# Sharp error bounds for complex floating-point inversion

1 ARIC - Arithmetic and Computing
Inria Grenoble - Rhône-Alpes, LIP - Laboratoire de l'Informatique du Parallélisme
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 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).
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Preprints, Working Papers, ...
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https://hal-ens-lyon.archives-ouvertes.fr/ensl-01195625
Contributor : Jean-Michel Muller <>
Submitted on : Tuesday, September 8, 2015 - 11:28:26 AM
Last modification on : Friday, April 20, 2018 - 3:44:26 PM
Long-term archiving on : Wednesday, December 9, 2015 - 10:54:20 AM

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inversion_complexe_HAL.pdf
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• HAL Id : ensl-01195625, version 1

### Citation

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

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