https://wikiskripta.fjfi.cvut.cz/wiki/index.php?title=01NUM1:Kapitola9&feed=atom&action=history 01NUM1:Kapitola9 - Historie editací 2024-03-28T12:24:11Z Historie editací této stránky MediaWiki 1.25.2 https://wikiskripta.fjfi.cvut.cz/wiki/index.php?title=01NUM1:Kapitola9&diff=7566&oldid=prev Kubuondr v 31. 1. 2017, 16:33 2017-01-31T16:33:48Z <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 31. 1. 2017, 16:33</td> </tr><tr><td colspan="2" class="diff-lineno" id="L56" >Řádka 56:</td> <td colspan="2" class="diff-lineno">Řádka 56:</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>\[\int_{a}^{b} L_1(x) dx = \frac{1}{2} h (f(a) + f(b))\]</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>\[\int_{a}^{b} L_1(x) dx = \frac{1}{2} h (f(a) + f(b))\]</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>\[E_1(f) = \int_{a}^{b} f(x) - f(x_0) - \frac{f(x_1) - f(x_0)}{x_1 - x_0} (x - x_0) dx \]</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>\[E_1(f) = \int_{a}^{b} f(x) - f(x_0) - \frac{f(x_1) - f(x_0)}{x_1 - x_0} (x - x_0) dx \]</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>\[E_1(f) = \int_{a}^{b} R_1(x) dx = \int_{a}^{b} \frac{f''(\xi)}{2} (x-x_0)(x-x_1)dx = \frac{c}{2} \int_{0}^{h} t (t-h) dt =&#160; \frac{c}{2} \bigg[\frac{t^3}{3} - h \frac{t^2}{2} \bigg]^h_0 = \frac{c}{2} (\frac{h^3}{3} - h \frac{h^<del class="diffchange diffchange-inline">3</del>}{2}) = \mathcal{O}(h^3)\]</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>\[E_1(f) = \int_{a}^{b} R_1(x) dx = \int_{a}^{b} \frac{f''(\xi)}{2} (x-x_0)(x-x_1)dx = \frac{c}{2} \int_{0}^{h} t (t-h) dt =&#160; \frac{c}{2} \bigg[\frac{t^3}{3} - h \frac{t^2}{2} \bigg]^h_0 = \frac{c}{2} (\frac{h^3}{3} - h \frac{h^<ins class="diffchange diffchange-inline">2</ins>}{2}) = \mathcal{O}(h^3)\]</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>Znovu jsme použili větu o střední hodnotě integrálu a odhad \(\lvert f''(\xi) \rvert \leq c\).</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>Znovu jsme použili větu o střední hodnotě integrálu a odhad \(\lvert f''(\xi) \rvert \leq c\).</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>Přednášku z ne úplně zjevného důvodu zakončila Cavalieri-Simpsonova formule:</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>Přednášku z ne úplně zjevného důvodu zakončila Cavalieri-Simpsonova formule:</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>\[I_2(f) = \frac{a-b}{6}\left(f(a) + 4f\left(\frac{a+b}{2}\right) + f(b)\right)\]</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>\[I_2(f) = \frac{a-b}{6}\left(f(a) + 4f\left(\frac{a+b}{2}\right) + f(b)\right)\]</div></td></tr> </table> Kubuondr https://wikiskripta.fjfi.cvut.cz/wiki/index.php?title=01NUM1:Kapitola9&diff=7565&oldid=prev Kubuondr v 31. 1. 2017, 16:33 2017-01-31T16:33:11Z <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, 16:33</td> </tr><tr><td colspan="2" class="diff-lineno" id="L54" >Řádka 54:</td> <td colspan="2" class="diff-lineno">Řádka 54:</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>Tedy odhad máme s přesností \(h^3\). Zkusíme se nyní přesunout k Lagrangeově polynomu vyššího řádu, vezměme \(n = 1\).</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>Tedy odhad máme s přesností \(h^3\). Zkusíme se nyní přesunout k Lagrangeově polynomu vyššího řádu, vezměme \(n = 1\).</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>\[L_1(x) = f(x_0) + \frac{f(x_1) - f(x_0)}{x_1 - x_0} (x - x_0)\]</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>\[L_1(x) = f(x_0) + \frac{f(x_1) - f(x_0)}{x_1 - x_0} (x - x_0)\]</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>\[\int_{a}^{b} L_1(x) dx = \frac{1}{2} h (f(a) + <del class="diffchange diffchange-inline">(</del>f(b))\]</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>\[\int_{a}^{b} L_1(x) dx = \frac{1}{2} h (f(a) + f(b))\]</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>\[E_1(f) = \int_{a}^{b} f(x) - f(x_0) - \frac{f(x_1) - f(x_0)}{x_1 - x_0} (x - x_0) dx \]</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>\[E_1(f) = \int_{a}^{b} f(x) - f(x_0) - \frac{f(x_1) - f(x_0)}{x_1 - x_0} (x - x_0) dx \]</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>\[E_1(f) = \int_{a}^{b} R_1(x) dx = \int_{a}^{b} \frac{f''(\xi)}{2} (x-x_0)(x-x_1)dx = \frac{c}{2} \int_{0}^{h} t (t-h) dt =&#160; \frac{c}{2} \bigg[\frac{t^3}{3} - h \frac{t^2}{2} \bigg]^h_0 = \frac{c}{2} (\frac{h^3}{3} - h \frac{h^3}{2}) = \mathcal{O}(h^3)\]</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>\[E_1(f) = \int_{a}^{b} R_1(x) dx = \int_{a}^{b} \frac{f''(\xi)}{2} (x-x_0)(x-x_1)dx = \frac{c}{2} \int_{0}^{h} t (t-h) dt =&#160; \frac{c}{2} \bigg[\frac{t^3}{3} - h \frac{t^2}{2} \bigg]^h_0 = \frac{c}{2} (\frac{h^3}{3} - h \frac{h^3}{2}) = \mathcal{O}(h^3)\]</div></td></tr> </table> Kubuondr https://wikiskripta.fjfi.cvut.cz/wiki/index.php?title=01NUM1:Kapitola9&diff=7564&oldid=prev Kubuondr: oprava překlepů v předchozí editaci. 2017-01-31T16:13:09Z <p>oprava překlepů v předchozí editaci.</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, 16:13</td> </tr><tr><td colspan="2" class="diff-lineno" id="L29" >Řádka 29:</td> <td colspan="2" class="diff-lineno">Řádka 29:</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>\[\frac{f(x_1)-f(x_0)}{h} = f'(x_0) + \frac{1}{2} f''(x_0)h + \frac{1}{3!}f'''(\xi_1)h^2 = f'(x_0) + \mathcal{O} (h)\]</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>\[\frac{f(x_1)-f(x_0)}{h} = f'(x_0) + \frac{1}{2} f''(x_0)h + \frac{1}{3!}f'''(\xi_1)h^2 = f'(x_0) + \mathcal{O} (h)\]</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>\[\frac{f(x_0)-f(x_{-1})}{h} = f'(x_0) + \mathcal{O} (h)\]</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>\[\frac{f(x_0)-f(x_{-1})}{h} = f'(x_0) + \mathcal{O} (h)\]</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>Za předpokladu spojité diferencovatelnosti druhého řádu jsme schopni aproximovat první derivaci s~přesností prvního řádu.</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>Za předpokladu spojité diferencovatelnosti druhého řádu jsme <ins class="diffchange diffchange-inline">tedy </ins>schopni aproximovat první derivaci s~přesností prvního řádu.</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>\[f(x_1) - f(x_{-1}) = 2h f'(x_0) + \underbrace{\left(\frac{1}{2}f''(x_0)h^2 - \frac{1}{2}f''(x_0)h^2\right)}_{=0} + \frac{1}{3!}\left(f'''(\xi_1)h^3+f'''(\xi_{-1})h^3\right)\]</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>\[f(x_1) - f(x_{-1}) = 2h f'(x_0) + \underbrace{\left(\frac{1}{2}f''(x_0)h^2 - \frac{1}{2}f''(x_0)h^2\right)}_{=0} + \frac{1}{3!}\left(f'''(\xi_1)h^3+f'''(\xi_{-1})h^3\right)\]</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>\[\frac{f(x_1) - f(x_{-1})}{2h} = f'(x_0)+\frac{f'''(\xi_1)+f'''(\xi_{-1})}{3!}h^2 = f'(x_0) + \mathcal{O} (h^2)\] za předpokladu spojité diferencovatelnosti do třetího řádu. Tvaru \(\frac{f(x_1) - f(x_{-1})}{2h}\) se říká centrální diference.</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>\[\frac{f(x_1) - f(x_{-1})}{2h} = f'(x_0)+\frac{f'''(\xi_1)+f'''(\xi_{-1})}{3!}h^2 = f'(x_0) + \mathcal{O} (h^2)\]</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 class="diffchange diffchange-inline"> </ins>za předpokladu spojité diferencovatelnosti do třetího řádu. Tvaru \(\frac{f(x_1) - f(x_{-1})}{2h}\) se říká centrální diference.</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>Protože \(\xi_1,\xi_{-1} \in \langle x_{-1};x_1\rangle\) a \(f\) je na \(\langle x_{-1};x_1\rangle\) spojitě diferencovatelná do třetího řádu, je \[\lvert \frac{1}{3!}(f'''(\xi_1)+f'''(\xi_{-1}) \rvert \leq C\] na \(\langle x_{-1};x_1\rangle\) tedy je omezená.