Inversion of all principal submatrices of a matrix

Let A/sub m/ be an m/spl times/m principal submatrix of an infinite-dimensional matrix A. We give a simple formula which expresses A/sub m+1//sup /spl minus/1/ in terms of A/sub m//sup /spl minus/1/, and based on this formula, an algorithm which computes the inverses of A/sub m/ for m=1, 2, 3, ..., n using only 2n/sup 3//spl minus/2n/sup 2/+n arithmetic operations. This is an improvement over the naive method of computing the inverses separately which would require /spl Sigma//sub m=1//sup n/ m/sup 3/=O(n/sup 4/) arithmetic operations.<
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