https://wikiskripta.fjfi.cvut.cz/wiki/index.php?title=01NUM1:Kapitola5&feed=atom&action=history 01NUM1:Kapitola5 - Historie editací 2024-03-29T14:13:22Z Historie editací této stránky MediaWiki 1.25.2 https://wikiskripta.fjfi.cvut.cz/wiki/index.php?title=01NUM1:Kapitola5&diff=7545&oldid=prev Kubuondr v 31. 1. 2017, 09:41 2017-01-31T09:41:37Z <p></p> <table class='diff diff-contentalign-left'> <col class='diff-marker' /> <col class='diff-content' /> <col class='diff-marker' /> <col class='diff-content' /> <tr style='vertical-align: top;'> <td colspan='2' style="background-color: white; color:black; text-align: center;">← Starší verze</td> <td colspan='2' style="background-color: white; color:black; text-align: center;">Verze z 31. 1. 2017, 09:41</td> </tr><tr><td colspan="2" class="diff-lineno" id="L469" >Řádka 469:</td> <td colspan="2" class="diff-lineno">Řádka 469:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Kde poslední rovnost plyne z Jordanovy věty a rozpisu nahoře. V absolutních hodnotách tedy máme: \[\lvert 1-\omega\rvert^n = \prod_{i=1}^n \lvert \lambda_i \rvert\]</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Kde poslední rovnost plyne z Jordanovy věty a rozpisu nahoře. V absolutních hodnotách tedy máme: \[\lvert 1-\omega\rvert^n = \prod_{i=1}^n \lvert \lambda_i \rvert\]</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Pak je splněno buď I) všechna \( \lambda_i \) jsou stejná, tj, \(\lvert 1-\omega\rvert^n = \lvert \lambda_i \rvert^n \) a tedy nastává rovnost \(\rho (\matice B_{\omega}) = \lvert \lambda_i \rvert \) &#160;</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Pak je splněno buď I) všechna \( \lambda_i \) jsou stejná, tj, \(\lvert 1-\omega\rvert^n = \lvert \lambda_i \rvert^n \) a tedy nastává rovnost \(\rho (\matice B_{\omega}) = \lvert \lambda_i \rvert \) &#160;</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; 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>anebo II) existuje takové \(\lambda_{i_1} \), že \(\lvert \lambda_{i_1} \rvert &lt; \lvert 1-\omega \rvert\), pak ale musí zároveň existovat takové \(\lambda_{i_2}\), že \(\lvert \lambda_{i_2} \rvert &gt; \lvert 1-\omega \rvert\) a nastává tudíž \(\rho (\matice B_{\omega}) <del class="diffchange diffchange-inline">&gt; </del>\lvert \lambda_{i_2} \rvert \). Z věty \ref{KSOR} pak plyne divergence metody pro \( \omega \in \mathbbm R \setminus \langle 0 , 2 \rangle&#160; \).</div></td><td class='diff-marker'>+</td><td style="color:black; 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>anebo II) existuje takové \(\lambda_{i_1} \), že \(\lvert \lambda_{i_1} \rvert &lt; \lvert 1-\omega \rvert\), pak ale musí zároveň existovat takové \(\lambda_{i_2}\), že \(\lvert \lambda_{i_2} \rvert &gt; \lvert 1-\omega \rvert\) a nastává tudíž \(\rho (\matice B_{\omega}) <ins class="diffchange diffchange-inline">\geq </ins>\lvert \lambda_{i_2} <ins class="diffchange diffchange-inline">\rvert&gt;\lvert \lambda_i </ins>\rvert \). Z věty \ref{KSOR} pak plyne divergence metody pro \( \omega \in \mathbbm R \setminus \langle 0 , 2 \rangle&#160; \).</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\end{proof}</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\end{proof}</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\end{theorem}</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\end{theorem}</div></td></tr> </table> Kubuondr https://wikiskripta.fjfi.cvut.cz/wiki/index.php?title=01NUM1:Kapitola5&diff=7544&oldid=prev Kubuondr v 31. 1. 2017, 09:40 2017-01-31T09:40:15Z <p></p> <table class='diff diff-contentalign-left'> <col class='diff-marker' /> <col class='diff-content' /> <col class='diff-marker' /> <col class='diff-content' /> <tr style='vertical-align: top;'> <td colspan='2' style="background-color: white; color:black; text-align: center;">← Starší verze</td> <td colspan='2' style="background-color: white; color:black; text-align: center;">Verze z 31. 1. 