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Micron Document

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<title>QZ-Algorithmus</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading">QZ-Algorithmus</h1>

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr">Der QZ-Algorithmus oder die QZ-Faktorisierung ist ein <a href="Numerische_Mathematik" title="Numerische Mathematik">numerisches</a> Verfahren zur Lösung des <a href="Verallgemeinertes_Eigenwertproblem" title="Verallgemeinertes Eigenwertproblem">verallgemeinerten Eigenwertproblems</a>.

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<annotation encoding="application/x-tex">{\displaystyle Ax=\lambda Bx^{\,}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/82d913cde0c85448c9d452f67c831c7a3ccd1ee8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.24ex; height:2.176ex;" alt="{\displaystyle Ax=\lambda Bx^{\,}}" loading="lazy"> , mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A,B\in \mathbb {R} ^{n\times n}}">

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<annotation encoding="application/x-tex">{\displaystyle A,B\in \mathbb {R} ^{n\times n}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/97284a00505184c117fc30195f79a0f87947bdef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.543ex; height:2.676ex;" alt="{\displaystyle A,B\in \mathbb {R} ^{n\times n}}" loading="lazy"> bzw. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A,B\in \mathbb {C} ^{n\times n}}">

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<annotation encoding="application/x-tex">{\displaystyle A,B\in \mathbb {C} ^{n\times n}}</annotation>

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Das <a href="Verallgemeinertes_Eigenwertproblem" title="Verallgemeinertes Eigenwertproblem">verallgemeinerte Eigenwertproblem</a> ist äquivalent zum Eigenwertproblem <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle AB^{-1}y=\lambda y}">

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<annotation encoding="application/x-tex">{\displaystyle AB^{-1}y=\lambda y}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d22c5f91a3d6ef09e57295f2d090a2975f049d9e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.605ex; height:3.009ex;" alt="{\displaystyle AB^{-1}y=\lambda y}" loading="lazy">, wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=Bx}">

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<annotation encoding="application/x-tex">{\displaystyle y=Bx}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fbf464a0667ca7e20975ca13a1832d1b60326cea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.348ex; height:2.509ex;" alt="{\displaystyle y=Bx}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

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<annotation encoding="application/x-tex">{\displaystyle B}</annotation>

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Es wird jedoch nicht explizit die Matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B^{-1}}">

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<annotation encoding="application/x-tex">{\displaystyle B^{-1}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2bef55d3c9b9256fe8540b097950ea123f5684a3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.097ex; height:2.676ex;" alt="{\displaystyle B^{-1}}" loading="lazy"> berechnet, um die <a href="Kondition_(Mathematik)" title="Kondition (Mathematik)">Kondition</a> des Problems nicht zu verschlechtern, sondern <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle B}</annotation>

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werden simultan durch <a href="%C3%84hnlichkeitstransformation" class="mw-redirect" title="Ähnlichkeitstransformation">Ähnlichkeitstransformationen</a> (<a href="Givens-Rotation" title="Givens-Rotation">Givens-Rotationen</a> und <a href="Householder-Spiegelung" class="mw-redirect" title="Householder-Spiegelung">Householder-Spiegelungen</a>) in verallgemeinerte <a href="Schurform" class="mw-redirect" title="Schurform">Schurform</a> gebracht.

Gegeben ist ein Matrixbüschel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A-\lambda B}">

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<annotation encoding="application/x-tex">{\displaystyle A-\lambda B}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9dff74047a9d0404122c79b5c0cfc9695f1e34dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.703ex; height:2.343ex;" alt="{\displaystyle A-\lambda B}" loading="lazy">.

Gesucht sind orthogonale Matrizen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q}">

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<annotation encoding="application/x-tex">{\displaystyle Q}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8752c7023b4b3286800fe3238271bbca681219ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z}">

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<annotation encoding="application/x-tex">{\displaystyle Z}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1cc6b75e09a8aa3f04d8584b11db534f88fb56bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="{\displaystyle Z}" loading="lazy">, so dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q^{T}(A-\lambda B)Z=T-\lambda S}">

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<annotation encoding="application/x-tex">{\displaystyle Q^{T}(A-\lambda B)Z=T-\lambda S}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f1ac8bd2a4d06b034312e0ef4d53960873098f45.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.85ex; height:3.176ex;" alt="{\displaystyle Q^{T}(A-\lambda B)Z=T-\lambda S}" loading="lazy"> von verallgemeinerter <a href="Schurform" class="mw-redirect" title="Schurform">Schurform</a> ist, d. h.

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<annotation encoding="application/x-tex">{\displaystyle T}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy"> ist von quasi-oberer <a href="Lineares_Gleichungssystem#Dreiecksform" title="Lineares Gleichungssystem">Dreiecksform</a> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

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<annotation encoding="application/x-tex">{\displaystyle S}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy"> ist von oberer Dreiecksform. Im Fall <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A,B\in \mathbb {C} ^{n\times n}}">

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<annotation encoding="application/x-tex">{\displaystyle A,B\in \mathbb {C} ^{n\times n}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5fab8e400f8f9009373e9b564a9d5b716b72a797.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.543ex; height:2.676ex;" alt="{\displaystyle A,B\in \mathbb {C} ^{n\times n}}" loading="lazy"> ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

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<annotation encoding="application/x-tex">{\displaystyle T}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy"> stets von oberer Dreiecksform. Aus der verallgemeinerten Schurform lassen sich dann die <a href="Eigenwert" class="mw-redirect" title="Eigenwert">Eigenwerte</a> und aus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q}">

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<annotation encoding="application/x-tex">{\displaystyle Q}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle Z}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1cc6b75e09a8aa3f04d8584b11db534f88fb56bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="{\displaystyle Z}" loading="lazy"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A,B)}">

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<annotation encoding="application/x-tex">{\displaystyle (A,B)}</annotation>

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Unterräume des Matrixbüschels <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A-\lambda B}">

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<annotation encoding="application/x-tex">{\displaystyle A-\lambda B}</annotation>

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<div class="mw-heading mw-heading2"><h2 id="Vortransformation">Vortransformation</h2></div>

