On the rate of growth of condition numbers for convolution matrices

Abstract
When analyzing linear systems of equations, the most im- portant indicator of potential instability is the condition number of the matrix. For a convolution matrix W formed from a series w (where Wij - wi-, + ,, 1 5 i - j + 1 5 k, W,j = 0 otherwise), this condition number defines the stabirity of the deconvolution process. For the larger con- volution matrices commonly encountered in practice, direct computa- tion of the condition number (e.g., by singular value decomposition) would be extremely time consuming. However, for convolution mat- rices, an upper bound for the condition number is defined by the ratio of the maximum to the minimum values of the amplitude spectrum of w. This bound is infinite for any series w with a zero value in its am- plitude spectrum; although for certain such series, the actual condition number for W may in fact be relatively small. In this paper we give a new simple derivation of the upper bound and present a means of de- fining the rate of growth of the condition number of W for a band- limited series by means of the higher order derivatives of the amplitude spectrum of w at its zeros. The rate of growth is shown to be propor- tional to mp, where m is the column dimension of Wand p is the order of the zero of the amplitude spectrum. -

This publication has 5 references indexed in Scilit: