An efficient parallel algorithm for the inverse of a banded matrix in the maximum entropy sense

This paper develops a new parallel algorithm for computing the inverse of a banded matrix when extended in its maximum entropy sense. The algorithm developed here computes the inverse in two parallel steps. The first parallel step uses a modified Schur´s complement technique to compute the individual inverses in each of the block matrices in parallel. The second parallel step then adds the overlapped sub-blocks inside the band. The parallel time complexity of our algorithm is O(bw³) requiring n/((bw-1)/2)-1 processors, where the matrix is of size n×n having a bandwidth of bw. The parallel time required is independent of the size of the matrix and only depends upon the bandwidth of the matrix if n/((bw-1)/2)-1 processors are employed. We also provide a multithreaded implementation of the algorithm for use in SMP machines so that the algorithm can be used without requiring n/((bw-1)/2)-1 number of processors. Even in the serial implementation, the algorithm developed here is considerably better than existing serial algorithms for computing the banded inverse in the maximum entropy sense.