A Class of Arbitrarily Ill Conditioned Floating-Point Matrices

Abstract

Let $\mathbb{F}$ be a floating-point number system with basis $\beta \geqq 2$ and an exponent range consisting of at least the exponents 1 and 2. A class of arbitrarily ill conditioned matrices is described, the coefficients of which are elements of $\mathbb{F}$. Due to the very rapidly increasing sensitivity of those matrices, they might be regarded as “almost” ill posed problems. The condition of those matrices and their sensitivity with respect to inversion is given by means of a closed formula. The condition is rapidly increasing with the dimension. For example, in the double precision of the IEEE 754 floating-point standard (base 2, 53 bits in the mantissa including implicit 1), matrices with $2n$ rows and columns are given with a condition number of approximately $4\cdot10^{32n} $.