http://www.openpipeflow.org/index.php?action=history&feed=atom&title=File%3AArnoldi.f 2026-10-09T05:37:08Z Revision history for this page on the wiki MediaWiki 1.39.1 http://www.openpipeflow.org/index.php?title=File:Arnoldi.f&diff=1115&oldid=prev 2025-11-20T12:07:45Z

<p></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en-GB"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 05:07, 20 November 2025</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l4">Line 4:</td> <td colspan="2" class="diff-lineno">Line 4:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The Arnoldi method is a method for calculating the eigenvalues and eigenvectors of a matrix, i.e. for calculating the scalar $\sigma$ and $n$-vectors $\vec{x}$ that satisfy</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The Arnoldi method is a method for calculating the eigenvalues and eigenvectors of a matrix, i.e. for calculating the scalar $\sigma$ and $n$-vectors $\vec{x}$ that satisfy</div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">\[</del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">$</ins></div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\sigma\,\vec{x} = A\,\vec{x}</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\sigma\,\vec{x} = A\,\vec{x}</div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">\]</del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">$</ins></div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>for a given $n\times n$ matrix $A$.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>for a given $n\times n$ matrix $A$.</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr> </table>

Apwillis http://www.openpipeflow.org/index.php?title=File:Arnoldi.f&diff=1052&oldid=prev 2021-10-21T10:22:42Z

<p></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en-GB"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 03:22, 21 October 2021</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l24">Line 24:</td> <td colspan="2" class="diff-lineno">Line 24:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>We apply the Arnoldi method to $B$, then recover $\sigma$ from $\tilde{\sigma}$.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>We apply the Arnoldi method to $B$, then recover $\sigma$ from $\tilde{\sigma}$.</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">Consider </del>the case of a perturbation $\vec{\delta x}$ linearised about a solution $\vec{x}_0$<del style="font-weight: bold; text-decoration: none;">, i.e</del>. $\partial_t \vec{\delta x} = A(\vec{x}_0)\,\vec{\delta x}$.  Let $\vec{X}(\vec{x})$ be the result of time integration of $\vec{x}$. For this system, the result of $B\,\vec{\delta x}$ for a given $\vec{\delta x}$ may be approximated by $\frac{1}{\epsilon}(\vec{X}(\vec{x}_0+\epsilon\,\vec{\delta x})-\vec{X}(\vec{x}_0))$ for some small value $\epsilon$.  (Although we expect that $\vec{X}(\vec{x}_0)=\vec{x}_0$, numerical accuracy is likely to be better with the given form.)  Note that to find the eigenvalues $\tilde{\sigma}=\mathrm{e}^{\sigma T}$ of $B$ with the Arnoldi method, only a routine for time integration of a given initial condition is required.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Now let&#039;s consider </ins>the case of a perturbation $\vec{\delta x}$ linearised about a solution $\vec{x}_0$. <ins style="font-weight: bold; text-decoration: none;"> Evolution of the perturbation is given by </ins>$\partial_t \vec{\delta x} = A(\vec{x}_0)\,\vec{\delta x}$.  Let $\vec{X}(\vec{x})$ be the result of time integration of $\vec{x}$. For this system, the result of $B\,\vec{\delta x}$ for a given $\vec{\delta x}$ may be approximated by $\frac{1}{\epsilon}(\vec{X}(\vec{x}_0+\epsilon\,\vec{\delta x})-\vec{X}(\vec{x}_0))$ for some small value $\epsilon$.  (Although we expect that $\vec{X}(\vec{x}_0)=<ins style="font-weight: bold; text-decoration: none;">\vec{x}_0$ for a solution $</ins>\vec{x}_0$, numerical accuracy is likely to be better with the given form.)  Note that to find the eigenvalues $\tilde{\sigma}=\mathrm{e}^{\sigma T}$ of $B$ with the Arnoldi method, only a routine for time integration of a given initial condition is required.</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== How to use the code ==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== How to use the code ==</div></td></tr> </table>

Apwillis http://www.openpipeflow.org/index.php?title=File:Arnoldi.f&diff=1051&oldid=prev 2021-10-21T10:01:42Z