2017, 09:40</td> </tr><tr><td colspan="2" class="diff-lineno" id="L469" >Řádka 469:</td> <td colspan="2" class="diff-lineno">Řádka 469:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Kde poslední rovnost plyne z Jordanovy věty a rozpisu nahoře. V absolutních hodnotách tedy máme: \[\lvert 1-\omega\rvert^n = \prod_{i=1}^n \lvert \lambda_i \rvert\]</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Kde poslední rovnost plyne z Jordanovy věty a rozpisu nahoře. V absolutních hodnotách tedy máme: \[\lvert 1-\omega\rvert^n = \prod_{i=1}^n \lvert \lambda_i \rvert\]</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Pak je splněno buď I) všechna \( \lambda_i \) jsou stejná, tj, \(\lvert 1-\omega\rvert^n = \lvert \lambda_i \rvert^n \) a tedy nastává rovnost \(\rho (\matice B_{\omega}) = \lvert \lambda_i \rvert \) &#160;</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Pak je splněno buď I) všechna \( \lambda_i \) jsou stejná, tj, \(\lvert 1-\omega\rvert^n = \lvert \lambda_i \rvert^n \) a tedy nastává rovnost \(\rho (\matice B_{\omega}) = \lvert \lambda_i \rvert \) &#160;</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; 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>anebo II) existuje takové \(\lambda_{i_1} \), že \(\lvert \lambda_{i_1} \rvert &lt; \lvert 1-\omega \rvert\), pak ale musí zároveň existovat takové \(\lambda_{i_2}\), že \(\lvert \lambda_{i_2} \rvert &gt; \lvert 1-\omega \rvert\) a nastává tudíž \(\rho (\matice B_{\omega}) &gt; \lvert \<del class="diffchange diffchange-inline">lambda_i </del>\rvert \). Z věty \ref{KSOR} pak plyne divergence metody pro \( \omega \in \mathbbm R \setminus \langle 0 , 2 \rangle&#160; \).</div></td><td class='diff-marker'>+</td><td style="color:black; 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>anebo II) existuje takové \(\lambda_{i_1} \), že \(\lvert \lambda_{i_1} \rvert &lt; \lvert 1-\omega \rvert\), pak ale musí zároveň existovat takové \(\lambda_{i_2}\), že \(\lvert \lambda_{i_2} \rvert &gt; \lvert 1-\omega \rvert\) a nastává tudíž \(\rho (\matice B_{\omega}) &gt; \lvert \<ins class="diffchange diffchange-inline">lambda_{i_2} </ins>\rvert \). Z věty \ref{KSOR} pak plyne divergence metody pro \( \omega \in \mathbbm R \setminus \langle 0 , 2 \rangle&#160; \).</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\end{proof}</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\end{proof}</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\end{theorem}</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\end{theorem}</div></td></tr> </table> Kubuondr https://wikiskripta.fjfi.cvut.cz/wiki/index.php?title=01NUM1:Kapitola5&diff=7543&oldid=prev Kubuondr v 31. 1. 2017, 09:38 2017-01-31T09:38:26Z <p></p> <table class='diff diff-contentalign-left'> <col class='diff-marker' /> <col class='diff-content' /> <col class='diff-marker' /> <col class='diff-content' /> <tr style='vertical-align: top;'> <td colspan='2' style="background-color: white; color:black; text-align: center;">← Starší verze</td> <td colspan='2' style="background-color: white; color:black; text-align: center;">Verze z 31. 1. 2017, 09:38</td> </tr><tr><td colspan="2" class="diff-lineno" id="L468" >Řádka 468:</td> <td colspan="2" class="diff-lineno">Řádka 468:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\[\det(\matice B_{\omega}) = \frac{1}{\prod_{i=1}^n \matice A_{ii}} \prod_{i=1}^n (1-\omega) \matice A_{ii}} = (1-\omega)^n = \prod_{i=1}^n \lambda_i\]</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\[\det(\matice B_{\omega}) = \frac{1}{\prod_{i=1}^n \matice A_{ii}} \prod_{i=1}^n (1-\omega) \matice A_{ii}} = (1-\omega)^n = \prod_{i=1}^n \lambda_i\]</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Kde poslední rovnost plyne z Jordanovy věty a rozpisu nahoře. V absolutních hodnotách tedy máme: \[\lvert 1-\omega\rvert^n = \prod_{i=1}^n \lvert \lambda_i \rvert\]</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Kde poslední rovnost plyne z Jordanovy věty a rozpisu nahoře. V absolutních hodnotách tedy máme: \[\lvert 1-\omega\rvert^n = \prod_{i=1}^n \lvert \lambda_i \rvert\]</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; 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>Pak je splněno buď I) všechna \( \lambda_i \) jsou stejná, tj, \(\lvert 1-\omega\rvert^n = \lvert \lambda_i \rvert<del class="diffchange diffchange-inline">\</del>^n \) a tedy nastává rovnost \(\rho (\matice B_{\omega}) = \lvert \lambda_i \rvert \) &#160;</div></td><td class='diff-marker'>+</td><td style="color:black; 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>Pak je splněno buď I) všechna \( \lambda_i \) jsou stejná, tj, \(\lvert 1-\omega\rvert^n = \lvert \lambda_i \rvert^n \) a tedy nastává rovnost \(\rho (\matice B_{\omega}) = \lvert \lambda_i \rvert \) &#160;</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>anebo II) existuje takové \(\lambda_{i_1} \), že \(\lvert \lambda_{i_1} \rvert &lt; \lvert 1-\omega \rvert\), pak ale musí zároveň existovat takové \(\lambda_{i_2}\), že \(\lvert \lambda_{i_2} \rvert &gt; \lvert 1-\omega \rvert\) a nastává tudíž \(\rho (\matice B_{\omega}) &gt; \lvert \lambda_i \rvert \). Z věty \ref{KSOR} pak plyne divergence metody pro \( \omega \in \mathbbm R \setminus \langle 0 , 2 \rangle&#160; \).