Numerical Analysis
Computing the matrix sign and absolute value functions
Comptes Rendus. Mathématique, Volume 346 (2008) no. 1-2, pp. 119-124.

We present two algorithms for the computation of the matrix sign and absolute value functions. Both algorithms avoid a complete diagonalisation of the matrix, but they however require some informations regarding the eigenvalues location. The first algorithm consists in a sequence of polynomial iterations based on appropriate estimates of the eigenvalues, and converging to the matrix sign if all the eigenvalues are real. Convergence is obtained within a finite number of steps when the eigenvalues are exactly known. Nevertheless, we present a second approach for the computation of the matrix sign and absolute value functions, when the eigenvalues are exactly known. This approach is based on the resolution of an interpolation problem, can handle the case of complex eigenvalues and appears to be faster than the iterative approach.

Nous présentons deux algorithmes pour le calcul des fonctions signe et valeur absolue matricielles. Ces algorithmes évitent une diagonalisation complète de la matrice, mais nécessitent des estimations de ses valeurs propres. Le premier algorithme consiste en une suite d'itérations polynomiales construite à partir d'approximations des valeurs propres, et convergeant vers le signe de la matrice lorsque les valeurs propres sont réelles. La convergence s'obtient en un nombre fini d'itérations lorsqu'elles sont connues exactement. Nous présentons cependant une seconde approche s'appuyant sur la résolution d'un problème d'interpolation polynomiale, pour le calcul des fonctions signe et valeur absolue matricielles dans le cas où les valeurs propres sont connues. Ce second algorithme se révèle plus rapide que le premier, et permet la prise en compte de matrices ayant des valeurs propres complexes.

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Accepted:
Published online:
DOI: 10.1016/j.crma.2007.11.028
Ndjinga, Michaël 1, 2

1 Commissariat à l'énergie atomique, centre de Saclay, CEA/DEN/DM2S/SFME, 91191 Gif-sur-Yvette, France
2 Laboratoire de mathématiques appliquées aux systèmes, École centrale Paris, 92295 Châtenay-Malabry cedex, France
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Ndjinga, Michaël. Computing the matrix sign and absolute value functions. Comptes Rendus. Mathématique, Volume 346 (2008) no. 1-2, pp. 119-124. doi : 10.1016/j.crma.2007.11.028. http://www.numdam.org/articles/10.1016/j.crma.2007.11.028/

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