<p></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en-GB"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 03:01, 21 October 2021</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l16">Line 16:</td> <td colspan="2" class="diff-lineno">Line 16:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>shares the same eigenvectors as the original problem, but its eigenvalues, $\tilde{\sigma}=\mathrm{e}^\sigma$, are often better suited to the Arnoldi method.  The negative eigenvalues $\sigma$, which typically correspond to modes of little interest, now correspond to eigenvalues $\tilde{\sigma}$ bunched close to the origin.  The Arnoldi method favours the $\tilde{\sigma}$ most separated in the complex plane, being the $\sigma$ with largest real parts.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>shares the same eigenvectors as the original problem, but its eigenvalues, $\tilde{\sigma}=\mathrm{e}^\sigma$, are often better suited to the Arnoldi method.  The negative eigenvalues $\sigma$, which typically correspond to modes of little interest, now correspond to eigenvalues $\tilde{\sigma}$ bunched close to the origin.  The Arnoldi method favours the $\tilde{\sigma}$ most separated in the complex plane, being the $\sigma$ with largest real parts.</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Note that for the system $\partial_t \vec{x} = A\,\vec{x}$, time integration corresponds to exponentiation: <del style="font-weight: bold; text-decoration: none;">Taking </del>eigenvector $\vec{x}$ with eigenvalue $\sigma$ <del style="font-weight: bold; text-decoration: none;">as an initial condition</del>, <del style="font-weight: bold; text-decoration: none;">the result of time integration from $0$ to $T$ is $</del>\<del style="font-weight: bold; text-decoration: none;">mathrm</del>{<del style="font-weight: bold; text-decoration: none;">e</del>}<del style="font-weight: bold; text-decoration: none;">^{</del>\<del style="font-weight: bold; text-decoration: none;">sigma T}</del>\vec{x}$.  <del style="font-weight: bold; text-decoration: none;">We </del>have <del style="font-weight: bold; text-decoration: none;">that </del>$\mathrm{e}^{\sigma T}\vec{x}=\int_0^T A\,\vec{x}\,dt<del style="font-weight: bold; text-decoration: none;">=</del>\mathrm{e}^{<del style="font-weight: bold; text-decoration: none;">AT</del>}\vec{x}\equiv B\,\vec{x}$.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Note that for the system $\partial_t \vec{x} = A\,\vec{x}$, time integration corresponds to exponentiation:  </div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">i.e.</del>, $\tilde{\sigma}=\mathrm{e}^{\sigma T<del style="font-weight: bold; text-decoration: none;">}$ is the eigenvalue of the time integration operator $B$ for the same eigenvector $\vec{x</del>}$. <del style="font-weight: bold; text-decoration: none;"> </del>We apply the Arnoldi method to $B$, then recover $\sigma$ from $\tilde{\sigma}$.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">First, for </ins>eigenvector $\vec{x}<ins style="font-weight: bold; text-decoration: none;">_0</ins>$ with eigenvalue $\sigma$ <ins style="font-weight: bold; text-decoration: none;">we may write $\sigma\</ins>, \<ins style="font-weight: bold; text-decoration: none;">vec</ins>{<ins style="font-weight: bold; text-decoration: none;">x</ins>}<ins style="font-weight: bold; text-decoration: none;">_0 = A</ins>\<ins style="font-weight: bold; text-decoration: none;">,</ins>\vec{x}<ins style="font-weight: bold; text-decoration: none;">_0</ins>$.   </div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Also, taking $\vec{x}_0$ as an intial condition, after a time $T$ it will </ins>have <ins style="font-weight: bold; text-decoration: none;">grown or decayed to </ins></div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>$<ins style="font-weight: bold; text-decoration: none;">\vec{x}_T=</ins>\mathrm{e}^{\sigma T}\vec{x}<ins style="font-weight: bold; text-decoration: none;">_0 </ins>=<ins style="font-weight: bold; text-decoration: none;">\vec{x}_0 + </ins>\int_0^T A\,\vec{x}\,dt<ins style="font-weight: bold; text-decoration: none;">$.  We may write this as</ins></div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">$</ins>\mathrm{e}^{<ins style="font-weight: bold; text-decoration: none;">\sigma T</ins>}\vec{x}<ins style="font-weight: bold; text-decoration: none;">_0 </ins>\equiv B\,\vec{x}<ins style="font-weight: bold; text-decoration: none;">_0</ins>$<ins style="font-weight: bold; text-decoration: none;">, where $B$ is the time integration operator</ins>.</div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Then $\vec{x}_0$ is an eigenvector of both $A$ and $B$</ins>, <ins style="font-weight: bold; text-decoration: none;">where its eigenvalue for $B$ is </ins>$\tilde{\sigma}=\mathrm{e}^{\sigma T}$.</div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>We apply the Arnoldi method to $B$, then recover $\sigma$ from $\tilde{\sigma}$.</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Consider the case of a perturbation $\vec{\delta x}$ linearised about a solution $\vec{x}_0$, i.e. $\partial_t \vec{\delta x} = A(\vec{x}_0)\,\vec{\delta x}$.  Let $\vec{X}(\vec{x})$ be the result of time integration of $\vec{x}$. For this system, the result of $B\,\vec{\delta x}$ for a given $\vec{\delta x}$ may be approximated by $\frac{1}{\epsilon}(\vec{X}(\vec{x}_0+\epsilon\,\vec{\delta x})-\vec{X}(\vec{x}_0))$ for some small value $\epsilon$.  (Although we expect that $\vec{X}(\vec{x}_0)=\vec{x}_0$, numerical accuracy is likely to be better with the given form.)  Note that to find the eigenvalues $\tilde{\sigma}=\mathrm{e}^{\sigma T}$ of $B$ with the Arnoldi method, only a routine for time integration of a given initial condition is required.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Consider the case of a perturbation $\vec{\delta x}$ linearised about a solution $\vec{x}_0$, i.e. $\partial_t \vec{\delta x} = A(\vec{x}_0)\,\vec{\delta x}$.  Let $\vec{X}(\vec{x})$ be the result of time integration of $\vec{x}$. For this system, the result of $B\,\vec{\delta x}$ for a given $\vec{\delta x}$ may be approximated by $\frac{1}{\epsilon}(\vec{X}(\vec{x}_0+\epsilon\,\vec{\delta x})-\vec{X}(\vec{x}_0))$ for some small value $\epsilon$.  (Although we expect that $\vec{X}(\vec{x}_0)=\vec{x}_0$, numerical accuracy is likely to be better with the given form.)  Note that to find the eigenvalues $\tilde{\sigma}=\mathrm{e}^{\sigma T}$ of $B$ with the Arnoldi method, only a routine for time integration of a given initial condition is required.</div></td></tr> </table>