</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>Protože \(\xi_1,\xi_{-1} \in \langle x_{-1};x_1\rangle\) a \(f\) je na \(\langle x_{-1};x_1\rangle\) spojitě diferencovatelná do třetího řádu, je \[\lvert \frac{1}{3!}(f'''(\xi_1)+f'''(\xi_{-1}) \rvert \leq C\] na \(\langle x_{-1};x_1\rangle\) tedy je omezená.</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>Přesuneme se k druhé derivaci. Rozepsání \(f(x_1)\) a \(f(x_{-1})\) tentokrát sečteme a dostaneme:</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>Přesuneme se k druhé derivaci. Rozepsání \(f(x_1)\) a \(f(x_{-1})\) tentokrát sečteme a dostaneme:</div></td></tr> <tr><td colspan="2" class="diff-lineno" id="L50" >Řádka 50:</td> <td colspan="2" class="diff-lineno">Řádka 51:</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>\[\int_{a}^{b} \left(f(x_0) + f'(x_0)(x-x_0) +\frac{1}{2} f''(\xi)(x-x_0)^2 - f(x_0)\right) dx = \int_{-\frac{h}{2}}^{\frac{h}{2}} f'(x_0)t +\frac{1}{2} f''(\xi)t^2 dt \leq \]</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>\[\int_{a}^{b} \left(f(x_0) + f'(x_0)(x-x_0) +\frac{1}{2} f''(\xi)(x-x_0)^2 - f(x_0)\right) dx = \int_{-\frac{h}{2}}^{\frac{h}{2}} f'(x_0)t +\frac{1}{2} f''(\xi)t^2 dt \leq \]</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>Za předpokladu \(f \in \mathcal C^{(2)}\) lze použít větu o střední hodnotě integrálu a odhadnout tak \(\lvert f''(\xi) \rvert \leq c\) (\(\xi\) totiž závisí na \(x\)):</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>Za předpokladu \(f \in \mathcal C^{(2)}\) lze použít větu o střední hodnotě integrálu a odhadnout tak \(\lvert f''(\xi) \rvert \leq c\) (\(\xi\) totiž závisí na \(x\)):</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>&#160; \[ \leq f'(x_0)\bigg[\frac{t^2}{2}\bigg]^\frac{h}{2}_-\frac{h}{2} + \frac{1}{2} c \bigg[\frac{t^3}{3}\bigg]^\frac{h}{2}_-\frac{h}{2} = \frac{1}{2} c \frac{h^3}{12} = \mathcal{O}(h^3)\]</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>&#160; \[ \leq f'(x_0)\bigg[\frac{t^2}{2}\bigg]^\frac{h}{2}_<ins class="diffchange diffchange-inline">{</ins>-\frac{h}{2<ins class="diffchange diffchange-inline">}</ins>} + \frac{1}{2} c \bigg[\frac{t^3}{3}\bigg]^\frac{h}{2}_<ins class="diffchange diffchange-inline">{</ins>-\frac{h}{2<ins class="diffchange diffchange-inline">}</ins>} = \frac{1}{2} c \frac{h^3}{12} = \mathcal{O}(h^3)\]</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>Tedy odhad máme s přesností \(h^3\). Zkusíme se nyní přesunout k Lagrangeově polynomu vyššího řádu, vezměme \(n = 1\).</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>Tedy odhad máme s přesností \(h^3\). Zkusíme se nyní přesunout k Lagrangeově polynomu vyššího řádu, vezměme \(n = 1\).</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>\[L_1(x) = f(x_0) + \frac{f(x_1) - f(x_0)}{x_1 - x_0} (x - x_0)\]</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>\[L_1(x) = f(x_0) + \frac{f(x_1) - f(x_0)}{x_1 - x_0} (x - x_0)\]</div></td></tr> </table> Kubuondr https://wikiskripta.fjfi.cvut.cz/wiki/index.php?title=01NUM1:Kapitola9&diff=7563&oldid=prev Kubuondr v 31. 1. 2017, 15:58 2017-01-31T15:58:41Z <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, 15:58</td> </tr><tr><td colspan="2" class="diff-lineno" id="L25" >Řádka 25:</td> <td colspan="2" class="diff-lineno">Řádka 25:</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>\[f(x_1) = f(x_0) + f'(x_0)\underbrace{(x_1-x_0)}_{h} + \frac{f''(x_0)}{2!}\underbrace{(x_1-x_0)^2}_{h^2} + \frac{f'''(\xi_1)}{3!}\underbrace{(x_1-x_0)^3}_{h^3}\]</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>\[f(x_1) = f(x_0) + f'(x_0)\underbrace{(x_1-x_0)}_{h} + \frac{f''(x_0)}{2!}\underbrace{(x_1-x_0)^2}_{h^2} + \frac{f'''(\xi_1)}{3!}\underbrace{(x_1-x_0)^3}_{h^3}\]</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>\[f(x_{-1}) = f(x_0) + f'(x_0)\underbrace{(x_{-1}-x_0)}_{-h} + \frac{f''(x_0)}{2!