</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>anebo II) existuje takové \(\lambda_{i_1} \), že \(\lvert \lambda_{i_1} \rvert &lt; \lvert 1-\omega \rvert\), pak ale musí zároveň existovat takové \(\lambda_{i_2}\), že \(\lvert \lambda_{i_2} \rvert &gt; \lvert 1-\omega \rvert\) a nastává tudíž \(\rho (\matice B_{\omega}) &gt; \lvert \lambda_i \rvert \). Z věty \ref{KSOR} pak plyne divergence metody pro \( \omega \in \mathbbm R \setminus \langle 0 , 2 \rangle&#160; \).</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\end{proof}</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\end{proof}</div></td></tr> </table> Kubuondr https://wikiskripta.fjfi.cvut.cz/wiki/index.php?title=01NUM1:Kapitola5&diff=7541&oldid=prev Kubuondr v 31. 1. 2017, 09:31 2017-01-31T09:31:42Z <p></p> <table class='diff diff-contentalign-left'> <col class='diff-marker' /> <col class='diff-content' /> <col class='diff-marker' /> <col class='diff-content' /> <tr style='vertical-align: top;'> <td colspan='2' style="background-color: white; color:black; text-align: center;">← Starší verze</td> <td colspan='2' style="background-color: white; color:black; text-align: center;">Verze z 31. 1. 2017, 09:31</td> </tr><tr><td colspan="2" class="diff-lineno" id="L464" >Řádka 464:</td> <td colspan="2" class="diff-lineno">Řádka 464:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\begin{proof}</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\begin{proof}</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Pro důkaz využijeme toho, že determinant nezávisí na volbě báze. Podle označení je \[\matice B_\omega = (\matice D - \omega \matice L)^{-1}\big[(1-\omega)\matice D + \omega \matice R\big] \]</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Pro důkaz využijeme toho, že determinant nezávisí na volbě báze. Podle označení je \[\matice B_\omega = (\matice D - \omega \matice L)^{-1}\big[(1-\omega)\matice D + \omega \matice R\big] \]</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; 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>Z Jordanovy věty zároveň víme: \[\matice A = (\matice T)^{-1} \matice {JT} \Rightarrow det(\matice A) = det(\matice J) = \prod_{i=1}^n \lambda_i\]</div></td><td class='diff-marker'>+</td><td style="color:black; 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>Z Jordanovy věty zároveň víme: \[\matice A = (\matice T)^{-1} \matice {JT} \Rightarrow <ins class="diffchange diffchange-inline">\</ins>det(\matice A) = <ins class="diffchange diffchange-inline">\</ins>det(\matice J) = \prod_{i=1}^n \lambda_i\]</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Aplikujeme tuto znalost na naši matici (členy determinantu za \(\matice L\) a \(\matice R\) vypadnou, neboť jsou to singulární matice:</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Aplikujeme tuto znalost na naši matici (členy determinantu za \(\matice L\) a \(\matice R\) vypadnou, neboť jsou to singulární matice:</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; 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>\[det(\matice B_{\omega}) = \frac{1}{\prod_{i=1}^n \matice A_{ii}} \prod_{i=1}^n (1-\omega) \matice A_{ii}} = (1-\omega)^n = \prod_{i=1}^n \lambda_i\]</div></td><td class='diff-marker'>+</td><td style="color:black; 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>\[<ins class="diffchange diffchange-inline">\</ins>det(\matice B_{\omega}) = \frac{1}{\prod_{i=1}^n \matice A_{ii}} \prod_{i=1}^n (1-\omega) \matice A_{ii}} = (1-\omega)^n = \prod_{i=1}^n \lambda_i\]</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Kde poslední rovnost plyne z Jordanovy věty a rozpisu nahoře. V absolutních hodnotách tedy máme: \[\lvert 1-\omega\rvert^n = \prod_{i=1}^n \lvert \lambda_i \rvert\]</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Kde poslední rovnost plyne z Jordanovy věty a rozpisu nahoře. V absolutních hodnotách tedy máme: \[\lvert 1-\omega\rvert^n = \prod_{i=1}^n \lvert \lambda_i \rvert\]</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Pak je splněno buď I) všechna \( \lambda_i \) jsou stejná, tj, \(\lvert 1-\omega\rvert^n = \lvert \lambda_i \rvert\^n \) a tedy nastává rovnost \(\rho (\matice B_{\omega}) = \lvert \lambda_i \rvert \) &#160;</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Pak je splněno buď I) všechna \( \lambda_i \) jsou stejná, tj, \(\lvert 1-\omega\rvert^n = \lvert \lambda_i \rvert\^n \) a tedy nastává rovnost \(\rho (\matice B_{\omega}) = \lvert \lambda_i \rvert \) &#160;</div></td></tr> </table> Kubuondr https://wikiskripta.fjfi.cvut.cz/wiki/index.php?title=01NUM1:Kapitola5&diff=7540&oldid=prev Kubuondr: poznámka k 5.11 2017-01-30T19:33:11Z <p>poznámka k 5.11</p> <table class='diff diff-contentalign-left'> <col class='diff-marker' /> <col class='diff-content' /> <col class='diff-marker' /> <col class='diff-content' /> <tr style='vertical-align: top;'> <td colspan='2' style="background-color: white; color:black; text-align: center;">← Starší verze</td> <td colspan='2' style="background-color: white; color:black; text-align: center;">Verze z 30. 1. 2017, 19:33</td> </tr><tr><td colspan="2" class="diff-lineno" id="L172" >Řádka 172:</td> <td colspan="2" class="diff-lineno">Řádka 172:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\end{proof}</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\end{proof}</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\end{theorem}</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\end{theorem}</div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color:black; 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;">\begin{remark*}</ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color:black; 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;">Nejspíše platí i pro nehermitovské matice. Všechny věty, které jsou v důkazu použité, platí bez ohledu na to, zda jsou vlastní čísla reálná či komplexní, pokud se nahradí interval kruhem v \(\matice C\) obsahujícím daný interval.</ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color:black; 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;">\end{remark*}</ins></div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\subsection{Předpodmíněná metoda postupných aproximací}</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\subsection{Předpodmíněná metoda postupných aproximací}</div></td></tr> </table> Kubuondr https://wikiskripta.fjfi.cvut.cz/wiki/index.php?title=01NUM1:Kapitola5&diff=7539&oldid=prev Kubuondr v 30. 1. 2017, 17:37 2017-01-30T17:37:48Z <p></p> <table class='diff diff-contentalign-left'> <col class='diff-marker' /> <col class='diff-content' /> <col class='diff-marker' /> <col class='diff-content' /> <tr style='vertical-align: top;'> <td colspan='2' style="background-color: white; color:black; text-align: center;">← Starší verze</td> <td colspan='2' style="background-color: white; color:black; text-align: center;">Verze z 30. 1. 2017, 17:37</td> </tr><tr><td colspan="2" class="diff-lineno" id="L413" >Řádka 413:</td> <td colspan="2" class="diff-lineno">Řádka 413:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\end{theorem}</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\end{theorem}</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; 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>\subsection{Super-relaxační metoda}</div></td><td class='diff-marker'>+</td><td style="color:black; 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>\subsection{Super-relaxační metoda <ins class="diffchange diffchange-inline">(SOR – Succesive Over Relaxation)</ins>}</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Upravíme předpis pro Gaussovu-Seidelovu metodu do tvaru</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Upravíme předpis pro Gaussovu-Seidelovu metodu do tvaru</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\[ \vec x_i^{( k + 1 )} = \vec x_i^{( k )} + \Delta \vec x_i^{( k )} \]</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\[ \vec x_i^{( k + 1 )} = \vec x_i^{( k )} + \Delta \vec x_i^{( k )} \]</div></td></tr> </table> Kubuondr https://wikiskripta.fjfi.cvut.cz/wiki/index.php?title=01NUM1:Kapitola5&diff=7538&oldid=prev Kubuondr: Důkaz 5.10 jinak (podle přednášky). 2017-01-30T16:21:00Z <p>Důkaz 5.10 jinak (podle přednášky).