Linear Algebra and Its Applications · Problem 7.4.1

Linear Programming and Game Theory — Gilbert Strang

Mathematics · iterative-methods-jacobi-gauss-seidel-sor

Problem

This matrix has eigenvalues

2-\sqrt{2}

2

, and

2+\sqrt{2}

A = \begin{bmatrix} 2 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 2 \end{bmatrix}.

Find the Jacobi matrix

D^{-1}(-L-U)

and the Gauss-Seidel matrix

(D+L)^{-1}(-U)

and their eigenvalues, and the numbers

\omega_{\text{opt}}

and

\lambda_{\max}

for SOR.