## Japan Journal of Industrial and Applied Mathematics

- Japan J. Indust. Appl. Math.
- Volume 26, Number 2-3 (2009), 249-277.

### Inversion of Extremely Ill-Conditioned Matrices in Floating-Point

#### Abstract

Let an $n\times n$ matrix $A$ of floating-point numbers in some format be given. Denote the relative rounding error unit of the given format by $\mathtt{eps}$. Assume $A$ to be extremely ill-conditioned, that is $\cond(A)\gg\mathtt{eps}^{-1}$. 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.

#### Article information

**Source**

Japan J. Indust. Appl. Math., Volume 26, Number 2-3 (2009), 249-277.

**Dates**

First available in Project Euclid: 1 February 2010

**Permanent link to this document**

https://projecteuclid.org/euclid.jjiam/1265033781

**Mathematical Reviews number (MathSciNet)**

MR2589475

**Zentralblatt MATH identifier**

1185.65050

**Keywords**

extremely ill-conditioned matrix condition number multiplicative correction accurate dot product accurate summation error-free transformations

#### Citation

Rump, Siegfried M. Inversion of Extremely Ill-Conditioned Matrices in Floating-Point. Japan J. Indust. Appl. Math. 26 (2009), no. 2-3, 249--277. https://projecteuclid.org/euclid.jjiam/1265033781