Inversion of Extremely Ill-Conditioned Matrices in Floating-Point

Let an n x n matrix A of floating-point numbers in some format be given. Denote the relative rounding error unit of the given format by eps. Assume A to be extremely ill-conditioned, that is cond(A) >> eps(-1). In about 1984 I developed an algorithm to calculate an approximate inverse of A solely using the given floating-point format. The key is a multiplicative correction rather than a Newton-type additive correction. I did not publish it because of lack of analysis. Recently, in [9] a modification of the algorithm was analyzed. The present paper has two purposes. The first is to present reasoning how and why the original algorithm works. The second is to discuss a quite unexpected feature of floating-point computations, namely, that an approximate inverse of an extraordinary ill-conditioned matrix still contains a lot of useful information. We will demonstrate this by inverting a matrix with condition number beyond 10(300) solely using double precision. This is a workout of the invited talk at the SCAN meeting 2006 in Duisburg.