Continuous analogues of matrix factorizations

Report

Analogues of QR, LU, SVD, and Cholesky factorizations are proposed for problems in which the usual discrete matrix is replaced by a “quasimatrix,” continuous in one dimension, or a “cmatrix,” continuous in both dimensions. Applications include Chebfun and similar computations involving functions of one or two variables. Two challenges arise: the generalization of the notions of triangular structure and row and column pivoting to continuous variables (required in all cases except the SVD), and the convergence of the infinite series that define the cmatrix factorizations. The generalizations of the factorizations work out neatly, but mathematical questions remain about convergence of the series. For example, our theorem about existence of an LU factorization of a cmatrix (a convergent infinite series) requires the cmatrix to be analytic in a “stadium” region of the complex plane.

Authors: Townsend, A; Trefethen, L

• Publication date: 2014-01-01
• Language: English
• Keywords: singular valve decomposition; Gaussian elimination; Chebfun; QR factorization