Structured matrices and Newton's iteration: unified approach

Abstract

Recent progress in the study of structured matrices shows advantages of unifying the treatment of various classes of such matrices. We recall some fundamental techniques for such a unification and then specify it in full details for Newton's iteration, which rapidly improves an initial approximation to the inverse matrix by performing two matrix multiplications per recursive step. The iteration is particularly suitable for n× n structured matrices, represented with O( n) entries of their short generators rather than with their own n 2 entries. Based on such a representation, matrix operations are performed much more rapidly and use much less memory space. A major problem is to control the length of the generators, which tends to grow quite rapidly in the iterative process. Two known methods solve this problem for Toeplitz-like and Cauchy-like matrices. We extend both methods to a more general class of structured matrices and estimate the convergence rate as well as the computational complexity. Some novel techniques are introduced in this study, in particular for the estimation of the norms of the inverse displacement operators.