Numerical Methods · Linear systems, least squares, and eigenproblems
Let A∈R^(n× n) be strictly row-diagonally dominant, and let G_J=D^(-1)(L+U) be the…
Problem
Let \(A\in\mathbb{R}^{n\times n}\) be strictly row-diagonally dominant, and let \(G_J=D^{-1}(L+U)\) be the Jacobi iteration matrix. Prove that \(\|G_J\|_\infty<1\), and deduce that Jacobi iteration converges for every \(\mathbf{b}\in\mathbb{R}^n\) and every initial vector \(\mathbf{x}^{(0)}\).
Hint
The infinity-norm of a matrix is the largest absolute row sum. For \(G_J\) that row sum is precisely the off-diagonal mass of row \(i\) of \(A\), scaled by \(|a_{ii}|\).
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