ExerciseLinear systemsadvanced

Exercise: convergence by spectral radius

A non-diagonally-dominant matrix where Jacobi diverges but Gauss-Seidel converges, decided by computing the spectral radius of each iteration matrix.

Deciding with ρ(H)

Example

For A=(321232123)A=\bigl(\begin{smallmatrix}3 & 2 & 1\\ 2 & 3 & 2\\ 1 & 2 & 3\end{smallmatrix}\bigr) (invertible but not diagonally dominant), decide which method converges by computing the spectral radius of its iteration matrices.

  1. Spectral radii of the iteration matrices:

    ρ(HJ)=1.1241>1,ρ(HGS)=0.6083<1\rho(H_J)=1.1241>1,\qquad \rho(H_{GS})=0.6083<1

By the theorem, Jacobi diverges and Gauss-Seidel converges; in fact Gauss-Seidel reaches the tolerance in 13 iterations while Jacobi does not converge in 300:

x(13)≈[0.6256, −0.5006, 0.1252]tx^{(13)}\approx[0.6256,\ -0.5006,\ 0.1252]^{t}