Ziel dieses Schrittes ist es, die Matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> durch orthogonale Transformationen auf obere Hessenbergform und die Matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

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<annotation encoding="application/x-tex">{\displaystyle B}</annotation>

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Durch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n-1}">

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<annotation encoding="application/x-tex">{\displaystyle n-1}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fbd0b0f32b28f51962943ee9ede4fb34198a2521.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\displaystyle n-1}" loading="lazy"> Householder-Spiegelungen von links wird <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

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<annotation encoding="application/x-tex">{\displaystyle B}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"> auf obere Dreiecksform transformiert. Wendet man die gleichen Transformationen gleichzeitig auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle A={\begin{pmatrix}*&*&*&*\\{*}&*&*&*\\{*}&*&*&*\\{*}&*&*&*\end{pmatrix}},B={\begin{pmatrix}*&*&*&*\\0&*&*&*\\0&0&*&*\\0&0&0&*\end{pmatrix}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4e9824a86b632bdda0f2e3d216b5c20fe4e8973c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:43.61ex; height:12.509ex;" alt="{\displaystyle A={\begin{pmatrix}*&*&*&*\\{*}&*&*&*\\{*}&*&*&*\\{*}&*&*&*\end{pmatrix}},B={\begin{pmatrix}*&*&*&*\\0&*&*&*\\0&0&*&*\\0&0&0&*\end{pmatrix}}}" loading="lazy">.

Man finde nun eine Givens-Rotation, die von links angewendet auf A folgende Matrix ergibt:

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<annotation encoding="application/x-tex">{\displaystyle A={\begin{pmatrix}*&*&*&*\\{*}&*&*&*\\{*}&*&*&*\\0&*&*&*\end{pmatrix}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ebf83632392bee4f3c37dfcaf91d04dd95c2a505.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:21.277ex; height:12.509ex;" alt="{\displaystyle A={\begin{pmatrix}*&*&*&*\\{*}&*&*&*\\{*}&*&*&*\\0&*&*&*\end{pmatrix}}}" loading="lazy">.

Damit erhält man für

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<annotation encoding="application/x-tex">{\displaystyle B={\begin{pmatrix}*&*&*&*\\0&*&*&*\\0&0&*&*\\0&0&*&*\end{pmatrix}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7befb34ad4a47ba3aa5398dd40c0a49842bc1269.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:21.298ex; height:12.509ex;" alt="{\displaystyle B={\begin{pmatrix}*&*&*&*\\0&*&*&*\\0&0&*&*\\0&0&*&*\end{pmatrix}}}" loading="lazy">.

Durch Anwendung einer Givens-Rotation von rechts kann die obere Dreiecksform von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

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<annotation encoding="application/x-tex">{\displaystyle B}</annotation>

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werden, ohne die Null an der linken unteren Position von A zu zerstören:

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<annotation encoding="application/x-tex">{\displaystyle A={\begin{pmatrix}*&*&*&*\\{*}&*&*&*\\{*}&*&*&*\\0&*&*&*\end{pmatrix}},B={\begin{pmatrix}*&*&*&*\\0&*&*&*\\0&0&*&*\\0&0&0&*\end{pmatrix}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/da72d73b076f8a79b6f0b1ae56c4adbf6ec3ad94.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:43.61ex; height:12.509ex;" alt="{\displaystyle A={\begin{pmatrix}*&*&*&*\\{*}&*&*&*\\{*}&*&*&*\\0&*&*&*\end{pmatrix}},B={\begin{pmatrix}*&*&*&*\\0&*&*&*\\0&0&*&*\\0&0&0&*\end{pmatrix}}}" loading="lazy">.

Durch analoges spaltenweises Erzeugen von Nullen in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle A={\begin{pmatrix}*&*&*&*\\{*}&*&*&*\\0&*&*&*\\0&*&*&*\end{pmatrix}},B={\begin{pmatrix}*&*&*&*\\0&*&*&*\\0&*&*&*\\0&0&0&*\end{pmatrix}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/facd84a68950597c577b1f1e2c7dc04fcc0f8398.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:43.61ex; height:12.509ex;" alt="{\displaystyle A={\begin{pmatrix}*&*&*&*\\{*}&*&*&*\\0&*&*&*\\0&*&*&*\end{pmatrix}},B={\begin{pmatrix}*&*&*&*\\0&*&*&*\\0&*&*&*\\0&0&0&*\end{pmatrix}}}" loading="lazy"></li>

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</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/154b0443a9dc6f6ede4137e02bb4e3ec337a1d44.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:43.61ex; height:12.509ex;" alt="{\displaystyle A={\begin{pmatrix}*&*&*&*\\{*}&*&*&*\\0&*&*&*\\0&*&*&*\end{pmatrix}},B={\begin{pmatrix}*&*&*&*\\0&*&*&*\\0&0&*&*\\0&0&0&*\end{pmatrix}}}" loading="lazy"></li>

<mrow>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A={\begin{pmatrix}*&*&*&*\\{*}&*&*&*\\0&*&*&*\\0&0&*&*\end{pmatrix}},B={\begin{pmatrix}*&*&*&*\\0&*&*&*\\0&0&*&*\\0&0&*&*\end{pmatrix}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8ba15225c80b95ec88cff74361c2226624f93b4f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:43.61ex; height:12.509ex;" alt="{\displaystyle A={\begin{pmatrix}*&*&*&*\\{*}&*&*&*\\0&*&*&*\\0&0&*&*\end{pmatrix}},B={\begin{pmatrix}*&*&*&*\\0&*&*&*\\0&0&*&*\\0&0&*&*\end{pmatrix}}}" loading="lazy"></li>

<mrow>

<mtr>

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</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a8aa58700fd5be117295cf6337a3f33723c03b01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:43.61ex; height:12.509ex;" alt="{\displaystyle A={\begin{pmatrix}*&*&*&*\\{*}&*&*&*\\0&*&*&*\\0&0&*&*\end{pmatrix}},B={\begin{pmatrix}*&*&*&*\\0&*&*&*\\0&0&*&*\\0&0&0&*\end{pmatrix}}}" loading="lazy">.</li></ol>

