Linear Algebra and Its Applications · Problem 7.4.13

Linear Programming and Game Theory — Gilbert Strang

Mathematics · jacobi-gauss-seidel-eigenvalues

Problem

Change the 2s to 3s and find the eigenvalues of

S^{-1}T

for both methods:

(\mathbf{J})\quad \begin{bmatrix} 3 & 0 \\ 0 & 3 \end{bmatrix} x_{k+1} = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} x_k + b \qquad (\mathbf{GS})\quad \begin{bmatrix} 3 & 0 \\ -1 & 3 \end{bmatrix} x_{k+1} = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix} x_k + b.

Does

|\lambda|_{\max}

for Gauss-Seidel equal

|\lambda|^2_{\max}

for Jacobi?