Apwillis http://www.openpipeflow.org/index.php?title=File:Arnoldi.f&diff=1050&oldid=prev 2021-10-21T09:35:05Z

<p></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en-GB"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 02:35, 21 October 2021</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l14">Line 14:</td> <td colspan="2" class="diff-lineno">Line 14:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The method finds the eigenvalues most separated in the complex plane first.  If $A$ is expected to have many negative eigenvalues of little interest, it may be better to work with $\tilde{A}=\mathrm{e}^A=1+A+\frac{1}{2!}A^2+...$, i.e. the eigenproblem $\mathrm{e}^\sigma\vec{x}=\mathrm{e}^A\vec{x}$.  This problem</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The method finds the eigenvalues most separated in the complex plane first.  If $A$ is expected to have many negative eigenvalues of little interest, it may be better to work with $\tilde{A}=\mathrm{e}^A=1+A+\frac{1}{2!}A^2+...$, i.e. the eigenproblem $\mathrm{e}^\sigma\vec{x}=\mathrm{e}^A\vec{x}$.  This problem</div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>shares the same eigenvectors as the original problem, but <del style="font-weight: bold; text-decoration: none;">often has more suitable </del>eigenvalues <del style="font-weight: bold; text-decoration: none;">for the Arnoldi method</del>, $\tilde{\sigma}=\mathrm{e}^\sigma$.  The negative eigenvalues $\sigma$ <del style="font-weight: bold; text-decoration: none;">then </del>correspond to eigenvalues $\tilde{\sigma}$ bunched close to the origin.  The Arnoldi method favours the $\tilde{\sigma}$ most separated in the complex plane, being the $\sigma$ with largest real parts.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>shares the same eigenvectors as the original problem, but <ins style="font-weight: bold; text-decoration: none;">its </ins>eigenvalues, $\tilde{\sigma}=\mathrm{e}^\sigma$<ins style="font-weight: bold; text-decoration: none;">, are often better suited to the Arnoldi method</ins>.  The negative eigenvalues $\sigma$<ins style="font-weight: bold; text-decoration: none;">, which typically correspond to modes of little interest, now </ins>correspond to eigenvalues $\tilde{\sigma}$ bunched close to the origin.  The Arnoldi method favours the $\tilde{\sigma}$ most separated in the complex plane, being the $\sigma$ with largest real parts.</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Note that for the system $\partial_t \vec{x} = A\,\vec{x}$, time integration corresponds to exponentiation: Taking eigenvector $\vec{x}$ with eigenvalue $\sigma$ as an initial condition, the result of time integration from $0$ to $T$ is $\mathrm{e}^{\sigma T}\vec{x}$.  We have that $\mathrm{e}^{\sigma T}\vec{x}=\int_0^T A\,\vec{x}\,dt=\mathrm{e}^{AT}\vec{x}\equiv B\,\vec{x}$.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Note that for the system $\partial_t \vec{x} = A\,\vec{x}$, time integration corresponds to exponentiation: Taking eigenvector $\vec{x}$ with eigenvalue $\sigma$ as an initial condition, the result of time integration from $0$ to $T$ is $\mathrm{e}^{\sigma T}\vec{x}$.  We have that $\mathrm{e}^{\sigma T}\vec{x}=\int_0^T A\,\vec{x}\,dt=\mathrm{e}^{AT}\vec{x}\equiv B\,\vec{x}$.</div></td></tr> </table>

