Some recursions on Arnoldiʹs method and IOM for large non-Hermitian linear systems

Arnoldiʹs method and the Incomplete Orthogonalization Method (IOM) for large non-Hermitian linear systems are studied. It is shown that the inverse of a general nonsingular j × j Hessenberg matrix can be updated in O(j2) flops from that of its (j − 1) × (j − 1) principal submatrix. The updating recursion of inverses of the Hessenberg matrices does not need any QR or LU decomposition as commonly used in the literature. Some updating recursions of the residual norms and the approximate solutions obtained by these two methods are derived. These results are appealing because they allow one to decide when the methods converge and show one how to compute approximate solutions very cheaply and easily.