Falls <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A,B)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (A,B)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ce67314185650d6f0deba39db7dcec9378f4d4d1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.35ex; height:2.843ex;" alt="{\displaystyle (A,B)}" loading="lazy">-invariante Unterräume berechnet werden sollen, so ist es notwendig, das <a href="Matrizenprodukt" class="mw-redirect" title="Matrizenprodukt">Produkt</a> der Transformationsmatrizen, die jeweils von links auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"> angewendet werden, in einer Matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Q}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8752c7023b4b3286800fe3238271bbca681219ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}" loading="lazy"> und das Produkt der Transformationsmatrizen, die von rechts angewendet werden, in einer Matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Z}</annotation>

</semantics>

<div class="mw-heading mw-heading2"><h2 id="QZ-Algorithmus_mit_impliziten_Shifts">QZ-Algorithmus mit impliziten Shifts</h2></div>

1. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q:=0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q:=0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f145519e2d0e911ca021766c92ccf38e6c78988f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.977ex; height:2.509ex;" alt="{\displaystyle q:=0}" loading="lazy">

2. while <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q<n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q<n}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7cafb59f70fb60999897da7e68802a2c123f5565.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.563ex; height:2.176ex;" alt="{\displaystyle q<n}" loading="lazy"> do

3. Bestimme alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j\in \{1,\cdots ,n-1\}}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle j\in \{1,\cdots ,n-1\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/40f139b87a47536f4c55d341ac1e02f7d0ac48e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.027ex; width:17.889ex; height:2.843ex;" alt="{\displaystyle j\in \{1,\cdots ,n-1\}}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |a_{j+1,j}|\leq \varepsilon (|a_{j,j}|+|a_{j+1,j+1}|)}">

</mrow>

<msub>

</mrow>

</msub>

</mrow>

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<msub>

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<annotation encoding="application/x-tex">{\displaystyle |a_{j+1,j}|\leq \varepsilon (|a_{j,j}|+|a_{j+1,j+1}|)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/58dbc8c75d572e0fc026b2f355542b6b447aa5f5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:28.837ex; height:3.009ex;" alt="{\displaystyle |a_{j+1,j}|\leq \varepsilon (|a_{j,j}|+|a_{j+1,j+1}|)}" loading="lazy">. Für diese <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle j}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2f461e54f5c093e92a55547b9764291390f0b5d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}" loading="lazy"> setze <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{j,j+1}=0}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a_{j,j+1}=0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8a9a9cc6dfc017112bacfb98b7e40a7b759e3ee5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.636ex; height:2.843ex;" alt="{\displaystyle a_{j,j+1}=0}" loading="lazy">.

4. Deflation: Finde minimales <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> und maximales

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p,q\in \{1,\cdots ,n\}}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p,q\in \{1,\cdots ,n\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ebc9c78150805ee94ce6eb9062ccefc596123520.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:16.263ex; height:2.843ex;" alt="{\displaystyle p,q\in \{1,\cdots ,n\}}" loading="lazy"> und definiere

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m:=n-p-q}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3448beb6619cc16e9da63f643d3f5155cfcbc3d9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.1ex; height:2.343ex;" alt="{\displaystyle m:=n-p-q}" loading="lazy">, so dass gilt:

<mrow>

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<mtd>

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<mtr>

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<mtd>

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</mrow>

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<mtr>

<mtd>

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<mtd>

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<mtd>

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</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A={\begin{pmatrix}A_{11}&A_{12}&A_{13}\\0&A_{22}&A_{23}\\0&0&A_{33}\end{pmatrix}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2840eae31fe8bd68868b1cd90a70f7eac0605c63.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:25.163ex; height:9.176ex;" alt="{\displaystyle A={\begin{pmatrix}A_{11}&A_{12}&A_{13}\\0&A_{22}&A_{23}\\0&0&A_{33}\end{pmatrix}}}" loading="lazy">, wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{11}\in \mathbb {R} ^{p\times p},A_{22}\in \mathbb {R} ^{m\times m},A_{33}\in \mathbb {R} ^{q\times q}}">

<msub>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

</msup>

<msub>

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</msub>

<msup>

</mrow>

</mrow>

</msup>

<msub>

</mrow>

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<msup>

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</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A_{11}\in \mathbb {R} ^{p\times p},A_{22}\in \mathbb {R} ^{m\times m},A_{33}\in \mathbb {R} ^{q\times q}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0ea3d89c251cfa67e519528cabd76e4fa7080f26.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:37.066ex; height:2.676ex;" alt="{\displaystyle A_{11}\in \mathbb {R} ^{p\times p},A_{22}\in \mathbb {R} ^{m\times m},A_{33}\in \mathbb {R} ^{q\times q}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{11}}">

<msub>

</mrow>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A_{11}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6c31b303d2cc87b9e65cebced16796c49e523565.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.619ex; height:2.509ex;" alt="{\displaystyle A_{11}}" loading="lazy"> von oberer <a href="Hessenbergform" class="mw-redirect" title="Hessenbergform">Hessenbergform</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{22}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A_{22}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fcfa7914515f7c23e481f3f4602c8df3d791384a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.619ex; height:2.509ex;" alt="{\displaystyle A_{22}}" loading="lazy"> von unreduzierter oberer Hessenbergform und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{33}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A_{33}}</annotation>

</semantics>

5. Partitioniere <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"> wie <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy">, d. h. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B={\begin{pmatrix}B_{11}&B_{12}&B_{13}\\0&B_{22}&B_{23}\\0&0&B_{33}\end{pmatrix}}}">

<mrow>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

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<mtr>

<mtd>

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<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B={\begin{pmatrix}B_{11}&B_{12}&B_{13}\\0&B_{22}&B_{23}\\0&0&B_{33}\end{pmatrix}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/72e2c7ab365d13265cf996ac1f22ad88d9bacbc9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:25.247ex; height:9.176ex;" alt="{\displaystyle B={\begin{pmatrix}B_{11}&B_{12}&B_{13}\\0&B_{22}&B_{23}\\0&0&B_{33}\end{pmatrix}}}" loading="lazy">, wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{11}\in \mathbb {R} ^{p\times p},B_{22}\in \mathbb {R} ^{m\times m},B_{33}\in \mathbb {R} ^{q\times q}}">

<msub>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B_{11}\in \mathbb {R} ^{p\times p},B_{22}\in \mathbb {R} ^{m\times m},B_{33}\in \mathbb {R} ^{q\times q}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a88c8372b63927443b720231a107afa5b66bc643.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:37.129ex; height:2.676ex;" alt="{\displaystyle B_{11}\in \mathbb {R} ^{p\times p},B_{22}\in \mathbb {R} ^{m\times m},B_{33}\in \mathbb {R} ^{q\times q}}" loading="lazy"> obere Dreiecksmatrizen sind.

