Convergence of Rump's 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. Rump's 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. Rump's method uses such inverses as preconditioners. Numerical experiments exhibit that Rump's method converges rapidly for various matrices with large condition numbers. Why Rump's 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 Rump's method.

Subjects

Accurate dot product

Ill-conditioned matrix

Matrix inversion

Precondition

DDC Class

004: Informatik

510: Mathematik

More Funding Information

This research was partially supported by Grant-in-Aid for Specially Promoted Research (No. 17002012: Establishment of Verified Numerical Computation) from the Ministry of Education, Science, Sports and Culture of Japan.

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Convergence of Rump's method for inverting arbitrarily ill-conditioned matrices