Algoritma Gauss-Newton - Wikipedia bahasa Indonesia, ensiklopedia bebas

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href="https://id-m-wikipedia-org.translate.goog/wiki/Wikipedia:Pedoman_gaya?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Wikipedia:Pedoman gaya">standar Wikipedia</a>.<span class="hide-when-compact"> Tidak ada alasan yang diberikan. Silakan <a class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://id.wikipedia.org/w/index.php?title%3DAlgoritma_Gauss-Newton%26action%3Dedit">kembangkan artikel</a> ini semampu Anda. Merapikan artikel dapat dilakukan dengan <a href="https://id-m-wikipedia-org.translate.goog/wiki/Bantuan:Wikifikasi?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Bantuan:Wikifikasi">wikifikasi</a> atau membagi artikel ke paragraf-paragraf. Jika sudah dirapikan, silakan hapus templat ini.</span><span class="hide-when-compact"><i> (<small><a href="https://id-m-wikipedia-org.translate.goog/wiki/Bantuan:Penghapusan_templat_pemeliharaan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Bantuan:Penghapusan templat pemeliharaan">Pelajari cara dan kapan saatnya untuk menghapus pesan templat ini</a></small>)</i></span> </div></td> </tr> </tbody> </table> <p>Dalam <a href="https://id-m-wikipedia-org.translate.goog/wiki/Matematika?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Matematika">matematika</a>, <b>algoritma Gauss-Newton</b> adalah penyelesaian yang digunakan untuk memecahkan masalah-masalah kuadrat terkecil. Algoritma ini merupakan sebuah perubahan dari metode Newton untuk mengoptimalkan sebuah fungsi. Tidak seperti metode Newton, algoritma Gauss-Newton hanya bisa digunakan untuk mengoptimumkan jumlah dari nilai fungsi kuadrat. Metode ini adalah hasil penemuannya matematikawan yang bernama <a href="https://id-m-wikipedia-org.translate.goog/wiki/Carl_Friedrich_Gauss?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Carl Friedrich Gauss">Carl Friedrich Gauss</a>.</p> <div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"> <input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none"> <div class="toctitle" lang="id" dir="ltr"> <h2 id="mw-toc-heading">Daftar isi</h2><span class="toctogglespan"><label class="toctogglelabel" for="toctogglecheckbox"></label></span> </div> <ul> <li class="toclevel-1 tocsection-1"><a href="https://id-m-wikipedia-org.translate.goog/wiki/Algoritma_Gauss-Newton?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Permasalahan"><span class="tocnumber">1</span> <span class="toctext">Permasalahan</span></a></li> <li class="toclevel-1 tocsection-2"><a href="https://id-m-wikipedia-org.translate.goog/wiki/Algoritma_Gauss-Newton?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Algoritme"><span class="tocnumber">2</span> <span class="toctext">Algoritme</span></a></li> <li class="toclevel-1 tocsection-3"><a href="https://id-m-wikipedia-org.translate.goog/wiki/Algoritma_Gauss-Newton?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Penelusuran_garis"><span class="tocnumber">3</span> <span class="toctext">Penelusuran garis</span></a></li> <li class="toclevel-1 tocsection-4"><a href="https://id-m-wikipedia-org.translate.goog/wiki/Algoritma_Gauss-Newton?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Derivasi_dari_metode_Newton"><span class="tocnumber">4</span> <span class="toctext">Derivasi dari metode Newton</span></a></li> <li class="toclevel-1 tocsection-5"><a href="https://id-m-wikipedia-org.translate.goog/wiki/Algoritma_Gauss-Newton?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Algoritme_lainnya"><span class="tocnumber">5</span> <span class="toctext">Algoritme lainnya</span></a></li> <li class="toclevel-1 tocsection-6"><a href="https://id-m-wikipedia-org.translate.goog/wiki/Algoritma_Gauss-Newton?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Referensi"><span class="tocnumber">6</span> <span class="toctext">Referensi</span></a></li> </ul> </div> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(1)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="Permasalahan">Permasalahan</h2><span class="mw-editsection"> <a role="button" href="https://id-m-wikipedia-org.translate.goog/w/index.php?title=Algoritma_Gauss-Newton&action=edit&section=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Sunting bagian: Permasalahan" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>sunting</span> </a> </span> </div> <section class="mf-section-1 collapsible-block" id="mf-section-1"> <p>Diberikan <i>m</i> fungsi <i>f</i><sub>1</sub>, ..., <i>f</i><sub><i>m</i></sub> of <i>n</i> parameters <i>p</i><sub>1</sub>, ..., <i>p</i><sub><i>n</i></sub> with <i>m</i>≥<i>n</i>, kita ingin meminimumkan jumlah</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 S(p)=\sum _{i=1}^{m}(f_{i}(p))^{2}.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> S </mi> <mo stretchy="false"> ( </mo> <mi> p </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> m </mi> </mrow> </munderover> <mo stretchy="false"> ( </mo> <msub> <mi> f </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> p </mi> <mo stretchy="false"> ) </mo> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle S(p)=\sum _{i=1}^{m}(f_{i}(p))^{2}.} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67eabdabd0ac77a5cfa21726c090d1f0adea8f01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:19.359ex; height:6.843ex;" alt="{\displaystyle S(p)=\sum _{i=1}^{m}(f_{i}(p))^{2}.}"> </noscript><span class="lazy-image-placeholder" style="width: 19.359ex;height: 6.843ex;vertical-align: -3.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67eabdabd0ac77a5cfa21726c090d1f0adea8f01" data-alt="{\displaystyle S(p)=\sum _{i=1}^{m}(f_{i}(p))^{2}.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <p>Disini, <i>p</i> adalah vektor kolom (<i>p</i><sub>1</sub>, ..., <i>p</i><sub><i>n</i></sub>)<sup>T</sup>.</p> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(2)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="Algoritme">Algoritme</h2><span class="mw-editsection"> <a role="button" href="https://id-m-wikipedia-org.translate.goog/w/index.php?title=Algoritma_Gauss-Newton&action=edit&section=2&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Sunting bagian: Algoritme" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>sunting</span> </a> </span> </div> <section class="mf-section-2 collapsible-block" id="mf-section-2"> <p>Algoritme Gauss-Newton merupakan prosedur <a href="https://id-m-wikipedia-org.translate.goog/wiki/Iterasi?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Iterasi">iterasi</a>. Ini berarti bahwa pengguna harus menetapkan sebuah penduga pertama untuk parameter vektor <i>p</i>, yang mana akan kita sebut <i>p</i><sup>0</sup>.</p> <p>Berikutnya penduga <i>p</i><sup><i>k</i></sup> untuk parameter vektor yang kemudian dihasilkan oleh perulangan hubungan</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 p^{k+1}=p^{k}-{\Big (}J_{f}(p^{k})^{\top }J_{f}(p^{k}){\Big )}^{-1}J_{f}(p^{k})^{\top }f(p^{k}),}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi> p </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> <mo> = </mo> <msup> <mi> p </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msup> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em"> ( </mo> </mrow> </mrow> <msub> <mi> J </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> f </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <msup> <mi> p </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msup> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ⊤ </mi> </mrow> </msup> <msub> <mi> J </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> f </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <msup> <mi> p </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msup> <mo stretchy="false"> ) </mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em"> ) </mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> <mn> 1 </mn> </mrow> </msup> <msub> <mi> J </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> f </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <msup> <mi> p </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msup> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ⊤ </mi> </mrow> </msup> <mi> f </mi> <mo stretchy="false"> ( </mo> <msup> <mi> p </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msup> <mo stretchy="false"> ) </mo> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle p^{k+1}=p^{k}-{\Big (}J_{f}(p^{k})^{\top }J_{f}(p^{k}){\Big )}^{-1}J_{f}(p^{k})^{\top }f(p^{k}),} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bdb15c13764d94fa2d8bee498d0c38900da7298" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; margin-left: -0.089ex; width:46.25ex; height:5.176ex;" alt="{\displaystyle p^{k+1}=p^{k}-{\Big (}J_{f}(p^{k})^{\top }J_{f}(p^{k}){\Big )}^{-1}J_{f}(p^{k})^{\top }f(p^{k}),}"> </noscript><span class="lazy-image-placeholder" style="width: 46.25ex;height: 5.176ex;vertical-align: -1.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bdb15c13764d94fa2d8bee498d0c38900da7298" data-alt="{\displaystyle p^{k+1}=p^{k}-{\Big (}J_{f}(p^{k})^{\top }J_{f}(p^{k}){\Big )}^{-1}J_{f}(p^{k})^{\top }f(p^{k}),}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <p>di mana <i>f</i>=(<i>f</i><sub>1</sub>, ..., <i>f</i><sub><i>m</i></sub>)<sup>T</sup> dan <i>J</i><sub>f</sub>(<i>p</i>) menunjukkan Jacobian dari <i>f</i> saat <i>p</i>.</p> <p>Matriks invers tidak pernah dihasilakan secara eksplisit dalam praktiknya. Sebagai pengganti, kita gunakan</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 p^{k+1}=p^{k}+\delta ^{k},\,}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi> p </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> <mo> = </mo> <msup> <mi> p </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msup> <mo> + </mo> <msup> <mi> δ </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msup> <mo> , </mo> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle p^{k+1}=p^{k}+\delta ^{k},\,} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce8c9d3a90b3a2790845197cb4bfa1ccd63f65ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:15.821ex; height:3.009ex;" alt="{\displaystyle p^{k+1}=p^{k}+\delta ^{k},\,}"> </noscript><span class="lazy-image-placeholder" style="width: 15.821ex;height: 3.