}\underbrace{(x_{-1}-x_0)^2}_{h^2} + \frac{f'''(\xi_{-1})}{3!}\underbrace{(x_{-1}-x_0)^3}_{-h^3}=\]</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>\[f(x_{-1}) = f(x_0) + f'(x_0)\underbrace{(x_{-1}-x_0)}_{-h} + \frac{f''(x_0)}{2!}\underbrace{(x_{-1}-x_0)^2}_{h^2} + \frac{f'''(\xi_{-1})}{3!}\underbrace{(x_{-1}-x_0)^3}_{-h^3}=\]</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>\[= f(x_0) - f'(x_0)<del class="diffchange diffchange-inline">\underbrace{(x_{-1}-x_0)}_{</del>h<del class="diffchange diffchange-inline">} </del>+ \frac{f''(x_0)}{2!}<del class="diffchange diffchange-inline">\underbrace{(x_{-1}-x_0)^2}_{</del>h^2<del class="diffchange diffchange-inline">} </del>- \frac{f'''(\xi_{-1})}{3!}<del class="diffchange diffchange-inline">\underbrace{(x_{-1}-x_0)^3}_{</del>h^3<del class="diffchange diffchange-inline">}</del>\]</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>\[= f(x_0) - f'(x_0)h + \frac{f''(x_0)}{2!}h^2 - \frac{f'''(\xi_{-1})}{3!}h^3\]</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>Vytvoříme dopředné, resp. zpětné diference prvního řádu:</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>Vytvoříme dopředné, resp. zpětné diference prvního řádu:</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>\[\frac{f(x_1)-f(x_0)}{h} = f'(x_0) + \frac{1}{2} f''(x_0)h + \frac{1}{3!}f'''(\xi_1)h^2 = f'(x_0) + \mathcal{O} (h)\]</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>\[\frac{f(x_1)-f(x_0)}{h} = f'(x_0) + \frac{1}{2} f''(x_0)h + \frac{1}{3!}f'''(\xi_1)h^2 = f'(x_0) + \mathcal{O} (h)\]</div></td></tr> </table> Kubuondr https://wikiskripta.fjfi.cvut.cz/wiki/index.php?title=01NUM1:Kapitola9&diff=7562&oldid=prev Kubuondr v 31. 1. 2017, 15:56 2017-01-31T15:56:52Z <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, 15:56</td> </tr><tr><td colspan="2" class="diff-lineno" id="L25" >Řádka 25:</td> <td colspan="2" class="diff-lineno">Řádka 25:</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>\[f(x_1) = f(x_0) + f'(x_0)\underbrace{(x_1-x_0)}_{h} + \frac{f''(x_0)}{2!}\underbrace{(x_1-x_0)^2}_{h^2} + \frac{f'''(\xi_1)}{3!}\underbrace{(x_1-x_0)^3}_{h^3}\]</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>\[f(x_1) = f(x_0) + f'(x_0)\underbrace{(x_1-x_0)}_{h} + \frac{f''(x_0)}{2!}\underbrace{(x_1-x_0)^2}_{h^2} + \frac{f'''(\xi_1)}{3!}\underbrace{(x_1-x_0)^3}_{h^3}\]</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>\[f(x_{-1}) = f(x_0) + f'(x_0)\underbrace{(x_{-1}-x_0)}_{-h} + \frac{f''(x_0)}{2!}\underbrace{(x_{-1}-x_0)^2}_{h^2} + \frac{f'''(\xi_{-1})}{3!}\underbrace{(x_{-1}-x_0)^3}_{-h^3}=\]</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>\[f(x_{-1}) = f(x_0) + f'(x_0)\underbrace{(x_{-1}-x_0)}_{-h} + \frac{f''(x_0)}{2!}\underbrace{(x_{-1}-x_0)^2}_{h^2} + \frac{f'''(\xi_{-1})}{3!}\underbrace{(x_{-1}-x_0)^3}_{-h^3}=\]</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>\[<del class="diffchange diffchange-inline">f(x_{-1}) </del>= f(x_0) - f'(x_0)\underbrace{(x_{-1}-x_0)}_{h} + \frac{f''(x_0)}{2!}\underbrace{(x_{-1}-x_0)^2}_{h^2} - \frac{f'''(\xi_{-1})}{3!}\underbrace{(x_{-1}-x_0)^3}_{h^3}<del class="diffchange diffchange-inline">=</del>\]</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>\[= f(x_0) - f'(x_0)\underbrace{(x_{-1}-x_0)}_{h} + \frac{f''(x_0)}{2!}\underbrace{(x_{-1}-x_0)^2}_{h^2} - \frac{f'''(\xi_{-1})}{3!}\underbrace{(x_{-1}-x_0)^3}_{h^3}\]</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>Vytvoříme dopředné, resp. zpětné diference prvního řádu:</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>Vytvoříme dopředné, resp. zpětné diference prvního řádu:</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>\[\frac{f(x_1)-f(x_0)}{h} = f'(x_0) + \frac{1}{2} f''(x_0)h + \frac{1}{3!}f'''(\xi_1)h^2 = f'(x_0) + \mathcal{O} (h)\]</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>\[\frac{f(x_1)-f(x_0)}{h} = f'(x_0) + \frac{1}{2} f''(x_0)h + \frac{1}{3!