Convergence of Rumpʹs method for inverting arbitrarily ill-conditioned matrices

In this paper, the problem of inverting regular matrices with arbitrarily large condition number is treated in double precision defined by IEEE 754 floating point standard. In about 1984, Rump derived a method for inverting arbitrarily ill-conditioned matrices. The method requires the possibility to calculate a dot product in higher precision. Rumpʹs method is of theoretical interest. Rump made it clear that inverting an arbitrarily ill-conditioned matrix in single or double precision does not produce meaningless numbers, but contains a lot of information in it. Rumpʹs method uses such inverses as preconditioners. Numerical experiments exhibit that Rumpʹs method converges rapidly for various matrices with large condition numbers. Why Rumpʹs method is so efficient for inverting arbitrarily ill-conditioned matrices is a little mysterious. Thus, to prove its convergence is an interesting problem in numerical error analysis. In this article, a convergence theorem is presented for a variant of Rumpʹs method.