Tall and skinny SVD — 2026-01-14

Recently, in my work I needed to construct a rational approximation of a matrix-valued function. The best algorithm I know is AAA¹ and that requires singular value decomposition of \(m \times n\) matrices with \(m \gg n\). For scalar functions, there is no problem with that since in most cases the matrix is reasonably sized to use svd call from standard Julia’s LinearAlgebra. To use AAA algorithm for \(r \times c\) matrix-valued function we have to deal with matrices \(r c\) times taller which is already painful. Continuous version of AAA algorithm² allows to increase number of rows adaptively, but even with that I had hard time even to store the full matrix in the memory: typical matrix size was millions times tens.

1. I. V. Gosea and S. Güttel, “Algorithms for the Rational Approximation of Matrix-Valued Functions,” SIAM Journal on Scientific Computing, vol. 43, no. 5, pp. A3033–A3054, Jan. 2021 ↩

2. T. A. Driscoll, Y. Nakatsukasa, and L. N. Trefethen, “AAA Rational Approximation on a Continuum,” SIAM Journal on Scientific Computing, vol. 46, no. 2, pp. A929–A952, Mar. 2024 ↩

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