}f'''(\xi_1)h^2 = f'(x_0) + \mathcal{O} (h)\]</div></td></tr> </table> Kubuondr https://wikiskripta.fjfi.cvut.cz/wiki/index.php?title=01NUM1:Kapitola9&diff=7561&oldid=prev Kubuondr: úprava předpisu pro zpětnou diferenci. 2017-01-31T15:56:05Z <p>úprava předpisu pro zpětnou diferenci.</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, 15:56</td> </tr><tr><td colspan="2" class="diff-lineno" id="L24" >Řádka 24:</td> <td colspan="2" class="diff-lineno">Řádka 24:</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>Rozvineme podle Taylora:</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>Rozvineme podle Taylora:</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>\[f(x_1) = f(x_0) + f'(x_0)\underbrace{(x_1-x_0)}_{h} + \frac{f''(x_0)}{2!}\underbrace{(x_1-x_0)^2}_{h^2} + \frac{f'''(\xi_1)}{3!}\underbrace{(x_1-x_0)^3}_{h^3}\]</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>\[f(x_1) = f(x_0) + f'(x_0)\underbrace{(x_1-x_0)}_{h} + \frac{f''(x_0)}{2!}\underbrace{(x_1-x_0)^2}_{h^2} + \frac{f'''(\xi_1)}{3!}\underbrace{(x_1-x_0)^3}_{h^3}\]</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>\[f(x_{-1}) = f(x_0) + f'(x_0)\underbrace{(x_{-1}-x_0)}_{-h} + \frac{f''(x_0)}{2!}\underbrace{(x_{-1}-x_0)^2}_{h^2} + \frac{f'''(\xi_{-1})}{3!}\underbrace{(x_{-1}-x_0)^3}_{-h^3}\]</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>\[f(x_{-1}) = f(x_0) + f'(x_0)\underbrace{(x_{-1}-x_0)}_{-h} + \frac{f''(x_0)}{2!}\underbrace{(x_{-1}-x_0)^2}_{h^2} + \frac{f'''(\xi_{-1})}{3!}\underbrace{(x_{-1}-x_0)^3}_{-h^3}<ins class="diffchange diffchange-inline">=\]</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 class="diffchange diffchange-inline">\[f(x_{-1}) = f(x_0) - f'(x_0)\underbrace{(x_{-1}-x_0)}_{h} + \frac{f''(x_0)}{2!}\underbrace{(x_{-1}-x_0)^2}_{h^2} - \frac{f'''(\xi_{-1})}{3!}\underbrace{(x_{-1}-x_0)^3}_{h^3}=</ins>\]</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>Vytvoříme dopředné, resp. zpětné diference prvního řádu:</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>Vytvoříme dopředné, resp. zpětné diference prvního řádu:</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>\[\frac{f(x_1)-f(x_0)}{h} = f'(x_0) + \frac{1}{2} f''(x_0)h + \frac{1}{3!}f'''(\xi_1)h^2 = f'(x_0) + \mathcal{O} (h)\]</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>\[\frac{f(x_1)-f(x_0)}{h} = f'(x_0) + \frac{1}{2} f''(x_0)h + \frac{1}{3!}f'''(\xi_1)h^2 = f'(x_0) + \mathcal{O} (h)\]</div></td></tr> </table> Kubuondr https://wikiskripta.fjfi.cvut.cz/wiki/index.php?title=01NUM1:Kapitola9&diff=7560&oldid=prev Kubuondr v 31. 1. 2017, 15:54 2017-01-31T15:54:08Z <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, 15:54</td> </tr><tr><td colspan="2" class="diff-lineno" id="L24" >Řádka 24:</td> <td colspan="2" class="diff-lineno">Řádka 24:</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>Rozvineme podle Taylora:</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>Rozvineme podle Taylora:</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>\[f(x_1) = f(x_0) + f'(x_0)\underbrace{(x_1-x_0)}_{h} + \frac{f''(x_0)}{2!}\underbrace{(x_1-x_0)^2}_{h^2} + \frac{f'''(\xi_1)}{3!}\underbrace{(x_1-x_0)^3}_{h^3}\]</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>\[f(x_1) = f(x_0) + f'(x_0)\underbrace{(x_1-x_0)}_{h} + \frac{f''(x_0)}{2!}\underbrace{(x_1-x_0)^2}_{h^2} + \frac{f'''(\xi_1)}{3!}\underbrace{(x_1-x_0)^3}_{h^3}\]</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>\[f(x_{-1}) = f(x_0) + f'(x_0)\underbrace{(x_{-1}-x_0)}_{h} + \frac{f''(x_0)}{2!}\underbrace{(x_{-1}-x_0)^2}_{h^2} + \frac{f'''(\xi_{-1})}{3!}\underbrace{(x_{-1}-x_0)^3}_{h^3}\]</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>\[f(x_{-1}) = f(x_0) + f'(x_0)\underbrace{(x_{-1}-x_0)}_{<ins class="diffchange diffchange-inline">-</ins>h} + \frac{f''(x_0)}{2!