Cholesky-Zerlegung symmetrischer Matrizen: A=L L[sup]T[/sup] [br][br][math]A=\left(\begin{array}{rrr}9&-15&3\\-15&41&23\\3&23&51\\\end{array}\right)[/math], [math]b=\left(\begin{array}{r}-3\\9\\5\\\end{array}\right)[/math][br][br]Allgemeine Form der Zerlegungsmatrix L[br][math]L_o \, := \, \left(\begin{array}{rrr}a11&0&0\\a12&a22&0\\a31&a32&a33\\\end{array}\right)[/math][br][br] A = L[sub]o[/sub] L[sub]o[/sub][sup]T[/sup] ===> GLS: L[sub]o[/sub] L[sub]o[/sub][sup]T[/sup]- A=0 ===>löse GLS[br][br][math]GLS0 \, := \, \left(\begin{array}{r}a11^{2} - 9\\a11 \; a21 + 15\\a11 \; a31 - 3\\a21^{2} + a22^{2} - 41\\a21 \; a31 + a22 \; a32 - 23\\a31^{2} + a32^{2} + a33^{2} - 51\\\end{array}\right)=0[/math] Obere Dreiecksmatrix: Einsetzen, von oben nach unten[br][br][math]L \, := \, \left(\begin{array}{rrr}3&0&0\\-5&4&0\\1&7&1\\\end{array}\right)[/math][br][br][math]A \; x = b[/math] [math]\to L \; L^{T} \; x = b[/math][br][br][math] L^{T} \; x = y \quad \to \quad L \; y = b [/math] [math] \to y1 = -1, y2 = 1, y3 = -1 [/math] [math] \to x1 = \frac{10}{3}, x2 = 2, x3 = -1 [/math][br]

A0 = {{16, -8, -20}, {-8, 40, 22}, {-20, 22, 45}}[br]A1 = {{9, 3, 0}, {3, 2, -5}, {0, -5, 50}}[br]A3={{1, 2, -4, 5}, {2, 40, -20, 10}, {-4, -20, 24, -20}, {5, 10, -20, 29}}[br]A4 = {{4, -8, 4, -4}, {-8, 17, -8, 13}, {4, -8, 5, -1}, {-4, 13, -1, 63}}[br]A5 = {{9, -15, 3}, {-15, 41, 23}, {3, 23, 51}}