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z5ONO;-v?VSfo=0(I}d-j<09C}tJ}p(+&u~QOo6?R!N_;8&jChxqEXh{&&mh5)3FhA zlWRyRDH9oAGGpZ;GapQ<5LucqsZ!+QqDfVv<18k5M0;mVsump_H>pPS!TPd&_*H@M3A3^mG!6Je33T19&b98cLVQrW1_5u|OGd4F0B_%~q FMhcdB_RjzS diff --git a/sose2020/num/uebungen/num8.tex b/sose2020/num/uebungen/num8.tex index cc6efab..4c7adc9 100644 --- a/sose2020/num/uebungen/num8.tex +++ b/sose2020/num/uebungen/num8.tex @@ -18,7 +18,7 @@ Es ist \begin{salign*} F(x) &= \frac{1}{2} \sum_{i,j=1}^{n} a_{ij}x_j x_i - \sum_{i=1}^{n} b_i x_i - \intertext{Damit folgt} + \intertext{Da $A$ symmetrisch, gilt $a_{ij} = a_{ji}$. Damit folgt} \frac{\partial F}{\partial x_i} &= \frac{1}{2} \left(2 \sum_{j=1, i\neq j}^{n} a_{ij}x_j + 2 a_{ii} x_i \right) - b_i \\ &= \sum_{j=1}^{n} a_{ij} x_j - b_i \\ @@ -183,10 +183,10 @@ \[ \frac{\partial F}{\partial x_j} = \sum_{i=1}^{n} 2 f_i(x) \frac{\partial f_i}{\partial x_j} = 2 \left( f(x), (J_f(x))_{j-\text{te Spalte}} \right)_2 - \implies \nabla F = 2 J_f(x^{k}) f(x^{k}) + \implies \nabla F = 2 J_f(x^{k})^{T} f(x^{k}) .\] Damit folgt \[ - x^{k+1} = x^{k} - 2 \alpha_{opt} J_f(x^{k}) f(x^{k}) + x^{k+1} = x^{k} - 2 \alpha_{opt} J_f(x^{k})^{T} f(x^{k}) .\] \item Newton-Verfahren: \[