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce8c9d3a90b3a2790845197cb4bfa1ccd63f65ba" data-alt="{\displaystyle p^{k+1}=p^{k}+\delta ^{k},\,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <p>Dan kita hitung perbaikan δ<sup><i>k</i></sup> dengan menyelesaikan sistem linear</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 J_{f}(p^{k})^{\top }J_{f}(p^{k})\,\delta ^{k}=-J_{f}(p^{k})^{\top }f(p^{k}),}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> J </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> f </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <msup> <mi> p </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msup> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ⊤ </mi> </mrow> </msup> <msub> <mi> J </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> f </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <msup> <mi> p </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msup> <mo stretchy="false"> ) </mo> <mspace width="thinmathspace"></mspace> <msup> <mi> δ </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msup> <mo> = </mo> <mo> − </mo> <msub> <mi> J </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> f </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <msup> <mi> p </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msup> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ⊤ </mi> </mrow> </msup> <mi> f </mi> <mo stretchy="false"> ( </mo> <msup> <mi> p </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msup> <mo stretchy="false"> ) </mo> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle J_{f}(p^{k})^{\top }J_{f}(p^{k})\,\delta ^{k}=-J_{f}(p^{k})^{\top }f(p^{k}),} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c0072a24c6ee7c433b3b5348fc7e2e7eb15ab4d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:35.933ex; height:3.343ex;" alt="{\displaystyle J_{f}(p^{k})^{\top }J_{f}(p^{k})\,\delta ^{k}=-J_{f}(p^{k})^{\top }f(p^{k}),}"> </noscript><span class="lazy-image-placeholder" style="width: 35.933ex;height: 3.343ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c0072a24c6ee7c433b3b5348fc7e2e7eb15ab4d" data-alt="{\displaystyle J_{f}(p^{k})^{\top }J_{f}(p^{k})\,\delta ^{k}=-J_{f}(p^{k})^{\top }f(p^{k}),}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>. </dd> </dl> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(3)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="Penelusuran_garis">Penelusuran garis</h2><span class="mw-editsection"> <a role="button" href="https://id-m-wikipedia-org.translate.goog/w/index.php?title=Algoritma_Gauss-Newton&action=edit&section=3&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Sunting bagian: Penelusuran garis" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>sunting</span> </a> </span> </div> <section class="mf-section-3 collapsible-block" id="mf-section-3"> <p>Sebuah implementasi yang baik dari algoritme Gauss-Newton juga menggunakan algoritme penelusuran garis: sebagai pengganti dari formula sebelumnya untuk <i>p</i><sup><i>k</i>+1</sup>, kita gunakan</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 p^{k+1}=p^{k}+\alpha ^{k}\,\delta ^{k},}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi> p </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> <mo> = </mo> <msup> <mi> p </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msup> <mo> + </mo> <msup> <mi> α </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msup> <mspace width="thinmathspace"></mspace> <msup> <mi> δ </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msup> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle p^{k+1}=p^{k}+\alpha ^{k}\,\delta ^{k},} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67a885fc76f9d70915ba9ab730a4ef91ed885e20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:18.398ex; height:3.009ex;" alt="{\displaystyle p^{k+1}=p^{k}+\alpha ^{k}\,\delta ^{k},}"> </noscript><span class="lazy-image-placeholder" style="width: 18.398ex;height: 3.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67a885fc76f9d70915ba9ab730a4ef91ed885e20" data-alt="{\displaystyle p^{k+1}=p^{k}+\alpha ^{k}\,\delta ^{k},}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <p>Dimana kita berusaha memilih sebuah nilai optimal untuk bilangan α<sup><i>k</i></sup>.</p> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(4)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="Derivasi_dari_metode_Newton">Derivasi dari metode Newton</h2><span class="mw-editsection"> <a role="button" href="https://id-m-wikipedia-org.translate.goog/w/index.php?title=Algoritma_Gauss-Newton&action=edit&section=4&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Sunting bagian: Derivasi dari metode Newton" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>sunting</span> </a> </span> </div> <section class="mf-section-4 collapsible-block" id="mf-section-4"> <p>Hubungan perulangan metode Newton untuk meminimumkan sebuah fungsi <i>S</i> adalah</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 p^{k+1}=p^{k}-[H_{S}(p^{k})]^{-1}\nabla S(p^{k}),\,}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi> p </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> <mo> = </mo> <msup> <mi> p </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msup> <mo> − </mo> <mo stretchy="false"> [ </mo> <msub> <mi> H </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> S </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <msup> <mi> p </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msup> <mo stretchy="false"> ) </mo> <msup> <mo stretchy="false"> ] </mo> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> <mn> 1 </mn> </mrow> </msup> <mi mathvariant="normal"> ∇ </mi> <mi> S </mi> <mo stretchy="false"> ( </mo> <msup> <mi> p </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msup> <mo stretchy="false"> ) </mo> <mo> , </mo> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle p^{k+1}=p^{k}-[H_{S}(p^{k})]^{-1}\nabla S(p^{k}),\,} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c983173e8f06d48deb16bdc8d08be97dcf8298cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:32.099ex; height:3.176ex;" alt="{\displaystyle p^{k+1}=p^{k}-[H_{S}(p^{k})]^{-1}\nabla S(p^{k}),\,}"> </noscript><span class="lazy-image-placeholder" style="width: 32.099ex;height: 3.176ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c983173e8f06d48deb16bdc8d08be97dcf8298cd" data-alt="{\displaystyle p^{k+1}=p^{k}-[H_{S}(p^{k})]^{-1}\nabla S(p^{k}),\,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <p>di mana <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 \nabla S}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal"> ∇ </mi> <mi> S </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \nabla S} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8f146cd25bfe02a0030d9cc1ff07417905a8719" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.435ex; height:2.176ex;" alt="{\displaystyle \nabla S}"> </noscript><span class="lazy-image-placeholder" style="width: 3.435ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8f146cd25bfe02a0030d9cc1ff07417905a8719" data-alt="{\displaystyle \nabla S}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> dan <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 H_{S}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> H </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> S </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle H_{S}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99d62a9b4028878b7d67a1b7006d8db189350a20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.224ex; height:2.509ex;" alt="{\displaystyle H_{S}}"> </noscript><span class="lazy-image-placeholder" style="width: 3.224ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99d62a9b4028878b7d67a1b7006d8db189350a20" data-alt="{\displaystyle H_{S}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> berarti gradien dan Hessian dari <i>S</i> . Sekarang kita misalkan S memiliki bentuk</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 S(p)=\sum _{i=1}^{m}(f_{i}(p))^{2}=f(p)^{\top }f(p)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> S </mi> <mo stretchy="false"> ( </mo> <mi> p </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> m </mi> </mrow> </munderover> <mo stretchy="false"> ( </mo> <msub> <mi> f </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> p </mi> <mo stretchy="false"> ) </mo> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> = </mo> <mi> f </mi> <mo stretchy="false"> ( </mo> <mi> p </mi> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ⊤ </mi> </mrow> </msup> <mi> f </mi> <mo stretchy="false"> ( </mo> <mi> p </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle S(p)=\sum _{i=1}^{m}(f_{i}(p))^{2}=f(p)^{\top }f(p)} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8448dd9465fd6e746a601f9503c8b8133138103c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:31.836ex; height:6.843ex;" alt="{\displaystyle S(p)=\sum _{i=1}^{m}(f_{i}(p))^{2}=f(p)^{\top }f(p)}"> </noscript><span class="lazy-image-placeholder" style="width: 31.836ex;height: 6.843ex;vertical-align: -3.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8448dd9465fd6e746a601f9503c8b8133138103c" data-alt="{\displaystyle S(p)=\sum _{i=1}^{m}(f_{i}(p))^{2}=f(p)^{\top }f(p)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <p>di mana <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}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> f </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle f} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> </noscript><span class="lazy-image-placeholder" style="width: 1.279ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" data-alt="{\displaystyle f}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> adalah sebuah nilai fungsi vector yang merupakan komponen <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_{i}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> f </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle f_{i}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65da883ca3d16b461e46c94777b0d9c4aa010e79" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.939ex; height:2.509ex;" alt="{\displaystyle f_{i}}"> </noscript><span class="lazy-image-placeholder" style="width: 1.939ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65da883ca3d16b461e46c94777b0d9c4aa010e79" data-alt="{\displaystyle f_{i}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>.