}_{n}({\boldsymbol {x}}_{n}-{\boldsymbol {x}}_{n-1})\simeq {\boldsymbol {f}}({\boldsymbol {x}}_{n})-{\boldsymbol {f}}({\boldsymbol {x}}_{n-1})}"></span></dd></dl> <p>ここで<span lang="en" class="texhtml"><i>n</i></span>はイテレーション指数である。 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {f}}_{n}={\boldsymbol {f}}({\boldsymbol {x}}_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">f</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">f</mi> </mrow> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {f}}_{n}={\boldsymbol {f}}({\boldsymbol {x}}_{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f5463b11b49d7f4bb95b7794eefe0841c960ad9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.647ex; height:2.843ex;" alt="{\displaystyle {\boldsymbol {f}}_{n}={\boldsymbol {f}}({\boldsymbol {x}}_{n})}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta {\boldsymbol {x}}_{n}={\boldsymbol {x}}_{n}-{\boldsymbol {x}}_{n-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta {\boldsymbol {x}}_{n}={\boldsymbol {x}}_{n}-{\boldsymbol {x}}_{n-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c5f3479dd5414d692a0dd512bc4f06d906bfa5b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.226ex; height:2.509ex;" alt="{\displaystyle \Delta {\boldsymbol {x}}_{n}={\boldsymbol {x}}_{n}-{\boldsymbol {x}}_{n-1}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta {\boldsymbol {f}}_{n}={\boldsymbol {f}}_{n}-{\boldsymbol {f}}_{n-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">f</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">f</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">f</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta {\boldsymbol {f}}_{n}={\boldsymbol {f}}_{n}-{\boldsymbol {f}}_{n-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4300e79d67eceaf8042c867b02fae7adbd8a6ad4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.592ex; height:2.843ex;" alt="{\displaystyle \Delta {\boldsymbol {f}}_{n}={\boldsymbol {f}}_{n}-{\boldsymbol {f}}_{n-1}}"></span></dd></dl> <p>のように定義すると、上式は以下のように簡潔に書ける。 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {J}}_{n}\Delta {\boldsymbol {x}}_{n}\simeq \Delta {\boldsymbol {f}}_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">J</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi mathvariant="normal">Δ</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>≃</mo> <mi mathvariant="normal">Δ</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">f</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {J}}_{n}\Delta {\boldsymbol {x}}_{n}\simeq \Delta {\boldsymbol {f}}_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2871c4ac4b2b291daa26f44749ca4bd3ecfc228e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.944ex; height:2.676ex;" alt="{\displaystyle {\boldsymbol {J}}_{n}\Delta {\boldsymbol {x}}_{n}\simeq \Delta {\boldsymbol {f}}_{n}}"></span></dd></dl> <p>上式は<span lang="en" class="texhtml"><i>k</i></span>が1より大きい場合は<span title="リンク先の項目はまだありません。新規の執筆や他言語版からの翻訳が望まれます。"><a href="/w/index.php?title=%E5%8A%A3%E6%B1%BA%E5%AE%9A%E7%B3%BB&action=edit&redlink=1" class="new" title="「劣決定系」 (存在しないページ)">劣決定系</a><span style="font-size: 0.77em; font-weight: normal;" class="noprint">（<a href="https://en.wikipedia.org/wiki/Underdetermined_system" class="extiw" title="en:Underdetermined system">英語版</a>）</span></span>となる。ブロイデンは、以下のようにヤコビアンの現状の推定値<span lang="en" class="texhtml"><i><b>J</b></i><sub><i>n</i>−1</sub></span>を最低限の変更により割線方程式を満たすよう改善することを提案した。 