2014/11/12 by Alex Townsend, Lloyd N. Trefethen · 1 citation
Computer Science · Engineering · Mathematics · #Algebra over a field #Algorithm #Applied mathematics #Cellular Automata and Applications #Cholesky decomposition #Combinatorics #Convergence (economics) #Dimension (graph theory) #Discrete mathematics #Eigenvalues and eigenvectors #Generalization #Mathematical analysis #Mathematics #Matrix (chemical analysis) #Matrix Theory and Algorithms #Matrix decomposition #Pure mathematics #QR decomposition #Series (stratigraphy) #Singular value decomposition #graph theory and CDMA systems
paper · pdf · doi:10.1098/rspa.2014.0585
openalex publication_date 2014/11/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
Analogues of singular value decomposition (SVD), QR, LU and Cholesky factorizations are presented for problems in which the usual discrete matrix is replaced by a 'quasimatrix', continuous in one dimension, or a 'cmatrix', continuous in both dimensions. 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 far from obvious), and the convergence of the infinite series that define the cmatrix factorizations. Our generalizations of triangularity and pivoting are based on a new notion of a 'triangular quasimatrix'. Concerning convergence of the series, we prove theorems asserting convergence provided the functions involved are sufficiently smooth.