Zur Konvergenz von Iterationsverfahren mit affinen Operatoren

Abstract

Many well-known numerical procedures for the iterative solution of linear equations in Banach spaces have the form $$ x{n+1} = \\mathbf A_n \\mathbf A{n-1} \\dots \\mathbf A_0x_0 (n = 0, 1, 2, \\dots) $$ with affine operators $\\mathbf A_n$. If the operators $\\mathbf A_n$ possess common fixed points, then there often exist affine projectors $\\mathbf P$ with the property $$ \\mathbf P = \\mathbf A_n \\mathbf P = \\mathbf {PA}\_n \\quad (n= 0, 1,2, \\dots). $$ In this paper some convergence theorems are proved for iteration methods having such operator sequences.