Sharp error bounds for complex floating-point inversion

Claude-Pierre Jeannerod (1, 2) , Nicolas Louvet (1, 2) , Jean-Michel Muller (1, 2) , Antoine Plet (1, 2, 3)

1 ARIC - Arithmetic and Computing
2 LIP - Laboratoire de l'Informatique du Parallélisme
3 ENS de Lyon - École normale supérieure de Lyon

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 in precision p, and we assume that the elementary arithmetic operations (+, −, ×, /) are rounded to nearest, so that the roundoff unit is u = 2 −p. 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 reasonably 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 significantly smaller bound 2.707131u holds (assuming p 24) for the normwise relative error, and we illustrate the sharpness of this bound using numerical examples for the basic IEEE 754 binary formats (p = 24, 53, 113).

Mots clés: floating-point arithmetic; numerical error; complex inversion; error analysis; roundings

Domaines: Autre [cs.OH]

Claude-Pierre Jeannerod, Nicolas Louvet, Jean-Michel Muller, Antoine Plet. Sharp error bounds for complex floating-point inversion. 2015. ⟨ensl-01195625v1⟩