6. Bringe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{33}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A_{33}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6d2a0dc8fe9b015a527e23f9f805efe27bca703f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.619ex; height:2.509ex;" alt="{\displaystyle A_{33}}" loading="lazy"> in obere <a href="Schurform" class="mw-redirect" title="Schurform">Schurform</a>: Finde orthogonale <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{33},Z_{33}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Q_{33},Z_{33}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ee327783051fff9a50283fae917bd41ce967f77f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.212ex; height:2.509ex;" alt="{\displaystyle Q_{33},Z_{33}}" loading="lazy"> so, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{33}:=Q_{33}^{T}A_{33}Z_{33}}">

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A_{33}:=Q_{33}^{T}A_{33}Z_{33}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0795dc4edd8185d3410909bbbd8542935e75e187.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.162ex; height:3.176ex;" alt="{\displaystyle A_{33}:=Q_{33}^{T}A_{33}Z_{33}}" loading="lazy"> in Schurform und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{33}:=Q_{33}^{T}B_{33}Z_{33}}">

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B_{33}:=Q_{33}^{T}B_{33}Z_{33}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fc7fadad1accf97a15eb329d631dc222d4be240e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.204ex; height:3.176ex;" alt="{\displaystyle B_{33}:=Q_{33}^{T}B_{33}Z_{33}}" loading="lazy"> obere Dreiecksmatrix ist.

Falls erforderlich: Aufdatieren von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Q}</annotation>

</semantics>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Z}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1cc6b75e09a8aa3f04d8584b11db534f88fb56bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="{\displaystyle Z}" loading="lazy">: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q:=Q\mathrm {diag} (I_{p},I_{m},Q_{33})}">

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Q:=Q\mathrm {diag} (I_{p},I_{m},Q_{33})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2da682f55568d517ebec2611ec3bd11fa6622dbc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.058ex; height:3.009ex;" alt="{\displaystyle Q:=Q\mathrm {diag} (I_{p},I_{m},Q_{33})}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z:=Z\mathrm {diag} (I_{p},I_{m},Z_{33})}">

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Z:=Z\mathrm {diag} (I_{p},I_{m},Z_{33})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/069061ceb66460270aeb0d396e23217457510b98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.492ex; height:3.009ex;" alt="{\displaystyle Z:=Z\mathrm {diag} (I_{p},I_{m},Z_{33})}" loading="lazy">.

7. if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q<n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q<n}</annotation>

</semantics>

if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle det(B_{22})=0}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle det(B_{22})=0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fa84355d1f72f9db0c81e81c4474aa0b58f9b231.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.849ex; height:2.843ex;" alt="{\displaystyle det(B_{22})=0}" loading="lazy">

Transformiere mithilfe einer Givens-Rotation von rechts <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{n-q,n-q-1}=0}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a_{n-q,n-q-1}=0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1cf62b9d7fddd22750c2071143c3fcf2b098dc30.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.323ex; height:2.843ex;" alt="{\displaystyle a_{n-q,n-q-1}=0}" loading="lazy">, um die Rang-Defizienz von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{22}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B_{22}}</annotation>

</semantics>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B_{33}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1560a8f16fb4f56dff0a77a6cb77941905666335.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.64ex; height:2.509ex;" alt="{\displaystyle B_{33}}" loading="lazy"> zu verschieben. Durch die Annullierung von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{n-q,n-q-1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a_{n-q,n-q-1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8df341ff48ef8287d90706d929f0b8436cf86872.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.062ex; height:2.343ex;" alt="{\displaystyle a_{n-q,n-q-1}}" loading="lazy"> ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{22}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A_{22}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fcfa7914515f7c23e481f3f4602c8df3d791384a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.619ex; height:2.509ex;" alt="{\displaystyle A_{22}}" loading="lazy"> keine unreduzierte Hessenbergmatrix mehr, somit wird <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q}</annotation>

</semantics>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B_{22}}</annotation>

</semantics>

else

Führe einen impliziten QZ-Schritt für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{22},B_{22}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A_{22},B_{22}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2a28c0de84009f8ab4dfcb00eeb93517f79bcdf3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.293ex; height:2.509ex;" alt="{\displaystyle A_{22},B_{22}}" loading="lazy"> aus:

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A_{22}:=Q_{22}^{T}A_{22}Z_{22},\quad {B_{22}}:=Q_{22}^{T}{B_{22}}Z_{22}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1fcacc9a05f7bde89d08a460d4479655e9f7220e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:39.723ex; height:3.176ex;" alt="{\displaystyle A_{22}:=Q_{22}^{T}A_{22}Z_{22},\quad {B_{22}}:=Q_{22}^{T}{B_{22}}Z_{22}}" loading="lazy">.

end if

8. end if

<div class="mw-heading mw-heading2"><h2 id="Wahl_der_Shifts">Wahl der Shifts</h2></div>

9. Bestimme Shifts <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a,b}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/181523deba732fda302fd176275a0739121d3bc8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.261ex; height:2.509ex;" alt="{\displaystyle a,b}" loading="lazy"> als <a href="Eigenwert" class="mw-redirect" title="Eigenwert">Eigenwerte</a> von

