Richardson's iteration for nonsymmetric matrices

Abstract

To solve the linear N × N system (1) Ax = a for any nonsingular matrix A , Richardson's iteration (2) x j +1 = x j -α j ( Ax j - a ), j =1,2,…, n , which is applied in a cyclic manner with cycle length n is investigated, where the α j are free parameters. The objective is to minimize the error | x n +1 - x |, where x is the solution of (1). If the spectrum of A is known to lie in a compact set S , one is led to the Chebyshev-type approximation problem (3) min p -1∈ V n max z ∈ S | p ( z )|, where V n is the linear span of z , z 2 ,…, z n . If p solves (3), then the reciprocals of the zeros of p are optimal iteration parameters α j . It is shown that for a real problem (1) the iteration (2) can be carried out with real arithmetic alone, even when there are complex α j . The stationary case n =1 is solved completely, i.e., for all compact sets S the problem (3) is solved explicitly. As a consequence, the converging stationary iteration processes can be characterized. For arbitrary closed disks S the problem (3) can be solved for all n ∈ N , and a simple proof is provided. The lemniscates associated with S are introduced. They appear as an important tool for studying the stability of the iteration (2).