On Fast Algorithms for Triangular and Dense Matrix Inversion

Abstract

We first propose in this paper a recursive algorithm for triangular matrix inversion (TMI) based on the ‘Divide and Conquer’ (D&C) paradigm. Different versions of an original sequential algorithm are presented. A theoretical performance study permits to establish an accurate comparison between the designed algorithms. Our implementation is designed to be used in place of dtrtri, the level 3 BLAS TMI. Afterwards, we generalize our approach for dense matrix inversion (DMI) based on LU factorization (LUF). This latter is used in Mathematical software libraries such as LAPACK xGETRI and MATLAB inv. $\mathrm{{A}}=\mathrm{{LU}}$ being the input dense matrix, xGETRI consists, once the factors L and U are known, in inverting U then solving the triangular matrix system $\mathrm{{XL}}=\mathrm{{U}}^{-1}$ (i.e. ${\mathrm{{L}}}^{\mathrm{{T}}}{\mathrm{{X}}}^{\mathrm{{T}}}=({\mathrm{{U}}}^{-1})^\mathrm{{T}}$, thus $\mathrm{{X}}={\mathrm{{A}}}^{-1})$. Two other alternatives may be derived here (L and U being known) : (i) first invert L, then solve the matrix system $\mathrm{{UX}}=\mathrm{{L}}^{-1}$ for X ; (ii) invert both L and U, then compute the product $\mathrm{{X}}={\mathrm{{U}}}^{-1}{\mathrm{{L}}}^{-1}$. Each of these three procedures involves at least one triangular matrix inversion (TMI). Our DMI implementation aims to be used in place of the level 3 BLAS TMI-DMI. Efficient results could be obtained through an experimental study achieved on a set of large sized randomly generated matrices.