<mrow>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

<msup>

<mrow>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}a_{m-1,m-1}&a_{m-1,m}\\a_{m,m-1}&a_{m,m}\end{pmatrix}}{\begin{pmatrix}b_{m-1,m-1}&b_{m-1,m}\\0&b_{m,m}\end{pmatrix}}^{-1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/97925e5e9bcd900f9a5e7c29de2b60a37ccd757a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:46.682ex; height:7.009ex;" alt="{\displaystyle {\begin{pmatrix}a_{m-1,m-1}&a_{m-1,m}\\a_{m,m-1}&a_{m,m}\end{pmatrix}}{\begin{pmatrix}b_{m-1,m-1}&b_{m-1,m}\\0&b_{m,m}\end{pmatrix}}^{-1}}" loading="lazy">

10. Bestimme <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A_{22}B_{22}^{-1}-aI)(A_{22}B_{22}^{-1}-bI)e_{1}={\begin{pmatrix}\alpha \\\beta \\\gamma \\0\\\vdots \\0\end{pmatrix}}}">

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

<mrow>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (A_{22}B_{22}^{-1}-aI)(A_{22}B_{22}^{-1}-bI)e_{1}={\begin{pmatrix}\alpha \\\beta \\\gamma \\0\\\vdots \\0\end{pmatrix}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/de57994dec960252c9f30ce660b3a4c0922bd60c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.505ex; width:40.845ex; height:20.176ex;" alt="{\displaystyle (A_{22}B_{22}^{-1}-aI)(A_{22}B_{22}^{-1}-bI)e_{1}={\begin{pmatrix}\alpha \\\beta \\\gamma \\0\\\vdots \\0\end{pmatrix}}}" loading="lazy">

<div class="mw-heading mw-heading2"><h2 id="Der_implizite_QZ-Schritt">Der implizite QZ-Schritt</h2></div>

11. Finde orthogonales <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Q_{1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a8ea6463cb36d8278ff71214fb4d13127039ae53.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.893ex; height:2.509ex;" alt="{\displaystyle Q_{1}}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{1}^{T}{\begin{pmatrix}\alpha \\\beta \\\gamma \end{pmatrix}}={\begin{pmatrix}*\\0\\0\end{pmatrix}}}">

</mrow>

</mrow>

</msubsup>

<mrow>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

<mrow>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Q_{1}^{T}{\begin{pmatrix}\alpha \\\beta \\\gamma \end{pmatrix}}={\begin{pmatrix}*\\0\\0\end{pmatrix}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/543198b9f032184b0e3d364c99bd84fdef67e99d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:18.613ex; height:9.509ex;" alt="{\displaystyle Q_{1}^{T}{\begin{pmatrix}\alpha \\\beta \\\gamma \end{pmatrix}}={\begin{pmatrix}*\\0\\0\end{pmatrix}}}" loading="lazy">

Für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{22}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B_{22}}</annotation>

</semantics>

<mrow>

<mtr>

<mtd>

</mrow>

</mrow>

</msubsup>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

<msub>

</mrow>

</msub>

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mrow>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mrow>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}Q_{1}^{T}&0\\0&I_{m-3}\end{pmatrix}}B_{22}={\begin{pmatrix}*&*&*&\cdots &\cdots &*\\{*}&*&*&\cdots &\cdots &*\\{*}&*&*&\cdots &\cdots &*\\0&0&0&\ddots &&\vdots \\\vdots &\vdots &\vdots &&\ddots &\vdots \\0&0&0&\cdots &\cdots &*\end{pmatrix}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e54d0699088bcc0aac3b98c50c16fc59118d1c3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.671ex; width:48.299ex; height:22.509ex;" alt="{\displaystyle {\begin{pmatrix}Q_{1}^{T}&0\\0&I_{m-3}\end{pmatrix}}B_{22}={\begin{pmatrix}*&*&*&\cdots &\cdots &*\\{*}&*&*&\cdots &\cdots &*\\{*}&*&*&\cdots &\cdots &*\\0&0&0&\ddots &&\vdots \\\vdots &\vdots &\vdots &&\ddots &\vdots \\0&0&0&\cdots &\cdots &*\end{pmatrix}}}" loading="lazy">.

Ziel ist es nun, die Dreiecksgestalt von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{22}}">

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12. Finde orthogonales <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z_{1}\in \mathbb {R} ^{3\times 3}}">

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<annotation encoding="application/x-tex">{\displaystyle Z_{1}\in \mathbb {R} ^{3\times 3}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/44c1239bdb346ef85ae9d78609ff8ade601ba116.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.315ex; height:3.009ex;" alt="{\displaystyle Z_{1}\in \mathbb {R} ^{3\times 3}}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{22}\mathrm {diag} (Z_{1},I_{m-3})={\begin{pmatrix}*&*&*&\cdots &\cdots &*\\{*}&*&*&\cdots &\cdots &*\\0&0&*&\cdots &\cdots &*\\0&0&0&\ddots &&\vdots \\\vdots &\vdots &\vdots &&\ddots &\vdots \\0&0&0&\cdots &\cdots &*\end{pmatrix}}}">

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<annotation encoding="application/x-tex">{\displaystyle B_{22}\mathrm {diag} (Z_{1},I_{m-3})={\begin{pmatrix}*&*&*&\cdots &\cdots &*\\{*}&*&*&\cdots &\cdots &*\\0&0&*&\cdots &\cdots &*\\0&0&0&\ddots &&\vdots \\\vdots &\vdots &\vdots &&\ddots &\vdots \\0&0&0&\cdots &\cdots &*\end{pmatrix}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ce437cc2e318c374597931692c31bdd2662c4890.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.671ex; width:48.325ex; height:22.509ex;" alt="{\displaystyle B_{22}\mathrm {diag} (Z_{1},I_{m-3})={\begin{pmatrix}*&*&*&\cdots &\cdots &*\\{*}&*&*&\cdots &\cdots &*\\0&0&*&\cdots &\cdots &*\\0&0&0&\ddots &&\vdots \\\vdots &\vdots &\vdots &&\ddots &\vdots \\0&0&0&\cdots &\cdots &*\end{pmatrix}}}" loading="lazy">. Finde dann orthogonales <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z_{1}'\in \mathbb {R} ^{2\times 2}}">

