On the LU factorization of infinite systems of semi-separable equations

LU-factorization has been an original motivation for the development of Semi-Separability (semi-separable systems of equations are sometimes called “quasi-separable”) theory, to reduce the computational complexity of matrix inversion. In the case of infinitely indexed matrices, it got side-tracked in favor of numerically more stable methods based on orthogonal transformations and structural “canonical forms”, in particular external (coprime) and outer–inner factorizations. This paper shows how these factorizations lead to what the author believes are new, closed and canonical expressions for the L and U factors, related existence theorems and a factorization algorithm for the case where the original system is invertible and the factors are required to have inverses of the same type themselves. The resulting algorithm is independent of the existence of the solution and has, in addition, the very nice property that it only uses orthogonal transformations. It succeeds in computing the subsequent partial Schur complements (the pivots) in a stable numerical way.