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {J}}_{n}={\boldsymbol {J}}_{n-1}+{\frac {\Delta {\boldsymbol {f}}_{n}-{\boldsymbol {J}}_{n-1}\Delta {\boldsymbol {x}}_{n}}{\|\Delta {\boldsymbol {x}}_{n}\|^{2}}}\Delta {\boldsymbol {x}}_{n}^{\top }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">J</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">J</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">f</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">J</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mi mathvariant="normal">Δ</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mrow> <mo fence="false" stretchy="false">‖</mo> <mi mathvariant="normal">Δ</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mi mathvariant="normal">Δ</mi> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {J}}_{n}={\boldsymbol {J}}_{n-1}+{\frac {\Delta {\boldsymbol {f}}_{n}-{\boldsymbol {J}}_{n-1}\Delta {\boldsymbol {x}}_{n}}{\|\Delta {\boldsymbol {x}}_{n}\|^{2}}}\Delta {\boldsymbol {x}}_{n}^{\top }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c67caf04c03415e43afb572c34e2f0184893a146" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:36.011ex; height:6.343ex;" alt="{\displaystyle {\boldsymbol {J}}_{n}={\boldsymbol {J}}_{n-1}+{\frac {\Delta {\boldsymbol {f}}_{n}-{\boldsymbol {J}}_{n-1}\Delta {\boldsymbol {x}}_{n}}{\|\Delta {\boldsymbol {x}}_{n}\|^{2}}}\Delta {\boldsymbol {x}}_{n}^{\top }}"></span></dd></dl> <p>これにより以下の<a href="/wiki/%E8%A1%8C%E5%88%97%E3%83%8E%E3%83%AB%E3%83%A0" title="行列ノルム">フロベニウスノルム</a>が最小化される。 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|{\boldsymbol {J}}_{n}-{\boldsymbol {J}}_{n-1}\|_{\rm {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">J</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">J</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|{\boldsymbol {J}}_{n}-{\boldsymbol {J}}_{n-1}\|_{\rm {F}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5438b84955f98606d5ee54f2c4f3b960d60b21ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.942ex; height:2.843ex;" alt="{\displaystyle \|{\boldsymbol {J}}_{n}-{\boldsymbol {J}}_{n-1}\|_{\rm {F}}}"></span></dd></dl> <p>これでNewton direction<sup class="noprint Inline-Template nowrap">[<i><a href="/wiki/Wikipedia:%E5%9F%B7%E7%AD%86%E3%83%BB%E7%BF%BB%E8%A8%B3%E8%80%85%E3%81%AE%E5%BA%83%E5%A0%B4" title="Wikipedia:執筆・翻訳者の広場"><span title="原文からの翻訳について、疑問が提出されています。（2024年9月）">訳語疑問点</span></a></i>]</sup>へ進むことができる。 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {x}}_{n+1}={\boldsymbol {x}}_{n}-{\boldsymbol {J}}_{n}^{-1}{\boldsymbol {f}}(\mathbf {x} _{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">J</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">f</mi> </mrow> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {x}}_{n+1}={\boldsymbol {x}}_{n}-{\boldsymbol {J}}_{n}^{-1}{\boldsymbol {f}}(\mathbf {x} _{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3d20fbc3e36683cba67c3a8ae344137671f25c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.419ex; height:3.176ex;" alt="{\displaystyle {\boldsymbol {x}}_{n+1}={\boldsymbol {x}}_{n}-{\boldsymbol {J}}_{n}^{-1}{\boldsymbol {f}}(\mathbf {x} _{n})}"></span></dd></dl> <p>ブロイデンは<span title="リンク先の項目はまだありません。新規の執筆や他言語版からの翻訳が望まれます。"><a href="/w/index.php?title=Sherman-Morrison%E3%81%AE%E5%85%AC%E5%BC%8F&action=edit&redlink=1" class="new" title="「Sherman-Morrisonの公式」 (存在しないページ)">Sherman-Morrisonの公式</a><span style="font-size: 0.77em; font-weight: normal;" class="noprint">（<a href="https://en.wikipedia.org/wiki/Sherman%E2%80%93Morrison_formula" class="extiw" title="en:Sherman–Morrison formula">英語版</a>）</span></span>を用いてヤコビアンの<a href="/wiki/%E9%80%86%E8%A1%8C%E5%88%97" class="mw-redirect" title="逆行列">逆行列</a>を直接更新することも提案している。 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {J}}_{n}^{-1}={\boldsymbol {J}}_{n-1}^{-1}+{\frac {\Delta {\boldsymbol {x}}_{n}-{\boldsymbol {J}}_{n-1}^{-1}\Delta {\boldsymbol {f}}_{n}}{\Delta {\boldsymbol {x}}_{n}^{\mathrm {T} }{\boldsymbol {J}}_{n-1}^{-1}\Delta {\boldsymbol {f}}_{n}}}\Delta {\boldsymbol {x}}_{n}^{\top }{\boldsymbol {J}}_{n-1}^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">J</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <mo>=</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">J</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">J</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <mi