Abstract
Let ann xn matrixA of floating-point numbers in some format be given. Denote the relative rounding error unit of the given format by eps. AssumeA to be extremely ill-conditioned, that is cond(A) ≫ eps−1. In about 1984 I developed an algorithm to calculate an approximate inverse ofA 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 10300 solely using double precision. This is a workout of the invited talk at the SCAN meeting 2006 in Duisburg.
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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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Rump, S.M. Inversion of extremely Ill-conditioned matrices in floating-point. Japan J. Indust. Appl. Math. 26, 249–277 (2009). https://doi.org/10.1007/BF03186534
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DOI: https://doi.org/10.1007/BF03186534