}\underbrace{(x_{-1}-x_0)^2}_{h^2} + \frac{f'''(\xi_{-1})}{3!}\underbrace{(x_{-1}-x_0)^3}_{<ins class="diffchange diffchange-inline">-</ins>h^3}\]</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>Vytvoříme <del class="diffchange diffchange-inline">vlastně </del>dopředné, <del class="diffchange diffchange-inline">respektive </del>zpětné diference prvního řádu:</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>Vytvoříme dopředné, <ins class="diffchange diffchange-inline">resp. </ins>zpětné diference prvního řádu:</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>\[\frac{f(x_1)-f(x_0)}{h} = f'(x_0) + \frac{1}{2} f''(x_0)h + \frac{1}{3!}f'''(\xi_1)h^2 = f'(x_0) + \mathcal{O} (h)\]</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>\[\frac{f(x_1)-f(x_0)}{h} = f'(x_0) + \frac{1}{2} f''(x_0)h + \frac{1}{3!}f'''(\xi_1)h^2 = f'(x_0) + \mathcal{O} (h)\]</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>\[\frac{f(x_0)-f(x_{-1})}{h} = f'(x_0) + \mathcal{O} (h)\]</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>\[\frac{f(x_0)-f(x_{-1})}{h} = f'(x_0) + \mathcal{O} (h)\]</div></td></tr> </table> Kubuondr https://wikiskripta.fjfi.cvut.cz/wiki/index.php?title=01NUM1:Kapitola9&diff=7528&oldid=prev Kubuondr v 29. 1. 2017, 15:27 2017-01-29T15:27:30Z <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 29. 1. 2017, 15:27</td> </tr><tr><td colspan="2" class="diff-lineno" id="L48" >Řádka 48:</td> <td colspan="2" class="diff-lineno">Řádka 48:</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>Rozvineme \(f(x)\) Taylorem:</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>Rozvineme \(f(x)\) Taylorem:</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>\[\int_{a}^{b} \left(f(x_0) + f'(x_0)(x-x_0) +\frac{1}{2} f''(\xi)(x-x_0)^2 - f(x_0)\right) dx = \int_{-\frac{h}{2}}^{\frac{h}{2}} f'(x_0)t +\frac{1}{2} f''(\xi)t^2 dt \leq \]</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>\[\int_{a}^{b} \left(f(x_0) + f'(x_0)(x-x_0) +\frac{1}{2} f''(\xi)(x-x_0)^2 - f(x_0)\right) dx = \int_{-\frac{h}{2}}^{\frac{h}{2}} f'(x_0)t +\frac{1}{2} f''(\xi)t^2 dt \leq \]</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>Za předpokladu \(f \in \mathcal C^{(2)}\) lze použít větu o střední hodnotě integrálu a odhadnout tak \(\lvert f''(\xi) \rvert \leq c\):</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>Za předpokladu \(f \in \mathcal C^{(2)}\) lze použít větu o střední hodnotě integrálu a odhadnout tak \(\lvert f''(\xi) \rvert \leq c\<ins class="diffchange diffchange-inline">) (\(\xi\) totiž závisí na \(x\)</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>&#160; \[ \leq f'(x_0)\bigg[\frac{t^2}{2}\bigg]^\frac{h}{2}_-\frac{h}{2} + \frac{1}{2} c \bigg[\frac{t^3}{3}\bigg]^\frac{h}{2}_-\frac{h}{2} = \frac{1}{2} c \frac{h^3}{12} = \mathcal{O}(h^3)\]</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>&#160; \[ \leq f'(x_0)\bigg[\frac{t^2}{2}\bigg]^\frac{h}{2}_-\frac{h}{2} + \frac{1}{2} c \bigg[\frac{t^3}{3}\bigg]^\frac{h}{2}_-\frac{h}{2} = \frac{1}{2} c \frac{h^3}{12} = \mathcal{O}(h^3)\]</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>Tedy odhad máme s přesností \(h^3\). Zkusíme se nyní přesunout k Lagrangeově polynomu vyššího řádu, vezměme \(n = 1\).</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>Tedy odhad máme s přesností \(h^3\). Zkusíme se nyní přesunout k Lagrangeově polynomu vyššího řádu, vezměme \(n = 1\).