Glossary

Spectral radius

The largest absolute value of a matrix's eigenvalues; it decides the convergence of linear iterations.

Definition

For the iteration matrix BB, it is ρ(B)=max⁡i∣λi∣\rho(B)=\max_i|\lambda_i|. A stationary iteration xk+1=Bxk+cx_{k+1}=Bx_k+c converges from any start if and only if ρ(B)<1\rho(B)<1, and the smaller ρ(B)\rho(B), the faster the convergence.

How it is used

Use it to compare methods on the same system: among Jacobi, Gauss-Seidel and SOR the smallest spectral radius wins, and −ln⁡ρ(B)-\ln\rho(B) estimates the speed.