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<annotation encoding="application/x-tex">{\displaystyle Z_{1}'\in \mathbb {R} ^{2\times 2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/096822d42547ff174dcbdd3d1404cae0db99e2da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.315ex; height:3.343ex;" alt="{\displaystyle Z_{1}'\in \mathbb {R} ^{2\times 2}}" loading="lazy">, so dass

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<annotation encoding="application/x-tex">{\displaystyle B_{22}\mathrm {diag} (Z_{1},I_{m-3})\mathrm {diag} (Z_{1}',I_{m-2})={\begin{pmatrix}*&*&*&\cdots &\cdots &*\\0&*&*&\cdots &\cdots &*\\0&0&*&\cdots &\cdots &*\\0&0&0&\ddots &&\vdots \\\vdots &\vdots &\vdots &&\ddots &\vdots \\0&0&0&\cdots &\cdots &*\end{pmatrix}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/873dd5d87d266bfe2db74e6da9498613e8986ef9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.671ex; width:62.873ex; height:22.509ex;" alt="{\displaystyle B_{22}\mathrm {diag} (Z_{1},I_{m-3})\mathrm {diag} (Z_{1}',I_{m-2})={\begin{pmatrix}*&*&*&\cdots &\cdots &*\\0&*&*&\cdots &\cdots &*\\0&0&*&\cdots &\cdots &*\\0&0&0&\ddots &&\vdots \\\vdots &\vdots &\vdots &&\ddots &\vdots \\0&0&0&\cdots &\cdots &*\end{pmatrix}}}" loading="lazy">.

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<annotation encoding="application/x-tex">{\displaystyle {\tilde {A}}_{22}:={A_{22}}\mathrm {diag} (Z_{1},I_{m-3})\mathrm {diag} (Z'_{1},I_{m-2})={\begin{pmatrix}*&*&*&\cdots &\cdots &\cdots &*\\{*}&*&*&\cdots &\cdots &\cdots &*\\{*}&*&*&\cdots &\cdots &\cdots &*\\{*}&*&*&\cdots &\cdots &\cdots &*\\0&0&0&\ddots &&&\vdots \\\vdots &\vdots &\vdots &&\ddots &&\vdots \\0&0&0&\cdots &0&*&*\end{pmatrix}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ff957e009edc928749fa6d8589559ca0942403f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -12.338ex; width:75.295ex; height:25.843ex;" alt="{\displaystyle {\tilde {A}}_{22}:={A_{22}}\mathrm {diag} (Z_{1},I_{m-3})\mathrm {diag} (Z'_{1},I_{m-2})={\begin{pmatrix}*&*&*&\cdots &\cdots &\cdots &*\\{*}&*&*&\cdots &\cdots &\cdots &*\\{*}&*&*&\cdots &\cdots &\cdots &*\\{*}&*&*&\cdots &\cdots &\cdots &*\\0&0&0&\ddots &&&\vdots \\\vdots &\vdots &\vdots &&\ddots &&\vdots \\0&0&0&\cdots &0&*&*\end{pmatrix}}}" loading="lazy">. D.h., die Hessenbergstruktur von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{22}}">

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13. Dieser Buckel kann durch elementäre, orthogonale Transformationen (z. B. Householder-Spiegelungen) von links eliminiert werden. Finde also orthogonales <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q''_{1}\in \mathbb {R} ^{3\times 3}}">

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<annotation encoding="application/x-tex">{\displaystyle Q''_{1}\in \mathbb {R} ^{3\times 3}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e1a9871ac13299a7ee20193ea9baf71e95e14042.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.649ex; height:3.343ex;" alt="{\displaystyle Q''_{1}\in \mathbb {R} ^{3\times 3}}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{1}'\in \mathbb {R} ^{3\times 3}}">

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<annotation encoding="application/x-tex">{\displaystyle Q_{1}'\in \mathbb {R} ^{3\times 3}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a2ff4b369d5d78779c4b83bb7641c048e9692a1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.566ex; height:3.343ex;" alt="{\displaystyle Q_{1}'\in \mathbb {R} ^{3\times 3}}" loading="lazy"> mit

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {diag} (1,Q'_{1},I_{m-4})^{T}\mathrm {diag} (I_{2},Q''_{1},I_{m-5})^{T}{{\tilde {A}}_{22}}={\begin{pmatrix}*&*&*&\cdots &\cdots &\cdots &*\\*&*&*&\cdots &\cdots &\cdots &*\\0&*&*&\ddots &\cdots &\cdots &*\\0&0&*&\ddots &\cdots &\cdots &*\\0&0&0&*&&\vdots \\\vdots &\vdots &\vdots &&\ddots &&\vdots \\0&0&0&\cdots &0&*&*\end{pmatrix}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ce22dc1d4fa779ec5895374dd0be6e9530fb059c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -13.671ex; width:76.601ex; height:28.509ex;" alt="{\displaystyle \mathrm {diag} (1,Q'_{1},I_{m-4})^{T}\mathrm {diag} (I_{2},Q''_{1},I_{m-5})^{T}{{\tilde {A}}_{22}}={\begin{pmatrix}*&*&*&\cdots &\cdots &\cdots &*\\*&*&*&\cdots &\cdots &\cdots &*\\0&*&*&\ddots &\cdots &\cdots &*\\0&0&*&\ddots &\cdots &\cdots &*\\0&0&0&*&&\vdots \\\vdots &\vdots &\vdots &&\ddots &&\vdots \\0&0&0&\cdots &0&*&*\end{pmatrix}}}" loading="lazy">. Es werden also nacheinander die Vektoren <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{pmatrix}a_{21}\\a_{31}\\a_{41}\end{pmatrix}}}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b3420b0809983608ec3540c038a9b5f5ecd50062.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:7.924ex; height:9.176ex;" alt="{\displaystyle {\begin{pmatrix}a_{21}\\a_{31}\\a_{41}\end{pmatrix}}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{pmatrix}a_{32}\\a_{42}\\a_{52}\end{pmatrix}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}a_{32}\\a_{42}\\a_{52}\end{pmatrix}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d8c997c59660aaa801425743791f03bd689a64c8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:7.924ex; height:9.176ex;" alt="{\displaystyle {\begin{pmatrix}a_{32}\\a_{42}\\a_{52}\end{pmatrix}}}" loading="lazy">auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{pmatrix}*\\0\\0\end{pmatrix}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}*\\0\\0\end{pmatrix}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2a4e3152f1afd84397501e8e74f709e88850d656.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:5.981ex; height:9.176ex;" alt="{\displaystyle {\begin{pmatrix}*\\0\\0\end{pmatrix}}}" loading="lazy">transformiert.

