Stochastic Matrices and Lp Norms : New Algorithms for Solving Ill-conditioned Linear Systems of Equations

Abstract

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 l 1 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 l p 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.