Rump, Siegfried M.Siegfried M.Rump2021-01-292021-01-292009-10Japan Journal of Industrial and Applied Mathematics 2/3 (26): 249-277 (2009)http://hdl.handle.net/11420/8630Let an n × 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 10300 solely using double precision. This is a workout of the invited talk at the SCAN meeting 2006 in Duisburg.en1868-937X20092/3249277Accurate dot productAccurate summationCondition numberError-free transformationsExtremely ill-conditioned matrixMultiplicative correctionInformatikMathematikJournal Article10.1007/BF03186534Other