Die Anwendung der Transformation auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {B}}_{22}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {diag} (1,Q'_{1},I_{m-4})^{T}\mathrm {diag} (I_{2},Q''_{1},I_{m-5})^{T}{{\tilde {B}}_{22}}={\begin{pmatrix}*&*&*&\cdots &\cdots &\cdots &*\\0&*&*&\cdots &\cdots &\cdots &*\\0&*&*&\ddots &\cdots &\cdots &*\\0&*&*&*&\ddots &\cdots &*\\0&0&0&0&*&\vdots \\\vdots &\vdots &\vdots &&\ddots &&\vdots \\0&0&0&\cdots &0&0&*\end{pmatrix}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/19a99e29eb53cd539380e90ff9c6f986473a530b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -13.671ex; width:76.59ex; height:28.509ex;" alt="{\displaystyle \mathrm {diag} (1,Q'_{1},I_{m-4})^{T}\mathrm {diag} (I_{2},Q''_{1},I_{m-5})^{T}{{\tilde {B}}_{22}}={\begin{pmatrix}*&*&*&\cdots &\cdots &\cdots &*\\0&*&*&\cdots &\cdots &\cdots &*\\0&*&*&\ddots &\cdots &\cdots &*\\0&*&*&*&\ddots &\cdots &*\\0&0&0&0&*&\vdots \\\vdots &\vdots &\vdots &&\ddots &&\vdots \\0&0&0&\cdots &0&0&*\end{pmatrix}}}" loading="lazy">, d. h. ein Buckel ist jetzt eine Position tiefer entlang der Diagonalen entstanden.

14. Man wiederhole die Schritte 11–13 so lange, bis <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{22}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle A_{22}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fcfa7914515f7c23e481f3f4602c8df3d791384a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.619ex; height:2.509ex;" alt="{\displaystyle A_{22}}" loading="lazy"> wieder in oberer Hessenberg- und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{22}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle B_{22}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1aef333fa4c092c88208c94d3b19ad2d603442df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.64ex; height:2.509ex;" alt="{\displaystyle B_{22}}" loading="lazy"> wieder in oberer Dreieckstruktur vorliegt. Diesen Prozess bezeichnet man, analog zum <a href="QR-Algorithmus" title="QR-Algorithmus">QR-Algorithmus</a>, auch als „Buckel-Jagen“ oder „Bulge-Chasing“. Die Eliminierung eines Buckels in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{22}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle B_{22}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1aef333fa4c092c88208c94d3b19ad2d603442df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.64ex; height:2.509ex;" alt="{\displaystyle B_{22}}" loading="lazy"> an der Diagonalposition j mit Transformationen von links führt zu einem Buckel an der entsprechenden Position in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{22}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle A_{22}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fcfa7914515f7c23e481f3f4602c8df3d791384a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.619ex; height:2.509ex;" alt="{\displaystyle A_{22}}" loading="lazy">. Wird dieser Buckel mit Transformationen von rechts eliminiert, führt das zu einem Buckel in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{22}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle B_{22}}</annotation>

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15. Nach <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m-2}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle m-2}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8af4a0e77fd467755d34b9dc34bec97accfa5874.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.043ex; height:2.343ex;" alt="{\displaystyle m-2}" loading="lazy"> Schritten wird das Ziel erreicht und es ergibt sich <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{22}^{T}=\mathrm {diag} (Q_{1},I_{m-3})^{T}\mathrm {diag} (1,Q'_{1},I_{m-4})^{T}\mathrm {diag} (I_{2},Q_{1}'',I_{m-5})^{T}\cdots \mathrm {diag} (I_{m-3},Q_{m-2})^{T}}">

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<annotation encoding="application/x-tex">{\displaystyle Q_{22}^{T}=\mathrm {diag} (Q_{1},I_{m-3})^{T}\mathrm {diag} (1,Q'_{1},I_{m-4})^{T}\mathrm {diag} (I_{2},Q_{1}'',I_{m-5})^{T}\cdots \mathrm {diag} (I_{m-3},Q_{m-2})^{T}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8908664c24c38d301d5f2cd4b8593425c8e1c223.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:83.174ex; height:3.343ex;" alt="{\displaystyle Q_{22}^{T}=\mathrm {diag} (Q_{1},I_{m-3})^{T}\mathrm {diag} (1,Q'_{1},I_{m-4})^{T}\mathrm {diag} (I_{2},Q_{1}'',I_{m-5})^{T}\cdots \mathrm {diag} (I_{m-3},Q_{m-2})^{T}}" loading="lazy">. Analog erhält man

<msub>

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<annotation encoding="application/x-tex">{\displaystyle Z_{22}=\mathrm {diag} (Z_{1},I_{m-3})\mathrm {diag} (Z_{1}',I_{m-2})\cdots \mathrm {diag} (I_{m-2},Z_{m-2})\mathrm {diag} (I_{m-2},Z_{m-2}')}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e337d4db285afb70c3ddb8ff600eec3456067d65.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:73.694ex; height:3.176ex;" alt="{\displaystyle Z_{22}=\mathrm {diag} (Z_{1},I_{m-3})\mathrm {diag} (Z_{1}',I_{m-2})\cdots \mathrm {diag} (I_{m-2},Z_{m-2})\mathrm {diag} (I_{m-2},Z_{m-2}')}" loading="lazy">.

