Direct fail-proof triangularization algorithms for AX + XB = C with Error-Free and parallel implementations

摘要

Two 0(mn3) inversion-free direct algorithms to compute a solution of the linear system AX +XB = C by triangularizing a Hessenberg matrix are presented. Without any loss of generality the matrix A is assumed upper Hessenberg and the order m of A ⩽ the order n of B. The algorithms have an in-built consistency check, are capable of pruning redundant rows and converting the resulting matrix into a full row rank matrix, and permit A and —B to be any square matrices with common or distinct eigenvalues. In addition, these algorithms can also solve the homogeneous system AX +XB = 0 (null matrix C). An error-free implementation of the solution X using multiple modulus residue arithmetic as well as a parallelization of the algorithms is discussed.