mathvariant="normal">Δ</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">f</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">Δ</mi> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mrow> </msubsup> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">J</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <mi mathvariant="normal">Δ</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">f</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mi mathvariant="normal">Δ</mi> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msubsup> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">J</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {J}}_{n}^{-1}={\boldsymbol {J}}_{n-1}^{-1}+{\frac {\Delta {\boldsymbol {x}}_{n}-{\boldsymbol {J}}_{n-1}^{-1}\Delta {\boldsymbol {f}}_{n}}{\Delta {\boldsymbol {x}}_{n}^{\mathrm {T} }{\boldsymbol {J}}_{n-1}^{-1}\Delta {\boldsymbol {f}}_{n}}}\Delta {\boldsymbol {x}}_{n}^{\top }{\boldsymbol {J}}_{n-1}^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c860f6f597b6f250a2fc89ba49f983eb868ba16e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:42.102ex; height:7.509ex;" alt="{\displaystyle {\boldsymbol {J}}_{n}^{-1}={\boldsymbol {J}}_{n-1}^{-1}+{\frac {\Delta {\boldsymbol {x}}_{n}-{\boldsymbol {J}}_{n-1}^{-1}\Delta {\boldsymbol {f}}_{n}}{\Delta {\boldsymbol {x}}_{n}^{\mathrm {T} }{\boldsymbol {J}}_{n-1}^{-1}\Delta {\boldsymbol {f}}_{n}}}\Delta {\boldsymbol {x}}_{n}^{\top }{\boldsymbol {J}}_{n-1}^{-1}}"></span></dd></dl> <p>この1つめの手法は「良いブロイデン法」とも呼ばれる。 </p><p>類似手法として、<span lang="en" class="texhtml"><i><b>J</b></i><sub><i>n</i>−1</sub></span>に若干異なる変更を加える手法も導出できる。この2つめの手法は「悪いブロイデン法」とも呼ばれる(ただし、<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>を参照)。 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {J}}_{n}^{-1}={\boldsymbol {J}}_{n-1}^{-1}+{\frac {\Delta {\boldsymbol {x}}_{n}-{\boldsymbol {J}}_{n-1}^{-1}\Delta {\boldsymbol {f}}_{n}}{\|\Delta {\boldsymbol {f}}_{n}\|^{2}}}\Delta {\boldsymbol {f}}_{n}^{\top }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">J</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <mo>=</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">J</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">J</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <mi mathvariant="normal">Δ</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">f</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mrow> <mo fence="false" stretchy="false">‖</mo> <mi mathvariant="normal">Δ</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">f</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mi mathvariant="normal">Δ</mi> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">f</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {J}}_{n}^{-1}={\boldsymbol {J}}_{n-1}^{-1}+{\frac {\Delta {\boldsymbol {x}}_{n}-{\boldsymbol {J}}_{n-1}^{-1}\Delta {\boldsymbol {f}}_{n}}{\|\Delta {\boldsymbol {f}}_{n}\|^{2}}}\Delta {\boldsymbol {f}}_{n}^{\top }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9220c6542323500550c79f5c67d9e3b48c2a4940" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:37.274ex; height:7.009ex;" alt="{\displaystyle {\boldsymbol {J}}_{n}^{-1}={\boldsymbol {J}}_{n-1}^{-1}+{\frac {\Delta {\boldsymbol {x}}_{n}-{\boldsymbol {J}}_{n-1}^{-1}\Delta {\boldsymbol {f}}_{n}}{\|\Delta {\boldsymbol {f}}_{n}\|^{2}}}\Delta {\boldsymbol {f}}_{n}^{\top }}"></span></dd></dl> <p>これは上とはことなる以下のフロベニウスノルムを最小化する。 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|{\boldsymbol {J}}_{n}^{-1}-{\boldsymbol {J}}_{n-1}^{-1}\|_{\rm {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">J</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <mo>−</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">J</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|{\boldsymbol {J}}_{n}^{-1}-{\boldsymbol {J}}_{n-1}^{-1}\|_{\rm {F}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93ba2361076e345b34169adf17dd07f9bbc911e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.246ex; height:3.343ex;" alt="{\displaystyle \|{\boldsymbol {J}}_{n}^{-1}-{\boldsymbol {J}}_{n-1}^{-1}\|_{\rm {F}}}"></span></dd></dl> <p>他にも多くの準ニュートン法が提案されており、これを用いてある関数の勾配の求根を行うことによりその関数の最大値または最小値をみつける、すなわち<a href="/wiki/%E6%95%B0%E7%90%86%E6%9C%80%E9%81%A9%E5%8C%96" title="数理最適化">最適化</a>を行うために活用されている。