The dual inverse scaling and squaring algorithm for the matrix logarithm

Abstract

The inverse scaling and squaring algorithm computes the logarithm of a square matrix $A$ by evaluating a rational approximant to the logarithm at the matrix $B:=A^{2^{-s}}$ for a suitable choice of $s$. We introduce a dual approach and approximate the logarithm of $B$ by solving the rational equation $r(X)=B$, where $r$ is a diagonal Padé approximant to the matrix exponential at $0$. This equation is solved by a substitution technique in the style of those developed by Fasi & Iannazzo (2020, Substitution algorithms for rational matrix equations. Elect. Trans. Num. Anal., 53, 500–521). The new method is tailored to the special structure of the diagonal Padé approximants to the exponential and in terms of computational cost is more efficient than the state-of-the-art inverse scaling and squaring algorithm.

Computing the Matrix Logarithm with the Romberg Integration Method

“…The applicability of the matrix logarithm in so many distinct areas has encouraged the development of different approaches to its evaluation. The traditionally proposed methods incorporate algorithms based on the inverse scaling and squaring technique [27], the Schur-Fréchet procedure [28], the Padé approximants [29][30][31][32][33][34][35][36], arithmetic-geometric mean iteration [37], numerical spectral and Jordan decomposition [38], contour integrals [39], or different quadrature formulas [40][41][42][43]. MATLAB incorporates logm as a built-in function that uses the algorithms described in [33,34] to compute the principal matrix logarithm.…”