Averaging operators for exponential splittings

Abstract

Averaging operations are considered in connection with exponential splitting methods. Toeplitz plus Hankel related matrices are resplit by applying appropriate averaging operators leading to a hierarchy of structured matrices. With the resulting parts, the option of using exponential splitting methods becomes available. A related, seemingly important group of unitary unipotents is looked at. Based on a formula due to Lenard, a very fast iterative method to find the nearest Toeplitz plus Hankel matrix in the Frobenius norm is devised.

Matrix subspaces and determinantal hypersurfaces

“…These are then the respective matrix subspaces V 1 and W defined as in (4). For another illustration, with iterative methods for solving very large linear systems, there arises the problem of approximating (16) AW V −1 1 ≈ I ( 12 ) For matrices from CI +F k , the exponential can be computed fast [20].…”

“…For a more tractable tool, scale the determinant function by setting (20) h Since the jth entry of r j /r jj is 1, write r j /r jj 2 =1+x j with x j ≥0. Then from the identity (22) …”