Apwillis http://www.openpipeflow.org/index.php?title=File:Arnoldi.f&diff=1006&oldid=prev 2020-01-29T00:29:16Z

<p></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en-GB"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 17:29, 28 January 2020</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l5">Line 5:</td> <td colspan="2" class="diff-lineno">Line 5:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The Arnoldi method is a method for calculating the eigenvalues and eigenvectors of a matrix, i.e. for calculating the scalar $\sigma$ and $n$-vectors $\vec{x}$ that satisfy</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The Arnoldi method is a method for calculating the eigenvalues and eigenvectors of a matrix, i.e. for calculating the scalar $\sigma$ and $n$-vectors $\vec{x}$ that satisfy</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\[</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\[</div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">  </del>\sigma\,\vec{x} = A\,\vec{x}</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>\sigma\,\vec{x} = A\,\vec{x}</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\]</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>for a given $n\times n$ matrix $A$.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>for a given $n\times n$ matrix $A$.</div></td></tr> </table>

Apwillis http://www.openpipeflow.org/index.php?title=File:Arnoldi.f&diff=931&oldid=prev 2019-07-17T11:57:35Z

<p></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en-GB"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 04:57, 17 July 2019</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l9">Line 9:</td> <td colspan="2" class="diff-lineno">Line 9:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>for a given $n\times n$ matrix $A$.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>for a given $n\times n$ matrix $A$.</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>The main advantage of the method is that it only requires calculations of multiplies by $A$ for a given $\vec{x}$ -- it does not need to know $A$ itself.  This means that $A$ need not even be stored, and could correspond to a very complex linear &#039;action&#039; on $\vec{x}$, e.g. a time integral with initial condition $\vec{x}$.  Given a starting vector $\vec{x}_0$, the method seeks eigenvectors $\vec{x}$ in $\mathrm{span}\{\vec{x}_0,\,A\vec{x}_0,\,A^2\vec{x}_0,...\}$, but uses Gram-Schmidt orthogonalisation to improve the suitability of this basis set.  The set of orthogonalised vectors is called the Krylov-subspace.  In practice we need to limit the size of this space to m vectors, but if this number is reached, it is possible to restart without losing information. Restarts are not implemented here; see the following which can dramatically reduce the number of multiplies by $A$ required.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>The main advantage of the method is that it only requires calculations of multiplies by $A$ for a given $\vec{x}$ -- it does not need to know $A$ itself.  This means that $A$ need not even be stored, and could correspond to a very complex linear &#039;action&#039; on $\vec{x}$, e.g. a time integral with initial condition $\vec{x}$.  Given a starting vector $\vec{x}_0$, the method seeks eigenvectors $\vec{x}$ in $\mathrm{span}\{\vec{x}_0,\,A\vec{x}_0,\,A^2\vec{x}_0,...\}$, but uses Gram-Schmidt orthogonalisation to improve the <ins style="font-weight: bold; text-decoration: none;">numerical </ins>suitability of this basis set.  The set of orthogonalised vectors is called the Krylov-subspace.  In practice we need to limit the size of this space to m vectors, but if this number is reached, it is possible to restart without losing information. Restarts are not implemented here; see the following which can dramatically reduce the number of multiplies by $A$ required.</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>===Time Integration and Exponentiation===</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>===Time Integration and Exponentiation===</div></td></tr> </table>