</p> <table class='diff diff-contentalign-left'> <col class='diff-marker' /> <col class='diff-content' /> <col class='diff-marker' /> <col class='diff-content' /> <tr style='vertical-align: top;'> <td colspan='2' style="background-color: white; color:black; text-align: center;">← Starší verze</td> <td colspan='2' style="background-color: white; color:black; text-align: center;">Verze z 30. 1. 2017, 16:21</td> </tr><tr><td colspan="2" class="diff-lineno" id="L155" >Řádka 155:</td> <td colspan="2" class="diff-lineno">Řádka 155:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\end{proof}</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\end{proof}</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\end{theorem}</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\end{theorem}</div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color:black; 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;">\begin{remark*}\textit{(Důkaz z přednášky)}</ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color:black; 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;">Z \ref{MocninaJordan} víme, že se při mocnění umocňuje spektrum.&#160; </ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color:black; 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;">S pomocí Jordanovy věty dále zjistíme, že pro násobení platí:</ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color:black; 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;">\[a^k\matice A^k=a^k \matice X^-1\matice J^k \matice X=\matice X^-1 (a\matice J)^k \matice X\]</ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color:black; 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;">Obdobně pro sčítání. Z toho je vidět, spektrum součtu matice obsahuje součet vlastních čísel. Složením těchto operací definuje polynom.</ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color:black; 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;">Platí tedy \(p(\matice A)=matice X^-1 p(\matice J) \matice X\), což dokazuje větu.</ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color:black; 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;">\end{remark*}</ins></div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\begin{theorem}</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\begin{theorem}</div></td></tr> </table> Kubuondr https://wikiskripta.fjfi.cvut.cz/wiki/index.php?title=01NUM1:Kapitola5&diff=7485&oldid=prev Kubuondr v 26. 1. 2017, 16:23 2017-01-26T16:23:26Z <p></p> <table class='diff diff-contentalign-left'> <col class='diff-marker' /> <col class='diff-content' /> <col class='diff-marker' /> <col class='diff-content' /> <tr style='vertical-align: top;'> <td colspan='2' style="background-color: white; color:black; text-align: center;">← Starší verze</td> <td colspan='2' style="background-color: white; color:black; text-align: center;">Verze z 26. 1. 2017, 16:23</td> </tr><tr><td colspan="2" class="diff-lineno" id="L318" >Řádka 318:</td> <td colspan="2" class="diff-lineno">Řádka 318:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Konvergence je navíc monotónní vzhledem k normě \( \lVert \, \cdot \, \rVert_{\matice A} \)</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Konvergence je navíc monotónní vzhledem k normě \( \lVert \, \cdot \, \rVert_{\matice A} \)</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\begin{proof}</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\begin{proof}</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; 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>Protože je matice \( \matice A \) hermitovská, jsou diagonální prvky \( \matice A_{ii} \in \mathbbm R \) (musí platit \( \matice A_{ii} = \overline{\matice A_{ii}} \)), tudíž \( \matice D = \matice D^* \). Protože je navíc pozitivné definitní, platí \( \vec x^* \matice A \vec x &gt; 0 \). Zvolíme–li \(\vec x = \vec e_{(i)} \), kde \( \vec e_{(i)} \) jsou vektory ze standardní báze, dostaneme \( \matice A_{ii} = \ matice D_{ii} = \vec e_{(i)}^* \matice A \vec e_{(i)} &gt; 0 \), tj. \(\matice D \) je pozitivně definitní.</div></td><td class='diff-marker'>+</td><td style="color:black; 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>Protože je matice \( \matice A \) hermitovská, jsou diagonální prvky \( \matice A_{ii} \in \mathbbm R \) (musí platit \( \matice A_{ii} = \overline{\matice A_{ii}} \)), tudíž \( \matice D = \matice D^* \). Protože je navíc pozitivné definitní, platí \( \vec x^* \matice A \vec x &gt; 0 \). Zvolíme–li \(\vec x = \vec e_{(i)} \), kde \( \vec e_{(i)} \) jsou vektory ze standardní báze, dostaneme \( \matice A_{ii} = \matice D_{ii} = \vec e_{(i)}^* \matice A \vec e_{(i)} &gt; 0 \), tj. \(\matice D \) je pozitivně definitní.