Computing the matrix sign and absolute value functions

Ndjinga, Michaël

Comptes Rendus. Mathématique, Tome 346 (2008) no. 1-2, pp. 119-124.

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.

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.

[1] Delhaye, J.M.; Hewitt, G.; Zuber, M. Numerical Benchmark Tests in Multiphase Science and Technology, vol. 3, Hemisphere, Washington, DC/New York, 1987

[2] Drew, D.A.; Passman, S.L. Theory of Multicomponents Fluids, Springer-Verlag, New York, 1999

[3] Gallouët, T.; Hérard, J.-M.; Seguin, N. Numerical modeling of two-phase flows using the two-fluid two-pressure approach, Math. Models Methods Appl. Sci., Volume 14 (2004) no. 5

[4] Ghidaglia, J.M.; Kumbaro, A.; Le Coq, G. On the numerical solution to two fluid models via a cell centered finite volume method, Eur. J. Mech. B/Fluids, Volume 20 (2001) no. 6

[5] Golub, G.H.; Van Loan, C.F. Matrix Computation, The John Hopkins University Press, 1996

[6] Kenney, C.S.; Laub, A.J. The matrix sign function, IEEE Trans. Automat. Control, Volume 40 (1995) no. 8, pp. 1330-1348

[7] LeVeque, R.J. Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, 2002

[8] Masella, J.M.; Faille, I.; Gallouët, T. On an approximate Godunov scheme, Int. J. Comput. Fluid Dynamics, Volume 12 (1999), pp. 133-149

[9] Ndjinga, M. Influence of interfacial pressure on the hyperbolicity of the two-fluid model, C. R. Acad. Sci. Paris, Ser. I, Volume 344 (2007), pp. 407-412

[10] M. Ndjinga, A. Kumbaro, P. Laurent-Gengoux, F. De Vuyst, 2006, Fast and robust computation of the matrix absolute value function. Application to Roe solver for the numerical simulation of two-phase flow models, in: 14th International Conference on Nuclear Energy, 17–20 July 2006, Miami, USA

[11] Roe, P.L. Approximate Riemannn solvers, parameter vectors and difference schemes, J. Comput. Phys., Volume 43 (1981)

[12] Stoer, J.; Bulirsch, R. Introduction to Numerical Analysis, Springer-Verlag, 2002

[13] Toumi, I.; Kumbaro, A. An approximate linearized Riemann solver for a two-fluid model, J. Comput. Phys., Volume 2 (1996), p. 124

[14] Wilkinson, J.H.; Reinsch, C. Handbook for Automatic Computation, vol. II, Linear Algebra, Springer-Verlag, New York, 1971