A direct incomplete factorization method for parallel solution of tridiagonal linear systems

Abstract

For the direct solution of tridiagonal linear systems Ax = d, the best known serial algorithm is based on LU-factorization of the coefficient matrix A. In the present paper we consider extending the idea to partitioned tridiagonal matrices. Let A be partitioned: A = (A (i, j)) so that the diagonal blocks A( i,i )are tridiagonal. We seek a factorization of A into L = (L (i j) and U = (U (i j) ), partitioned conformally. For the diagonal blocks of A we require the classical factorization: A( ii ) = L (i,i) U (i,i) , L (i,i) unit lower bidiagonal and U (i,i) upper bidiagonal. But, because of the presence of a non-zero element in each of the off-diagonal blocks of A, it is necessary to have Lupper block bidiagonal and U lower block bidiagonal, with only last row of L(i,i+1) and last column of U(i,i-1) filled. To avoid any interlocking/updating during/after the factorization stage, each of these last row and column in each block are required to have their last elements as zeros. On completion of the determinat...