</p> <p>Dalam kasus ini, gradien diberikan oleh</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 \nabla S(p)=2J_{f}(p)^{\top }f(p)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal"> ∇ </mi> <mi> S </mi> <mo stretchy="false"> ( </mo> <mi> p </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <mn> 2 </mn> <msub> <mi> J </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> f </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> p </mi> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ⊤ </mi> </mrow> </msup> <mi> f </mi> <mo stretchy="false"> ( </mo> <mi> p </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \nabla S(p)=2J_{f}(p)^{\top }f(p)} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ea6c5d2094dec58a65a42a96c122286243c699a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.848ex; height:3.343ex;" alt="{\displaystyle \nabla S(p)=2J_{f}(p)^{\top }f(p)}"> </noscript><span class="lazy-image-placeholder" style="width: 21.848ex;height: 3.343ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ea6c5d2094dec58a65a42a96c122286243c699a" data-alt="{\displaystyle \nabla S(p)=2J_{f}(p)^{\top }f(p)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <p>di mana <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 J_{f}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> J </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> f </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle J_{f}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63d25ef7b611edb6008ece6655389f3a95082d5b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.427ex; height:2.843ex;" alt="{\displaystyle J_{f}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.427ex;height: 2.843ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63d25ef7b611edb6008ece6655389f3a95082d5b" data-alt="{\displaystyle J_{f}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> adalah Jacobian dari <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}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> f </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle f} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> </noscript><span class="lazy-image-placeholder" style="width: 1.279ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" data-alt="{\displaystyle f}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>, dan Hessian diberikan oleh</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 H_{S}(p)=2J_{f}(p)^{\top }J_{f}(p)+2\sum _{i=1}^{m}f_{i}(p)\,H_{f_{i}}(p)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> H </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> S </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> p </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <mn> 2 </mn> <msub> <mi> J </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> f </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> p </mi> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ⊤ </mi> </mrow> </msup> <msub> <mi> J </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> f </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> p </mi> <mo stretchy="false"> ) </mo> <mo> + </mo> <mn> 2 </mn> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> m </mi> </mrow> </munderover> <msub> <mi> f </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> p </mi> <mo stretchy="false"> ) </mo> <mspace width="thinmathspace"></mspace> <msub> <mi> H </mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi> f </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> p </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle H_{S}(p)=2J_{f}(p)^{\top }J_{f}(p)+2\sum _{i=1}^{m}f_{i}(p)\,H_{f_{i}}(p)} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8a9ba08469cca669ad0ee78b80e9501ee6b2782" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:42.794ex; height:6.843ex;" alt="{\displaystyle H_{S}(p)=2J_{f}(p)^{\top }J_{f}(p)+2\sum _{i=1}^{m}f_{i}(p)\,H_{f_{i}}(p)}"> </noscript><span class="lazy-image-placeholder" style="width: 42.794ex;height: 6.843ex;vertical-align: -3.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8a9ba08469cca669ad0ee78b80e9501ee6b2782" data-alt="{\displaystyle H_{S}(p)=2J_{f}(p)^{\top }J_{f}(p)+2\sum _{i=1}^{m}f_{i}(p)\,H_{f_{i}}(p)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <p>di mana <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 H_{f_{i}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> H </mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi> f </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle H_{f_{i}}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee08a2d7682e8f6b165caa54b61922705e743175" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.594ex; height:2.843ex;" alt="{\displaystyle H_{f_{i}}}"> </noscript><span class="lazy-image-placeholder" style="width: 3.594ex;height: 2.843ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee08a2d7682e8f6b165caa54b61922705e743175" data-alt="{\displaystyle H_{f_{i}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> adalah Hessian dari <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_{i}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> f </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle f_{i}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65da883ca3d16b461e46c94777b0d9c4aa010e79" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.939ex; height:2.509ex;" alt="{\displaystyle f_{i}}"> </noscript><span class="lazy-image-placeholder" style="width: 1.939ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65da883ca3d16b461e46c94777b0d9c4aa010e79" data-alt="{\displaystyle f_{i}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>.