</div></td></tr> <tr><td colspan="2" class="diff-lineno" id="L54" >Řádka 54:</td> <td colspan="2" class="diff-lineno">Řádka 54:</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>\[\int_{a}^{b} L_1(x) dx = \frac{1}{2} h (f(a) + (f(b))\]</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>\[\int_{a}^{b} L_1(x) dx = \frac{1}{2} h (f(a) + (f(b))\]</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>\[E_1(f) = \int_{a}^{b} f(x) - f(x_0) - \frac{f(x_1) - f(x_0)}{x_1 - x_0} (x - x_0) dx \]</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>\[E_1(f) = \int_{a}^{b} f(x) - f(x_0) - \frac{f(x_1) - f(x_0)}{x_1 - x_0} (x - x_0) dx \]</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>\[E_1(f) = \int_{a}^{b} R_1(x) dx = \int_{a}^{b} \frac{f''(\xi)}{2} (x-x_0)(x-x_1)dx = \frac{<del class="diffchange diffchange-inline">f''(\xi)</del>}{2} \int_{0}^{h} t (t-h) dt =&#160; \frac{<del class="diffchange diffchange-inline">f''(\xi)</del>}{2} \bigg[\frac{t^3}{3} - h \frac{t^2}{2} \bigg]^h_0 = \frac{<del class="diffchange diffchange-inline">f''(\xi)</del>}{2} (\frac{h^3}{3} - h \frac{h^3}{2}) = \mathcal{O}(h^3)\]</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>\[E_1(f) = \int_{a}^{b} R_1(x) dx = \int_{a}^{b} \frac{f''(\xi)}{2} (x-x_0)(x-x_1)dx = \frac{<ins class="diffchange diffchange-inline">c</ins>}{2} \int_{0}^{h} t (t-h) dt =&#160; \frac{<ins class="diffchange diffchange-inline">c</ins>}{2} \bigg[\frac{t^3}{3} - h \frac{t^2}{2} \bigg]^h_0 = \frac{<ins class="diffchange diffchange-inline">c</ins>}{2} (\frac{h^3}{3} - h \frac{h^3}{2}) = \mathcal{O}(h^3)\]</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>&#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><ins class="diffchange diffchange-inline">Znovu jsme použili větu o střední hodnotě integrálu a odhad \(\lvert f''(\xi) \rvert \leq c\).</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>Přednášku z ne úplně zjevného důvodu zakončila Cavalieri-Simpsonova formule:</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>Přednášku z ne úplně zjevného důvodu zakončila Cavalieri-Simpsonova formule:</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>\[I_2(f) = \frac{a-b}{6}\left(f(a) + 4f\left(\frac{a+b}{2}\right) + f(b)\right)\]</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>\[I_2(f) = \frac{a-b}{6}\left(f(a) + 4f\left(\frac{a+b}{2}\right) + f(b)\right)\]</div></td></tr> </table> Kubuondr https://wikiskripta.fjfi.cvut.cz/wiki/index.php?title=01NUM1:Kapitola9&diff=7527&oldid=prev Kubuondr v 29. 1. 2017, 15:24 2017-01-29T15:24:19Z <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 29. 1. 2017, 15:24</td> </tr><tr><td colspan="2" class="diff-lineno" id="L49" >Řádka 49:</td> <td colspan="2" class="diff-lineno">Řádka 49:</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>\[\int_{a}^{b} \left(f(x_0) + f'(x_0)(x-x_0) +\frac{1}{2} f''(\xi)(x-x_0)^2 - f(x_0)\right) dx = \int_{-\frac{h}{2}}^{\frac{h}{2}} f'(x_0)t +\frac{1}{2} f''(\xi)t^2 dt \leq \]</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>\[\int_{a}^{b} \left(f(x_0) + f'(x_0)(x-x_0) +\frac{1}{2} f''(\xi)(x-x_0)^2 - f(x_0)\right) dx = \int_{-\frac{h}{2}}^{\frac{h}{2}} f'(x_0)t +\frac{1}{2} f''(\xi)t^2 dt \leq \]</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>Za předpokladu \(f \in \mathcal C^{(2)}\) lze použít větu o střední hodnotě integrálu a odhadnout tak \(\lvert f''(\xi) \rvert \leq c\):</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>Za předpokladu \(f \in \mathcal C^{(2)}\) lze použít větu o střední hodnotě integrálu a odhadnout tak \(\lvert f''(\xi) \rvert \leq c\):</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>&#160; \[ \leq f'(x_0)\bigg[\frac{t^2}{2}\bigg]^\frac{h}{2}_\frac{<del class="diffchange diffchange-inline">-</del>h}{2} + \frac{1}{2} c \bigg[\frac{t^3}{3}\bigg]^\frac{h}{2}_\frac{<del class="diffchange diffchange-inline">-</del>h}{2} = \frac{1}{2} c \frac{h^3}{12} = \mathcal{O}(h^3)\]</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>&#160; \[ \leq f'(x_0)\bigg[\frac{t^2}{2}\bigg]^\frac{h}{2}_<ins class="diffchange diffchange-inline">-</ins>\frac{h}{2} + \frac{1}{2} c \bigg[\frac{t^3}{3}\bigg]^\frac{h}{2}_<ins class="diffchange diffchange-inline">-</ins>\frac{h}{2} = \frac{1}{2} c \frac{h^3}{12} = \mathcal{O}(h^3)\]</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>Tedy odhad máme s přesností \(h^3\). Zkusíme se nyní přesunout k Lagrangeově polynomu vyššího řádu, vezměme \(n = 1\).