A parallel algorithm for the inversion of matrices with simultaneously diagonalizable blocks

Abstract

Block matrices with simultaneously diagonalizable blocks arise in diverse application areas, including, e.g., numerical methods for engineering based on partial differential equations as well as network synchronization, cryptography and control theory. In the present paper, we develop a parallel algorithm for the inversion of $m\times m$ block matrices with simultaneously-diagonalizable blocks of order n. First, a sequential version of the algorithm is presented and its computational complexity is determined. Then, a parallelization of the algorithm is implemented and analyzed. The complexity of the derived parallel algorithm is expressed as a function of m and n as well as of the number μ of utilized CPU threads. Results of numerical experiments demonstrate the CPU time superiority of the parallel algorithm versus the respective sequential version and a standard inversion method applied to the original block matrix. An efficient parallelizable procedure to compute the determinants of such block matrices is also described. Numerical examples are presented for using the developed serial and parallel inversion algorithms for boundary-value problems involving transmission problems for the Helmholtz partial differential equation in piecewise homogeneous media.