勾配のヤコビアンは<a href="/wiki/%E3%83%98%E3%83%83%E3%82%BB%E8%A1%8C%E5%88%97" title="ヘッセ行列">ヘッシアン</a>と呼ばれ、<a href="/wiki/%E5%AF%BE%E7%A7%B0%E8%A1%8C%E5%88%97" title="対称行列">対称行列</a>であるため更新式にさらなる制約が追加される。 </p> <div class="mw-heading mw-heading2"><h2 id="Broyden_Classの手法"><span id="Broyden_Class.E3.81.AE.E6.89.8B.E6.B3.95"></span>Broyden Classの手法</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E3%83%96%E3%83%AD%E3%82%A4%E3%83%87%E3%83%B3%E6%B3%95&action=edit&section=4" title="節を編集: Broyden Classの手法"><span>編集</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>上述の2つの手法に加え、ブロイデンは関連する手法の1群を定義した<sup id="cite_ref-Broyden_1965_1-1" class="reference"><a href="#cite_note-Broyden_1965-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="reference" style="white-space:nowrap;">:578</sup>。一般に、Broyden Class<sup class="noprint Inline-Template nowrap">[<i><a href="/wiki/Wikipedia:%E5%9F%B7%E7%AD%86%E3%83%BB%E7%BF%BB%E8%A8%B3%E8%80%85%E3%81%AE%E5%BA%83%E5%A0%B4" title="Wikipedia:執筆・翻訳者の広場"><span title="原文からの翻訳について、疑問が提出されています。（2024年9月）">訳語疑問点</span></a></i>]</sup>の手法は以下の形式で与えられる<sup id="cite_ref-Nocedal_2006_4-0" class="reference"><a href="#cite_note-Nocedal_2006-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup class="reference" style="white-space:nowrap;">:150</sup>。<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {J}}_{k+1}={\boldsymbol {J}}_{k}-{\frac {{\boldsymbol {J}}_{k}s_{k}s_{k}^{\top }{\boldsymbol {J}}_{k}}{s_{k}^{\top }{\boldsymbol {J}}_{k}s_{k}}}+{\frac {y_{k}y_{k}^{\top }}{y_{k}^{T}s_{k}}}+\phi _{k}\left(s_{k}^{\top }{\boldsymbol {J}}_{k}s_{k}\right)v_{k}v_{k}^{\top }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">J</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">J</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">J</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msubsup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msubsup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">J</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mrow> <msubsup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msubsup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">J</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msubsup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msubsup> </mrow> <mrow> <msubsup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msubsup> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>+</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msubsup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">J</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {J}}_{k+1}={\boldsymbol {J}}_{k}-{\frac {{\boldsymbol {J}}_{k}s_{k}s_{k}^{\top }{\boldsymbol {J}}_{k}}{s_{k}^{\top }{\boldsymbol {J}}_{k}s_{k}}}+{\frac {y_{k}y_{k}^{\top }}{y_{k}^{T}s_{k}}}+\phi _{k}\left(s_{k}^{\top }{\boldsymbol {J}}_{k}s_{k}\right)v_{k}v_{k}^{\top }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25af1d0df94c604cd88b21ce92ea4064ec7afbb5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:52.863ex; height:7.176ex;" alt="{\displaystyle {\boldsymbol {J}}_{k+1}={\boldsymbol {J}}_{k}-{\frac {{\boldsymbol {J}}_{k}s_{k}s_{k}^{\top }{\boldsymbol {J}}_{k}}{s_{k}^{\top }{\boldsymbol {J}}_{k}s_{k}}}+{\frac {y_{k}y_{k}^{\top }}{y_{k}^{T}s_{k}}}+\phi _{k}\left(s_{k}^{\top }{\boldsymbol {J}}_{k}s_{k}\right)v_{k}v_{k}^{\top }}"></span>ここで、<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{k}:={\boldsymbol {f}}({\boldsymbol {x}}_{k+1})-{\boldsymbol {f}}({\boldsymbol {x}}_{k})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>:=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">f</mi> </mrow> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">f</mi> </mrow> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{k}:={\boldsymbol {f}}({\boldsymbol {x}}_{k+1})-{\boldsymbol {f}}({\boldsymbol {x}}_{k})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60415062a9f5eaad91e0d8d67226b4eaeca54a31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.675ex; height:2.843ex;" alt="{\displaystyle y_{k}:={\boldsymbol {f}}({\boldsymbol {x}}_{k+1})-{\boldsymbol {f}}({\boldsymbol {x}}_{k})}"></span>および<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{k}:={\boldsymbol {x}}_{k+1}-{\boldsymbol {x}}_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>:=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{k}:={\boldsymbol {x}}_{k+1}-{\boldsymbol {x}}_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5cf1fdd4acfe646f36cf244c841964369b98db63" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.106ex; height:2.343ex;" alt="{\displaystyle s_{k}:={\boldsymbol {x}}_{k+1}-{\boldsymbol {x}}_{k}}"></span>、<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{k}=\left[{\frac {y_{k}}{y_{k}^{T}s_{k}}}-{\frac {\mathbf {J} _{k}s_{k}}{s_{k}^{T}\mathbf {J} _{k}s_{k}}}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mrow> <msubsup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msubsup> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">J</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mrow> <msubsup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msubsup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">J</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{k}=\left[{\frac {y_{k}}{y_{k}^{T}s_{k}}}-{\frac {\mathbf {J} _{k}s_{k}}{s_{k}^{T}\mathbf {J} _{k}s_{k}}}\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86ea285a0a417b332719af47748bd88cdaf630a5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:24.395ex; height:7.509ex;" alt="{\displaystyle v_{k}=\left[{\frac {y_{k}}{y_{k}^{T}s_{k}}}-{\frac {\mathbf {J} _{k}s_{k}}{s_{k}^{T}\mathbf {J} _{k}s_{k}}}\right]}"></span>であり、<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=1,2,...}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=1,2,...}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8452c9b1148ad30f1cd2747b3d21cc3c2fc87c1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.417ex; height:2.509ex;" alt="{\displaystyle k=1,2,...}"></span>に対して各<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi _{k}\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi _{k}\in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e24dd832f502d9db7a2415e7d073906a1df7f8a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.993ex; height:2.509ex;" alt="{\displaystyle \phi _{k}\in \mathbb {R} }"></span>を定めることによりその手法が決定される。 </p><p>Broyden classに分類できる手法のいくつかは他の著者により提案されている。 </p> <ul><li><a href="/wiki/DFP%E6%B3%95" title="DFP法">DFP法</a>はBroyden classに分類できる手法のうち、先述の2手法がブロイデンにより提案されるようりも前に発表されていた唯一の手法である<sup id="cite_ref-Broyden_1965_1-2" class="reference"><a href="#cite_note-Broyden_1965-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="reference" style="white-space:nowrap;">:582</sup>。DFP法は<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi _{k}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi _{k}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a847ae61879ed00c1f776caf714d010a62aaa43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.735ex; height:2.509ex;" alt="{\displaystyle \phi _{k}=1}"></span>を用いる<sup id="cite_ref-Nocedal_2006_4-1" class="reference"><a href="#cite_note-Nocedal_2006-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup class="reference" style="white-space:nowrap;">:150</sup>。</li> <li>Schubert's algorithm<sup class="noprint Inline-Template nowrap">[<i><a href="/wiki/Wikipedia:%E5%9F%B7%E7%AD%86%E3%83%BB%E7%BF%BB%E8%A8%B3%E8%80%85%E3%81%AE%E5%BA%83%E5%A0%B4" title="Wikipedia:執筆・翻訳者の広場"><span title="原文からの翻訳について、疑問が提出されています。