Falls <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A,B)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (A,B)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ce67314185650d6f0deba39db7dcec9378f4d4d1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.35ex; height:2.843ex;" alt="{\displaystyle (A,B)}" loading="lazy">-invarianten Unterräume benötigt werden, ist es notwendig die Matrizen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {Q}}">

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<annotation encoding="application/x-tex">{\displaystyle {Q}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6d07aa40096410d8a45588baa42452ae706d0156.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle {Q}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z}">

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<annotation encoding="application/x-tex">{\displaystyle Z}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle Q:=Q\mathrm {diag} (I_{p},Q_{22},I_{q})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ddb9a6cba3994a538338d49f3226a05fb055ab36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.372ex; height:3.009ex;" alt="{\displaystyle Q:=Q\mathrm {diag} (I_{p},Q_{22},I_{q})}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z:=Z\mathrm {diag} (I_{p},Z_{22},I_{q})}">

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<annotation encoding="application/x-tex">{\displaystyle Z:=Z\mathrm {diag} (I_{p},Z_{22},I_{q})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/149f865bed6c4f993fb9568dbbcb0fa7fadeea9e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.805ex; height:3.009ex;" alt="{\displaystyle Z:=Z\mathrm {diag} (I_{p},Z_{22},I_{q})}" loading="lazy">

16. end while

<div class="mw-heading mw-heading2"><h2 id="Bestimmung_der_Eigenwerte">Bestimmung der Eigenwerte</h2></div>

In den meisten Fällen konvergiert <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A,B)}">

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<annotation encoding="application/x-tex">{\displaystyle (A,B)}</annotation>

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Für skalare Diagonalblöcke in A gilt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{i}={\frac {a_{ii}}{b_{ii}}}:b_{ii}\neq 0}">

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<annotation encoding="application/x-tex">{\displaystyle \lambda _{i}={\frac {a_{ii}}{b_{ii}}}:b_{ii}\neq 0}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f1e71f3dd482fd1fa4bdc49b940cc9af9b7f07ce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:17.249ex; height:5.176ex;" alt="{\displaystyle \lambda _{i}={\frac {a_{ii}}{b_{ii}}}:b_{ii}\neq 0}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{i}=\infty }">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle \lambda _{i}=\infty }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/814f68dbd4f19e3810c8eb4c1ee91c2cab3d6d90.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.577ex; height:2.509ex;" alt="{\displaystyle \lambda _{i}=\infty }" loading="lazy"> falls <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{ii}=0}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle b_{ii}=0}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b3dda76090aa22abfb479a07eb6a7c7ba7ea6bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.626ex; height:2.509ex;" alt="{\displaystyle b_{ii}=0}" loading="lazy">. Falls ein

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<annotation encoding="application/x-tex">{\displaystyle i}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"> existiert, für das <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{ii}=b_{ii}=0}">

<msub>

</mrow>

</msub>

<msub>

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<annotation encoding="application/x-tex">{\displaystyle a_{ii}=b_{ii}=0}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5ca0a30eea386499773c7fa3f3b155b1778dac7e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.321ex; height:2.509ex;" alt="{\displaystyle a_{ii}=b_{ii}=0}" loading="lazy"> ist, so ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda (A,B)=\mathbb {C} }">

</mrow>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \Lambda (A,B)=\mathbb {C} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4ace01e82cf5d5f0fbd5283be2edd2d1b5a6abec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.74ex; height:2.843ex;" alt="{\displaystyle \Lambda (A,B)=\mathbb {C} }" loading="lazy">. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\times 2}">

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<annotation encoding="application/x-tex">{\displaystyle 2\times 2}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f8a0e3400ffb97d67c00267ed50cddfe824cbe80.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.165ex; height:2.176ex;" alt="{\displaystyle 2\times 2}" loading="lazy"> Diagonalblöcke von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> beziehen sich (analog zum QR-Algorithmus) auf Paare komplex konjugierter Eigenwerte <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda ,{\overline {\lambda }}=\Lambda \left({\begin{pmatrix}a_{ii}&a_{i,i+1}\\a_{i+1,i}&a_{i+1,i+1}\end{pmatrix}},{\begin{pmatrix}b_{ii}&b_{i,i+1}\\0&b_{i+1,i+1}\end{pmatrix}}\right)}">

<mover>

</mover>

</mrow>

<mrow>

<mrow>

<mrow>

<mtr>

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<mtr>

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<annotation encoding="application/x-tex">{\displaystyle \lambda ,{\overline {\lambda }}=\Lambda \left({\begin{pmatrix}a_{ii}&a_{i,i+1}\\a_{i+1,i}&a_{i+1,i+1}\end{pmatrix}},{\begin{pmatrix}b_{ii}&b_{i,i+1}\\0&b_{i+1,i+1}\end{pmatrix}}\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/143d4d326dc981f728ff80f212c19966aacfe71a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:48.201ex; height:6.509ex;" alt="{\displaystyle \lambda ,{\overline {\lambda }}=\Lambda \left({\begin{pmatrix}a_{ii}&a_{i,i+1}\\a_{i+1,i}&a_{i+1,i+1}\end{pmatrix}},{\begin{pmatrix}b_{ii}&b_{i,i+1}\\0&b_{i+1,i+1}\end{pmatrix}}\right)}" loading="lazy">.

<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>

<ul><li>Gene H. Golub, Charles F. Van Loan: Matrix Computations. Johns Hopkins University Press, 1996, ISBN 0-8018-5414-8.</li>

<li>G. W. Stewart: Matrix Algorithms. Band II: Eigensystems. SIAM 2001, ISBN 0-89871-503-2.</li></ul></div><div><div class="zim-footer">

Dieser Artikel wurde von <a class="external text" title="Zuletzt bearbeitet am 2025-08-14" href="https://de.wikipedia.org/wiki/?title=QZ-Algorithmus&oldid=258845276">Wikipedia</a> herausgegeben. Der Text ist unter <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.de">Creative Commons Attribution-Share Alike 4.0</a> verfügbar, sofern nicht anders angegeben. Für die Mediendateien können zusätzliche Bedingungen gelten.

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