Another alternative for bordered systems

Abstract

This partitioned system, [EQUATION] is to be solved many times with the same n by n matrix, A, but differing n-vectors, b, c, and f, and differing scalars d and g. By using a bordering algorithm, A need be inverted once only. The remaining computation, done often, uses 2n2+3n+constant long operations (multiplications + divisions). Govaerts and Pryce (1987) consider the situation when A is nearly singular although (1) itself is well conditioned, reporting that using a bordering computation followed by iterative refinement of the initial poor solution works well, being both simple and practical.