The Schur method for computing a matrix square root reduces the matrix to the Schur triangular form and then computes a square root of the triangular matrix. We show that by using a recursive blocking technique the computation of the square root of the triangular matrix can be made rich in matrix multiplication. Numerical experiments making appropriate use of level 3 BLAS show significant speed...

Given a complex matrix H, we consider the decomposition H = QRP∗, where R is upper triangular and Q and P have orthonormal columns. Special instances of this decomposition include the singular value decomposition (SVD) and the Schur decomposition where R is an upper triangular matrix with the eigenvalues of H on the diagonal. We show that any diagonal for R can be achieved that satisfies Weyl’s...

Let R be a ring and V be a matrix valuation on R. It is shown that, there exists a correspondence between matrix valuations on R and some special subsets ?(MVPR) of the set of all square matrices over R, analogous to the correspondence between invariant valuation rings and abelian valuation functions on a division ring. Furthermore, based on Malcolmson’s localization, an alternative proof for t...

This research investigates the implementation mechanism of block-wise ILU(k) preconditioner on GPU. The block-wise ILU(k) algorithm requires both the level k and the block size to be designed as variables. A decoupled ILU(k) algorithm consists of a symbolic phase and a factorization phase. In the symbolic phase, a ILU(k) nonzero pattern is established from the point-wise structure extracted fro...