Apwillis http://www.openpipeflow.org/index.php?title=File:Arnoldi.f&diff=848&oldid=prev 2019-02-25T13:38:21Z

<p></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en-GB"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 06:38, 25 February 2019</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l17">Line 17:</td> <td colspan="2" class="diff-lineno">Line 17:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Note that for the system $\partial_t \vec{x} = A\,\vec{x}$, time integration corresponds to exponentiation: Taking eigenvector $\vec{x}$ with eigenvalue $\sigma$ as an initial condition, the result of time integration from $0$ to $T$ is $\mathrm{e}^{\sigma T}\vec{x}$.  We have that $\mathrm{e}^{\sigma T}\vec{x}=\int_0^T A\,\vec{x}\,dt=\mathrm{e}^{AT}\vec{x}\equiv B\,\vec{x}$.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Note that for the system $\partial_t \vec{x} = A\,\vec{x}$, time integration corresponds to exponentiation: Taking eigenvector $\vec{x}$ with eigenvalue $\sigma$ as an initial condition, the result of time integration from $0$ to $T$ is $\mathrm{e}^{\sigma T}\vec{x}$.  We have that $\mathrm{e}^{\sigma T}\vec{x}=\int_0^T A\,\vec{x}\,dt=\mathrm{e}^{AT}\vec{x}\equiv B\,\vec{x}$.</div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>i.e., $\mathrm{e}^{\sigma T}$ is the eigenvalue of the time integration operator $B$ for the same eigenvector $\vec{x}$, from <del style="font-weight: bold; text-decoration: none;">which it is straight forward to recover </del>$\sigma$.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>i.e., $<ins style="font-weight: bold; text-decoration: none;">\tilde{\sigma}=</ins>\mathrm{e}^{\sigma T}$ is the eigenvalue of the time integration operator $B$ for the same eigenvector $\vec{x}<ins style="font-weight: bold; text-decoration: none;">$.  We apply the Arnoldi method to $B</ins>$, <ins style="font-weight: bold; text-decoration: none;">then recover $\sigma$ </ins>from $<ins style="font-weight: bold; text-decoration: none;">\tilde{</ins>\sigma<ins style="font-weight: bold; text-decoration: none;">}</ins>$.</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Consider the case of a perturbation $\vec{\delta x}$ linearised about a solution $\vec{x}_0$, i.e. $\partial_t \vec{\delta x} = A(\vec{x}_0)\,\vec{\delta x}$.  Let $\vec{X}(\vec{x})$ be the result of time integration of $\vec{x}$. For this system, the result of $B\,\vec{\delta x}$ for a given $\vec{\delta x}$ may be approximated by $\frac{1}{\epsilon}(\vec{X}(\vec{x}_0+\epsilon\,\vec{\delta x})-\vec{X}(\vec{x}_0))$ for some small value $\epsilon$.  (Although we expect that $\vec{X}(\vec{x}_0)=\vec{x}_0$, numerical accuracy is likely to be better with the given form.)  Note that to find the eigenvalues $\tilde{\sigma}=\mathrm{e}^{\sigma T}$ of $B$ with the Arnoldi method, only a routine for time integration of a given initial condition is required.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Consider the case of a perturbation $\vec{\delta x}$ linearised about a solution $\vec{x}_0$, i.e. $\partial_t \vec{\delta x} = A(\vec{x}_0)\,\vec{\delta x}$.  Let $\vec{X}(\vec{x})$ be the result of time integration of $\vec{x}$. For this system, the result of $B\,\vec{\delta x}$ for a given $\vec{\delta x}$ may be approximated by $\frac{1}{\epsilon}(\vec{X}(\vec{x}_0+\epsilon\,\vec{\delta x})-\vec{X}(\vec{x}_0))$ for some small value $\epsilon$.  (Although we expect that $\vec{X}(\vec{x}_0)=\vec{x}_0$, numerical accuracy is likely to be better with the given form.)  Note that to find the eigenvalues $\tilde{\sigma}=\mathrm{e}^{\sigma T}$ of $B$ with the Arnoldi method, only a routine for time integration of a given initial condition is required.</div></td></tr> </table>

