Sharp error bounds for complex floating-point inversion

Abstract

We study the accuracy of the classic algorithm for inverting a complex number given by its real and imaginary parts as floating-point numbers. Our analyses are done in binary floating-point arithmetic, with an unbounded exponent range and in precision p; we also assume that the basic arithmetic operations (+, −, ×, /) are rounded to nearest, so that the roundoff unit is u = 2^−p. We bound the largest relative error in the computed inverse either in the componentwise or in the normwise sense. We prove the componentwise relative error bound 3u for the complex inversion algorithm (assuming p ≥ 4), and we show that this bound is asymptotically optimal (as p → ∞) when p is even, and sharp when using one of the basic IEEE 754 binary formats with an odd precision (p = 53, 113). This componentwise bound obviously leads to the same bound 3u for the normwise relative error. However, we prove that the smaller bound 2.707131u holds (assuming p ≥ 24) for the normwise relative error, and we illustrate the sharpness of this bound for the basic IEEE 754 binary formats (p = 24,53,113) using numerical examples.