The Mathematical Institute, University of Oxford, Eprints Archive

Continuous analogues of matrix factorizations

Townsend, Alex and Trefethen, Lloyd N. (2014) Continuous analogues of matrix factorizations. Technical Report. SIAM Review. (Submitted)

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Abstract

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.

Item Type:Technical Report (Technical Report)
Subjects:H - N > Numerical analysis
Research Groups:Numerical Analysis Group
ID Code:1766
Deposited By: Quentin Parsons
Deposited On:05 Feb 2014 09:45
Last Modified:29 May 2015 19:27

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