</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\begin{enumerate}</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\begin{enumerate}</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\item[( \( \Leftarrow \) )] Upravíme předpis Jacobiho metody do tvaru</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\item[( \( \Leftarrow \) )] Upravíme předpis Jacobiho metody do tvaru</div></td></tr> </table> Kubuondr https://wikiskripta.fjfi.cvut.cz/wiki/index.php?title=01NUM1:Kapitola5&diff=7484&oldid=prev Kubuondr v 26. 1. 2017, 13:48 2017-01-26T13:48:29Z <p></p> <table class='diff diff-contentalign-left'> <col class='diff-marker' /> <col class='diff-content' /> <col class='diff-marker' /> <col class='diff-content' /> <tr style='vertical-align: top;'> <td colspan='2' style="background-color: white; color:black; text-align: center;">← Starší verze</td> <td colspan='2' style="background-color: white; color:black; text-align: center;">Verze z 26. 1. 2017, 13:48</td> </tr><tr><td colspan="2" class="diff-lineno" id="L110" >Řádka 110:</td> <td colspan="2" class="diff-lineno">Řádka 110:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>platí</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>platí</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\[ \left\lVert \vec x^{(k)} - \vec x \right\rVert \leq \lVert \matice B \rVert^k \left( \left\lVert \vec x^{(0)} \right\rVert + \frac{\lVert \vec c \rVert}{1 - \lVert \matice B \rVert} \right) \]</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\[ \left\lVert \vec x^{(k)} - \vec x \right\rVert \leq \lVert \matice B \rVert^k \left( \left\lVert \vec x^{(0)} \right\rVert + \frac{\lVert \vec c \rVert}{1 - \lVert \matice B \rVert} \right) \]</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; 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>kde používaná vektorová norma je souhlasná s normou maticovou.</div></td><td class='diff-marker'>+</td><td style="color:black; 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>kde používaná vektorová norma je <ins class="diffchange diffchange-inline">\textbf{</ins>souhlasná<ins class="diffchange diffchange-inline">} </ins>s normou maticovou.</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\begin{proof}</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\begin{proof}</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\[ \vec x^{(k)} = \matice B \vec x^{(k - 1)} + \vec c = \dots = \matice B^k \vec x^{(0)} + \sum_{i = 0}^{k - 1} \matice B^i \vec c \]</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\[ \vec x^{(k)} = \matice B \vec x^{(k - 1)} + \vec c = \dots = \matice B^k \vec x^{(0)} + \sum_{i = 0}^{k - 1} \matice B^i \vec c \]</div></td></tr> </table> Kubuondr https://wikiskripta.fjfi.cvut.cz/wiki/index.php?title=01NUM1:Kapitola5&diff=7483&oldid=prev Kubuondr: změněn nadpis. 2017-01-26T12:38:42Z <p>změněn nadpis.</p> <table class='diff diff-contentalign-left'> <col class='diff-marker' /> <col class='diff-content' /> <col class='diff-marker' /> <col class='diff-content' /> <tr style='vertical-align: top;'> <td colspan='2' style="background-color: white; color:black; text-align: center;">← Starší verze</td> <td colspan='2' style="background-color: white; color:black; text-align: center;">Verze z 26. 1. 2017, 12:38</td> </tr><tr><td colspan="2" class="diff-lineno" id="L1" >Řádka 1:</td> <td colspan="2" class="diff-lineno">Řádka 1:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>%\wikiskriptum{01NUM1}</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>%\wikiskriptum{01NUM1}</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; 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>\section{Iterativní metody}</div></td><td class='diff-marker'>+</td><td style="color:black; 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>\section{Iterativní metody <ins class="diffchange diffchange-inline">– úvod a soustavy lineárních rovnic</ins>}</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\subsection{Iterativní metody obecně}</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\subsection{Iterativní metody obecně}</div></td></tr> </table> Kubuondr