Apwillis http://www.openpipeflow.org/index.php?title=File:Arnoldi.f&diff=838&oldid=prev 2018-11-13T14:02:36Z

<p></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en-GB"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 07:02, 13 November 2018</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l9">Line 9:</td> <td colspan="2" class="diff-lineno">Line 9:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>for a given $n\times n$ matrix $A$.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>for a given $n\times n$ matrix $A$.</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>The main advantage of the method is that it only requires calculations of multiplies by $A$ for a given $\vec{x}$ -- it does not need to know $A$ itself.  This means that $A$ need not even be stored, and could correspond to a very complex linear &#039;action&#039; on $\vec{x}$, e.g. a time integral with initial condition $\vec{x}$.  <del style="font-weight: bold; text-decoration: none;">The </del>method seeks eigenvectors in $\mathrm{span}\{\vec{x},\,A\vec{x},\,A^2\vec{x},...\}$, but uses Gram-Schmidt orthogonalisation to improve the suitability of this basis set.  The set of orthogonalised vectors is called the Krylov-subspace<del style="font-weight: bold; text-decoration: none;">, and m is </del>the <del style="font-weight: bold; text-decoration: none;">maximum number </del>of vectors <del style="font-weight: bold; text-decoration: none;">stored.  It </del>is possible to restart without losing information<del style="font-weight: bold; text-decoration: none;">; </del>not implemented here.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>The main advantage of the method is that it only requires calculations of multiplies by $A$ for a given $\vec{x}$ -- it does not need to know $A$ itself.  This means that $A$ need not even be stored, and could correspond to a very complex linear &#039;action&#039; on $\vec{x}$, e.g. a time integral with initial condition $\vec{x}$.  <ins style="font-weight: bold; text-decoration: none;">Given a starting vector $\vec{x}_0$, the </ins>method seeks eigenvectors <ins style="font-weight: bold; text-decoration: none;">$\vec{x}$ </ins>in $\mathrm{span}\{\vec{x}<ins style="font-weight: bold; text-decoration: none;">_0</ins>,\,A\vec{x}<ins style="font-weight: bold; text-decoration: none;">_0</ins>,\,A^2\vec{x}<ins style="font-weight: bold; text-decoration: none;">_0</ins>,...\}$, but uses Gram-Schmidt orthogonalisation to improve the suitability of this basis set.  The set of orthogonalised vectors is called the Krylov-subspace<ins style="font-weight: bold; text-decoration: none;">.  In practice we need to limit </ins>the <ins style="font-weight: bold; text-decoration: none;">size </ins>of <ins style="font-weight: bold; text-decoration: none;">this space to m </ins>vectors<ins style="font-weight: bold; text-decoration: none;">, but if this number is reached, it </ins>is possible to restart without losing information<ins style="font-weight: bold; text-decoration: none;">. Restarts are </ins>not implemented here<ins style="font-weight: bold; text-decoration: none;">; see the following which can dramatically reduce the number of multiplies by $A$ required</ins>.</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>===Time Integration and Exponentiation===</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>===Time Integration and Exponentiation===</div></td></tr> </table>

Apwillis http://www.openpipeflow.org/index.php?title=File:Arnoldi.f&diff=837&oldid=prev 2018-11-13T13:54:14Z

