Geodesic
Stochastic Matrices and Lp Norms : New Algorithms for Solving Ill-conditioned Linear Systems of Equations
Riadh Zorgati1, 2 ; Wim van Ackooij2 ; Marc Lambert1
1 Supelec, Plateau de Moulon, 3 rue Joliot-Curie, F-91192 Gif-sur-Yvette Cedex FRANCE
2 EDF R&D. 1, avenue du Général de Gaulle, F-92141 Clamart Cedex FRANCE
ESAIM. Proceedings, Tome 18 (2007), pp. 70-86.

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We propose new iterative algorithms for solving a system of linear equations, possibly singular and inconsistent, presenting outstanding performances regarding ill-conditioning and error propagation. The basis of our approach is constructing with the l1 norm, a preconditioning matrix C (an approximation of a generalized inverse of the matrix) such that the preconditioned matrix CA is stochastic. This property allows us to retrieve, in an original way, the Schultz-Hotelling-Bodewig's algorithm of iterative refinement of the approximate inverse of a matrix. The approach, valid for non-negative matrices, is then generalized to any complex, rectangular matrix. We are then able to compute a generalized inverse of any matrix and this inverse is fit for use in classical solving schemes such as : Richardson-Tanabe, Schultz-Hotelling-Bodewig, preconditioned conjugate gradients and also in the Kaczmarz scheme (that we have generalized using lp norms). Regarding the obtained results on pathological well-known test-cases such as Hilbert and Nakasaka matrices, some of the proposed algorithms are empirically shown to be more efficient than the known classical techniques.
DOI : 10.1051/proc:071807

Riadh Zorgati 1, 2 ; Wim van Ackooij 2 ; Marc Lambert 1

1 Supelec, Plateau de Moulon, 3 rue Joliot-Curie, F-91192 Gif-sur-Yvette Cedex FRANCE
2 EDF R&D. 1, avenue du Général de Gaulle, F-92141 Clamart Cedex FRANCE 
@article{EP_2007_18_a7,
     author = {Riadh Zorgati and Wim van Ackooij and Marc Lambert},
     title = {Stochastic {Matrices} and {Lp} {Norms} : {New} {Algorithms} for {Solving} {Ill-conditioned} {Linear} {Systems} of {Equations}},
     journal = {ESAIM. Proceedings},
     pages = {70--86},
     publisher = {mathdoc},
     volume = {18},
     year = {2007},
     doi = {10.1051/proc:071807},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/proc:071807/}
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Riadh Zorgati; Wim van Ackooij; Marc Lambert. Stochastic Matrices and Lp Norms : New Algorithms for Solving Ill-conditioned Linear Systems of Equations. ESAIM. Proceedings, Tome 18 (2007), pp. 70-86. doi : 10.1051/proc:071807. http://geodesic.mathdoc.fr/articles/10.1051/proc:071807/

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