</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>Tedy odhad máme s přesností \(h^3\). Zkusíme se nyní přesunout k Lagrangeově polynomu vyššího řádu, vezměme \(n = 1\).</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>\[L_1(x) = f(x_0) + \frac{f(x_1) - f(x_0)}{x_1 - x_0} (x - x_0)\]</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>\[L_1(x) = f(x_0) + \frac{f(x_1) - f(x_0)}{x_1 - x_0} (x - x_0)\]</div></td></tr> </table> Kubuondr https://wikiskripta.fjfi.cvut.cz/wiki/index.php?title=01NUM1:Kapitola9&diff=7526&oldid=prev Kubuondr: Další opravy v poslední kapitole. 2017-01-29T15:07:42Z <p>Další opravy v poslední kapitole.</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 29. 1. 2017, 15:07</td> </tr><tr><td colspan="2" class="diff-lineno" id="L47" >Řádka 47:</td> <td colspan="2" class="diff-lineno">Řádka 47:</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>\[E_0(f) = \int_{a}^{b} f(x) dx - \int_{a}^{b} f(x_0) dx = \int_{a}^{b} \underbrace{(f(x) - \underbrace{f(x_0)}_{L_0(x)}) dx}_{R_0(x)}\] &#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>\[E_0(f) = \int_{a}^{b} f(x) dx - \int_{a}^{b} f(x_0) dx = \int_{a}^{b} \underbrace{(f(x) - \underbrace{f(x_0)}_{L_0(x)}) dx}_{R_0(x)}\] &#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>Rozvineme \(f(x)\) Taylorem:</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>Rozvineme \(f(x)\) Taylorem:</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>\[\int_{a}^{b} \left(f(x_0) + f'(x_0)(x-x_0) +\frac{1}{2} f''(\xi)(x-x_0)^2 - f(x_0)\right) dx = \int_{-\frac{h}{2}}^{\frac{h}{2}} f'(x_0)t +\frac{1}{2} f''(\xi)t^2 dt <del class="diffchange diffchange-inline">= </del>\] \[ f'(x_0)\bigg[\frac{t^2}{2}\bigg]^\frac{h}{2}_\frac{h}{2} + \frac{1}{2} <del class="diffchange diffchange-inline">f''(\xi) </del>\bigg[\frac{t^3}{3}\bigg]^\frac{h}{2}_\frac{h}{2} = \frac{1}{2} <del class="diffchange diffchange-inline">f''(\xi) </del>\frac{h^3}{12} = \mathcal{O}(h^3)\]</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>\[\int_{a}^{b} \left(f(x_0) + f'(x_0)(x-x_0) +\frac{1}{2} f''(\xi)(x-x_0)^2 - f(x_0)\right) dx = \int_{-\frac{h}{2}}^{\frac{h}{2}} f'(x_0)t +\frac{1}{2} f''(\xi)t^2 dt <ins class="diffchange diffchange-inline">\leq </ins>\]</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 class="diffchange diffchange-inline">Za předpokladu \(f \in \mathcal C^{(2)}\) lze použít větu o střední hodnotě integrálu a odhadnout tak \(\lvert f''(\xi) \rvert \leq c\):</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 class="diffchange diffchange-inline"> </ins>\[ <ins class="diffchange diffchange-inline">\leq </ins>f'(x_0)\bigg[\frac{t^2}{2}\bigg]^\frac{h}{2}_\frac{<ins class="diffchange diffchange-inline">-</ins>h}{2} + \frac{1}{2} <ins class="diffchange diffchange-inline">c </ins>\bigg[\frac{t^3}{3}\bigg]^\frac{h}{2}_\frac{<ins class="diffchange diffchange-inline">-</ins>h}{2} = \frac{1}{2} <ins class="diffchange diffchange-inline">c </ins>\frac{h^3}{12} = \mathcal{O}(h^3)\]</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>Tedy odhad máme s přesností \(h^3\). Zkusíme se nyní přesunout k Lagrangeově polynomu vyššího řádu, vezměme \(n = 1\).</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>Tedy odhad máme s přesností \(h^3\). Zkusíme se nyní přesunout k Lagrangeově polynomu vyššího řádu, vezměme \(n = 1\).</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>\[L_1(x) = f(x_0) + \frac{f(x_1) - f(x_0)}{x_1 - x_0} (x - x_0)\]</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>\[L_1(x) = f(x_0) + \frac{f(x_1) - f(x_0)}{x_1 - x_0} (x - x_0)\]</div></td></tr> </table> Kubuondr