<p></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en-GB"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 06:54, 13 November 2018</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l5">Line 5:</td> <td colspan="2" class="diff-lineno">Line 5:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The Arnoldi method is a method for calculating the eigenvalues and eigenvectors of a matrix, i.e. for calculating the scalar $\sigma$ and $n$-vectors $\vec{x}$ that satisfy</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The Arnoldi method is a method for calculating the eigenvalues and eigenvectors of a matrix, i.e. for calculating the scalar $\sigma$ and $n$-vectors $\vec{x}$ that satisfy</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\[</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\[</div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>   \sigma\,<del style="font-weight: bold; text-decoration: none;">A </del>= A\,\vec{x}</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>   \sigma\,<ins style="font-weight: bold; text-decoration: none;">\vec{x} </ins>= A\,\vec{x}</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\]</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>for a given $n\times n$ matrix $A$.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>for a given $n\times n$ matrix $A$.</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>The main advantage of the method is that it only requires calculations of multiplies by $A$ for a given $\vec{x}$ -- it does not need to know $A$ itself.  This means that $A$ need not even be stored, and could correspond to a very complex linear &#039;action&#039; on $\vec{x}$, e.g. a time integral with initial condition $\vec{x}$.  The method seeks eigenvectors in $\mathrm{span}\{\vec{x},\,A\vec{x},\,A^2\vec{x},...\}$, but uses Gram-Schmidt orthogonalisation to improve the suitability of <del style="font-weight: bold; text-decoration: none;">the </del>basis.  The set of orthogonalised vectors is called the Krylov-subspace, and m is the maximum number of vectors stored.  It is possible to restart without losing information; not implemented here.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>The main advantage of the method is that it only requires calculations of multiplies by $A$ for a given $\vec{x}$ -- it does not need to know $A$ itself.  This means that $A$ need not even be stored, and could correspond to a very complex linear &#039;action&#039; on $\vec{x}$, e.g. a time integral with initial condition $\vec{x}$.  The method seeks eigenvectors in $\mathrm{span}\{\vec{x},\,A\vec{x},\,A^2\vec{x},...\}$, but uses Gram-Schmidt orthogonalisation to improve the suitability of <ins style="font-weight: bold; text-decoration: none;">this </ins>basis <ins style="font-weight: bold; text-decoration: none;">set</ins>.  The set of orthogonalised vectors is called the Krylov-subspace, and m is the maximum number of vectors stored.  It is possible to restart without losing information; not implemented here.</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>===Time Integration and Exponentiation===</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>===Time Integration and Exponentiation===</div></td></tr> </table>

Apwillis http://www.openpipeflow.org/index.php?title=File:Arnoldi.f&diff=784&oldid=prev 2018-03-08T13:53:39Z

<p><span dir="auto"><span class="autocomment">Time Integration and Exponentiation</span></span></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en-GB"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 06:53, 8 March 2018</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l19">Line 19:</td> <td colspan="2" class="diff-lineno">Line 19:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>i.e., $\mathrm{e}^{\sigma T}$ is the eigenvalue of the time integration operator $B$ for the same eigenvector $\vec{x}$, from which it is straight forward to recover $\sigma$.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>i.e., $\mathrm{e}^{\sigma T}$ is the eigenvalue of the time integration operator $B$ for the same eigenvector $\vec{x}$, from which it is straight forward to recover $\sigma$.</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Consider the case of a perturbation $\vec{\delta x}$ linearised about a solution $\vec{x}_0$, i.e. $\partial_t \vec{\delta x} = A(\vec{x}_0)\,\vec{\delta x}$.  Let $\vec{X}(\vec{x})$ be the result of time integration of $\vec{x}$. For this system, the result of $B\,\vec{\delta x}$ for a given $\vec{\delta x}$ may be approximated by $\frac{1}{\epsilon}(\vec{X}(\vec{x}_0+\epsilon\,\vec{\delta x})-\vec{X}(\vec{x}_0))$ for some small value $\epsilon$.  (Although we expect that $\vec{X}(\vec{x}_0)=\vec{x}_0$, numerical accuracy is likely to be better with the given form.)  Note that to find the eigenvalues of $<del style="font-weight: bold; text-decoration: none;">A</del>$ with the Arnoldi method, <del style="font-weight: bold; text-decoration: none;">in this case </del>only a routine for time integration of a given initial condition is required.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Consider the case of a perturbation $\vec{\delta x}$ linearised about a solution $\vec{x}_0$, i.e. $\partial_t \vec{\delta x} = A(\vec{x}_0)\,\vec{\delta x}$.  Let $\vec{X}(\vec{x})$ be the result of time integration of $\vec{x}$. For this system, the result of $B\,\vec{\delta x}$ for a given $\vec{\delta x}$ may be approximated by $\frac{1}{\epsilon}(\vec{X}(\vec{x}_0+\epsilon\,\vec{\delta x})-\vec{X}(\vec{x}_0))$ for some small value $\epsilon$.  (Although we expect that $\vec{X}(\vec{x}_0)=\vec{x}_0$, numerical accuracy is likely to be better with the given form.)  Note that to find the eigenvalues <ins style="font-weight: bold; text-decoration: none;">$\tilde{\sigma}=\mathrm{e}^{\sigma T}$ </ins>of $<ins style="font-weight: bold; text-decoration: none;">B</ins>$ with the Arnoldi method, only a routine for time integration of a given initial condition is required.</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== How to use the code ==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== How to use the code ==</div></td></tr> </table>

Apwillis