</p> <p>Catatan bahwa syarat kedua dalam ekspresi ini untuk v <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 H_{S}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> H </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> S </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle H_{S}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99d62a9b4028878b7d67a1b7006d8db189350a20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.224ex; height:2.509ex;" alt="{\displaystyle H_{S}}"> </noscript><span class="lazy-image-placeholder" style="width: 3.224ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99d62a9b4028878b7d67a1b7006d8db189350a20" data-alt="{\displaystyle H_{S}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> menuju nol sama <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(p)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> S </mi> <mo stretchy="false"> ( </mo> <mi> p </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle S(p)} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/433ae27e87adabe259edf4777bd53d84243c6192" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.478ex; height:2.843ex;" alt="{\displaystyle S(p)}"> </noscript><span class="lazy-image-placeholder" style="width: 4.478ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/433ae27e87adabe259edf4777bd53d84243c6192" data-alt="{\displaystyle S(p)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> menuju nol. Jadi jika nilai minimum dari S(p) tertutup untuk nol, dan nilai percobaan dari p adalah tertutup untuk minimum, kemudian kita bisa mengira Hessian dengan:</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 H_{S}(p)=2J_{f}(p)^{\top }J_{f}(p)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> H </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> S </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> p </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <mn> 2 </mn> <msub> <mi> J </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> f </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> p </mi> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ⊤ </mi> </mrow> </msup> <msub> <mi> J </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> f </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> p </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle H_{S}(p)=2J_{f}(p)^{\top }J_{f}(p)} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f36a94ace98491b8269fd489c4f3a32bbd6fbef4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.785ex; height:3.343ex;" alt="{\displaystyle H_{S}(p)=2J_{f}(p)^{\top }J_{f}(p)}"> </noscript><span class="lazy-image-placeholder" style="width: 22.785ex;height: 3.343ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f36a94ace98491b8269fd489c4f3a32bbd6fbef4" data-alt="{\displaystyle H_{S}(p)=2J_{f}(p)^{\top }J_{f}(p)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <p>Dengan memasukkan ekspresi ini untuk gradeien dan Hessian kedalam hubungan perulangan sebelumnya kita memiliki</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 p^{k+1}=p^{k}-\left(J_{f}(p^{k})^{\top }J_{f}(p^{k})\right)^{-1}J_{f}(p^{k})^{\top }f(p^{k}).}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi> p </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> <mo> = </mo> <msup> <mi> p </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msup> <mo> − </mo> <msup> <mrow> <mo> ( </mo> <mrow> <msub> <mi> J </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> f </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <msup> <mi> p </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msup> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ⊤ </mi> </mrow> </msup> <msub> <mi> J </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> f </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <msup> <mi> p </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msup> <mo stretchy="false"> ) </mo> </mrow> <mo> ) </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> <mn> 1 </mn> </mrow> </msup> <msub> <mi> J </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> f </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <msup> <mi> p </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msup> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ⊤ </mi> </mrow> </msup> <mi> f </mi> <mo stretchy="false"> ( </mo> <msup> <mi> p </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msup> <mo stretchy="false"> ) </mo> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle p^{k+1}=p^{k}-\left(J_{f}(p^{k})^{\top }J_{f}(p^{k})\right)^{-1}J_{f}(p^{k})^{\top }f(p^{k}).} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b469e882f6cea1b5e7e815eb4c3b8f347cbeedb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.089ex; width:45.604ex; height:3.843ex;" alt="{\displaystyle p^{k+1}=p^{k}-\left(J_{f}(p^{k})^{\top }J_{f}(p^{k})\right)^{-1}J_{f}(p^{k})^{\top }f(p^{k}).}"> </noscript><span class="lazy-image-placeholder" style="width: 45.604ex;height: 3.843ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b469e882f6cea1b5e7e815eb4c3b8f347cbeedb" data-alt="{\displaystyle p^{k+1}=p^{k}-\left(J_{f}(p^{k})^{\top }J_{f}(p^{k})\right)^{-1}J_{f}(p^{k})^{\top }f(p^{k}).}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(5)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="Algoritme_lainnya">Algoritme lainnya</h2><span class="mw-editsection"> <a role="button" href="https://id-m-wikipedia-org.translate.goog/w/index.php?title=Algoritma_Gauss-Newton&action=edit&section=5&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Sunting bagian: Algoritme lainnya" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>sunting</span> </a> </span> </div> <section class="mf-section-5 collapsible-block" id="mf-section-5"> <p>Metode lain untuk menyelesaikan masalah kuadrat terkecil hanya menggunakan derivatif pertama adalah <a href="https://id-m-wikipedia-org.translate.goog/wiki/Penurunan_gradien?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Penurunan gradien">penurunan gradien</a>. Bagaimanapun, metode ini tidak memasukkan nilai maupun perhitungan derivatif kedua dengan perkiraan yang sama. Karenanya, metode ini tidak efisien untuk fungsi-fungsi tertentu, seperti fungsi Rosenbrock.</p> <p>Pada kasus di mana minimum lebih besar dari nol, pengabaian syarat/ketentuan pada Hessian bisa jadi signifikan. Pada kasus ini, salah satunya bisa menggunakan algoritme Levenberg-Marquardt, yang merupakan kombinasi dari Gauss-Newton dan <a href="https://id-m-wikipedia-org.translate.goog/wiki/Penurunan_gradien?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Penurunan gradien">penurunan gradien</a>.</p> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(6)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="Referensi">Referensi</h2><span class="mw-editsection"> <a role="button" href="https://id-m-wikipedia-org.translate.goog/w/index.php?title=Algoritma_Gauss-Newton&action=edit&section=6&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Sunting bagian: Referensi" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>sunting</span> </a> </span> </div> <section class="mf-section-6 collapsible-block" id="mf-section-6"> <ul> <li><cite class="citation book">Nocedal, Jorge (1999). <i>Numerical optimization</i>. New York: Springer. <a href="https://id-m-wikipedia-org.translate.goog/wiki/International_Standard_Book_Number?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="International Standard Book Number">ISBN</a> <a href="https://id-m-wikipedia-org.translate.goog/wiki/Istimewa:Sumber_buku/0387987932?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Istimewa:Sumber buku/0387987932">0387987932</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Numerical+optimization&rft.pub=New+York%3A+Springer&rft.date=1999&rft.isbn=0387987932&rft.aulast=Nocedal&rft.aufirst=Jorge&rfr_id=info%3Asid%2Fid.wikipedia.org%3AAlgoritma+Gauss-Newton" class="Z3988"><span style="display:none;"> </span></span> <span style="display:none;font-size:100%" class="error citation-comment">Parameter <code style="color:inherit; border:inherit; padding:inherit;">|coauthors=</code> yang tidak diketahui mengabaikan (<code style="color:inherit; border:inherit; padding:inherit;">|author=</code> yang disarankan) (<a href="https://id-m-wikipedia-org.translate.goog/wiki/Bantuan:Galat_CS1?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#parameter_ignored_suggest" title="Bantuan:Galat CS1">bantuan</a>)</span></li> </ul> <ul> <li><cite class="citation book">Deuflhard, P. (2003). <i>Numerical analysis in modern scientific computing: an introduction</i> (edisi ke-2nd ed). New York: Springer. <a href="https://id-m-wikipedia-org.translate.goog/wiki/International_Standard_Book_Number?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="International Standard Book Number">ISBN</a> <a href="https://id-m-wikipedia-org.translate.goog/wiki/Istimewa:Sumber_buku/0387954104?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Istimewa:Sumber buku/0387954104">0387954104</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Numerical+analysis+in+modern+scientific+computing%3A+an+introduction&rft.edition=2nd+ed&rft.pub=New+York%3A+Springer&rft.date=2003&rft.isbn=0387954104&rft.aulast=Deuflhard&rft.aufirst=P.&rfr_id=info%3Asid%2Fid.wikipedia.org%3AAlgoritma+Gauss-Newton" class="Z3988"><span style="display:none;"> </span></span> <span style="display:none;font-size:100%" class="error citation-comment">Parameter <code style="color:inherit; border:inherit; padding:inherit;">|coauthors=</code> yang 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