Numerical Methods20825 marksNumerical: Cholesky decompositionCholesky decomposition method
Solve the following system of linear equation using Cholesky decomposition method:
Solve the following system of linear equation using Cholesky decomposition method:
$$ \begin{aligned} 4x + 6y - 8z &= -8 \ 6x + 13y - 11z &= -1 \ -8x - 11y + 29z &= 57 \end{aligned} $$
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$$A = \begin{bmatrix} 4 & 6 & -8 \ 6 & 13 & -11 \ -8 & -11 & 29 \end{bmatrix}, \quad B = \begin{bmatrix} -8 \ -1 \ 57 \end{bmatrix}$$ $A$ is symmetric, so $A = LL^T$ applies. Column 1: $$l{11} = \sqrt{4} = 2$$ $$l{21} = \frac{6}{2} = 3, \quad l{31} = \frac{-8}{2} = -4$$ Column 2: $$l{22} = \s...