（2024年9月）">訳語疑問点</span></a></i>]</sup>または疎ブロイデン法は<a href="/wiki/%E7%96%8E%E8%A1%8C%E5%88%97" title="疎行列">疎</a>なヤコビアン向けの修正版である<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup>。</li> <li>Klement (2014) は多方程式系の求根を少ないイテレーションで解く<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup>。</li></ul> <div class="mw-heading mw-heading2"><h2 id="関連項目"><span id=".E9.96.A2.E9.80.A3.E9.A0.85.E7.9B.AE"></span>関連項目</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E3%83%96%E3%83%AD%E3%82%A4%E3%83%87%E3%83%B3%E6%B3%95&action=edit&section=5" title="節を編集: 関連項目"><span>編集</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/%E5%89%B2%E7%B7%9A%E6%B3%95" title="割線法">割線法</a></li> <li><a href="/wiki/%E3%83%8B%E3%83%A5%E3%83%BC%E3%83%88%E3%83%B3%E6%B3%95" title="ニュートン法">ニュートン法</a></li> <li><a href="/wiki/%E6%BA%96%E3%83%8B%E3%83%A5%E3%83%BC%E3%83%88%E3%83%B3%E6%B3%95" title="準ニュートン法">準ニュートン法</a></li> <li><span title="リンク先の項目はまだありません。新規の執筆や他言語版からの翻訳が望まれます。"><a href="/w/index.php?title=%E3%83%8B%E3%83%A5%E3%83%BC%E3%83%88%E3%83%B3%E6%B3%95%E3%81%AB%E3%82%88%E3%82%8B%E6%9C%80%E9%81%A9%E5%8C%96&action=edit&redlink=1" class="new" title="「ニュートン法による最適化」 (存在しないページ)">ニュートン法による最適化</a><span style="font-size: 0.77em; font-weight: normal;" class="noprint">（<a href="https://en.wikipedia.org/wiki/Newton%27s_method_in_optimization" class="extiw" title="en:Newton's method in optimization">英語版</a>）</span></span></li> <li><a href="/wiki/DFP%E6%B3%95" title="DFP法">DFP法</a></li> <li><a href="/wiki/BFGS%E6%B3%95" title="BFGS法">BFGS法</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="出典"><span id=".E5.87.BA.E5.85.B8"></span>出典</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E3%83%96%E3%83%AD%E3%82%A4%E3%83%87%E3%83%B3%E6%B3%95&action=edit&section=6" title="節を編集: 出典"><span>編集</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="reflist" style="list-style-type: decimal;"> <ol class="references"> <li id="cite_note-Broyden_1965-1">^ <a href="#cite_ref-Broyden_1965_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Broyden_1965_1-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Broyden_1965_1-2"><sup><i><b>c</b></i></sup></a> <span class="reference-text"><cite style="font-style:normal" class="citation journal">Broyden, C. G. (October 1965). “A Class of Methods for Solving Nonlinear Simultaneous Equations”. <i>Mathematics of Computation</i> (American Mathematical Society) <b>19</b> (92): 577–593. <a href="/wiki/Doi_(%E8%AD%98%E5%88%A5%E5%AD%90)" class="mw-redirect" title="Doi (識別子)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0025-5718-1965-0198670-6">10.1090/S0025-5718-1965-0198670-6</a>. <a href="/wiki/JSTOR" title="JSTOR">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2003941">2003941</a>.</cite><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=A+Class+of+Methods+for+Solving+Nonlinear+Simultaneous+Equations&rft.jtitle=Mathematics+of+Computation&rft.aulast=Broyden&rft.aufirst=C.+G.&rft.au=Broyden%2C%26%2332%3BC.+G.&rft.date=October+1965&rft.volume=19&rft.issue=92&rft.pages=577%E2%80%93593&rft.pub=American+Mathematical+Society&rft_id=info:doi/10.1090%2FS0025-5718-1965-0198670-6&rft.jstor=2003941&rfr_id=info:sid/ja.wikipedia.org:%E3%83%96%E3%83%AD%E3%82%A4%E3%83%87%E3%83%B3%E6%B3%95"><span style="display: none;"> </span></span></span> </li> <li id="cite_note-2"><b><a href="#cite_ref-2">^</a></b> <span class="reference-text"><cite style="font-style:normal" class="citation journal">Gay, D. M. 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(1970-01-01). <a rel="nofollow" class="external text" href="https://www.ams.org/mcom/1970-24-109/S0025-5718-1970-0258276-9/">“Modification of a quasi-Newton method for nonlinear equations with a sparse Jacobian”</a>. <i>Mathematics of Computation</i> <b>24</b> (109): 27–30. <a href="/wiki/Doi_(%E8%AD%98%E5%88%A5%E5%AD%90)" class="mw-redirect" title="Doi (識別子)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0025-5718-1970-0258276-9">10.1090/S0025-5718-1970-0258276-9</a>. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r101121245"><a href="/wiki/ISSN" title="ISSN">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/ja/search?fq=x0:jrnl&q=n2:0025-5718">0025-5718</a><span style="display:none;">. <a rel="nofollow" class="external free" href="https://www.ams.org/mcom/1970-24-109/S0025-5718-1970-0258276-9/">https://www.ams.org/mcom/1970-24-109/S0025-5718-1970-0258276-9/</a></span>.</cite><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Modification+of+a+quasi-Newton+method+for+nonlinear+equations+with+a+sparse+Jacobian&rft.jtitle=Mathematics+of+Computation&rft.aulast=Schubert&rft.aufirst=L.+K.&rft.au=Schubert%2C%26%2332%3BL.+K.&rft.date=1970-01-01&rft.volume=24&rft.issue=109&rft.pages=27%E2%80%9330&rft_id=info:doi/10.1090%2FS0025-5718-1970-0258276-9&rft.issn=0025-5718&rft_id=https%3A%2F%2Fwww.ams.org%2Fmcom%2F1970-24-109%2FS0025-5718-1970-0258276-9%2F&rfr_id=info:sid/ja.wikipedia.org:%E3%83%96%E3%83%AD%E3%82%A4%E3%83%87%E3%83%B3%E6%B3%95"><span style="display: none;"> </span></span></span> </li> <li id="cite_note-6"><b><a href="#cite_ref-6">^</a></b> <span class="reference-text"><cite style="font-style:normal" class="citation journal">Klement, Jan (2014-11-23). <a rel="nofollow" class="external text" href="http://www.jatm.com.br/ojs/index.php/jatm/article/view/373">“On Using Quasi-Newton Algorithms of the Broyden Class for Model-to-Test Correlation”</a> (英語). <i>Journal of Aerospace Technology and Management</i> <b>6</b> (4): 407–414. <a href="/wiki/Doi_(%E8%AD%98%E5%88%A5%E5%AD%90)" class="mw-redirect" title="Doi (識別子)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.5028%2Fjatm.v6i4.373">10.5028/jatm.v6i4.373</a>. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r101121245"><a href="/wiki/ISSN" title="ISSN">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/ja/search?fq=x0:jrnl&q=n2:2175-9146">2175-9146</a><span style="display:none;">. <a rel="nofollow" class="external free" href="http://www.jatm.com.br/ojs/index.php/jatm/article/view/373">http://www.jatm.com.br/ojs/index.php/jatm/article/view/373</a></span>.</cite><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=On+Using+Quasi-Newton+Algorithms+of+the+Broyden+Class+for+Model-to-Test+Correlation&rft.jtitle=Journal+of+Aerospace+Technology+and+Management&rft.aulast=Klement&rft.aufirst=Jan&rft.au=Klement%2C%26%2332%3BJan&rft.date=2014-11-23&rft.volume=6&rft.issue=4&rft.pages=407%E2%80%93414&rft_id=info:doi/10.5028%2Fjatm.v6i4.373&rft.issn=2175-9146&rft_id=http%3A%2F%2Fwww.jatm.com.br%2Fojs%2Findex.php%2Fjatm%2Farticle%2Fview%2F373&rfr_id=info:sid/ja.wikipedia.org:%E3%83%96%E3%83%AD%E3%82%A4%E3%83%87%E3%83%B3%E6%B3%95"><span style="display: none;"> </span></span></span> </li> <li id="cite_note-7"><b><a href="#cite_ref-7">^</a></b> <span class="reference-text"><cite class="citation web" style="font-style:normal">“<a rel="nofollow" class="external text" href="http://www.mathworks.com/matlabcentral/fileexchange/55251-broyden-class-methods">Broyden class methods – File Exchange – MATLAB Central</a>”. <i>www.mathworks.com</i>. <span title="">2016年2月4日</span>閲覧。</cite></span> </li> </ol></div><div class="reflist" style="list-style-type: decimal;"> </div> <div class="mw-heading mw-heading2"><h2 id="関連文献"><span id=".E9.96.A2.E9.80.A3.E6.96.87.E7.8C.AE"></span>関連文献</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E3%83%96%E3%83%AD%E3%82%A4%E3%83%87%E3%83%B3%E6%B3%95&action=edit&section=7" title="節を編集: 関連文献"><span>編集</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><cite style="font-style:normal" class="citation book"><a href="/w/index.php?title=John_E._Dennis&action=edit&redlink=1" class="new" title="「John E. 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