Computing functions of $A^{-1}B$ where $A$ and $B$ are Hermitian matrices

TL;DR

This work addresses the numerical evaluation of $\varphi(A,B)=Af(A^{-1}B)$ for Hermitian $A\succ0$ and Hermitian $B$ by comparing square-root and Cholesky--Schur based algorithms. It provides a conditioning analysis of $\varphi$ via the Fréchet derivative and develops detailed rounding-error analyses for the square-root and Cholesky-based schemes, including cost estimates. Numerical experiments show that the Cholesky--Schur variants offer superior accuracy and efficiency, especially for ill-conditioned $A$, making them the recommended approach for moderate-sized problems. The results connect to pencils and primary matrix functions and point to future work on large sparse problems using Krylov methods or quadrature.

Abstract

We consider the numerical evaluation of the quantity $Af(A^{-1}B)$, where $A$ is Hermitian positive definite, $B$ is Hermitian, and $f$ is a function defined on the spectrum of $A^{-1}B$. We study the conditioning of the problem, and we introduce several algorithms that combine the Schur decomposition with either the matrix square root or the Cholesky factorization. We study the numerical behavior of these algorithms in floating-point arithmetic, assess their computational costs, and compare their numerical performance. Our analysis suggests that the algorithms based on the Cholesky factorization will be more accurate and efficient than those based on the matrix square root. This is confirmed by our numerical experiments.

Figures (3)

• Figure 1: Relative forward error \ref{['eq:rel-fwd-err']} of the four implementations on random positive definite $n\times n$ matrices of increasing size $n$, with $f(x)=\log(x)$. On the left, $A$ is ill conditioned ($\mathop{\mathrm{\mu}}\nolimits(A)=10^7$), while $B$ is not ($\mathop{\mathrm{\mu}}\nolimits(B)=10$); on the right, both matrices are ill conditioned ($\mathop{\mathrm{\mu}}\nolimits(A)=\mathop{\mathrm{\mu}}\nolimits(B)=10^7$). The values reported are the arithmetic mean of 100 measurements with different matrices.
• Figure 2: Relative forward error \ref{['eq:rel-fwd-err']} of the four implementations on random positive definite $n\times n$ matrices $A$ and $B$ with varying condition numbers for $n=10$ and $f(x) = \log(x)$, together with the quantity $u \cdot \hbox{cond}(\varphi; A, B)$. On the left, the condition number of $A$ grows ($\mathop{\mathrm{\mu}}\nolimits(A)=\mu$) while $B$ remains well conditioned ($\mathop{\mathrm{\mu}}\nolimits(B) = 10)$; on the right, the condition number of both matrices grows ($\mathop{\mathrm{\mu}}\nolimits(A)=\mathop{\mathrm{\mu}}\nolimits(B)=\mu$). The accuracy is measured using the relative forward error \ref{['eq:rel-fwd-err']}. The values reported are the arithmetic mean of 100 measurements with different matrices.
• Figure 3: Timings of the four implementations on $n\times n$ random positive definite matrices of increasing size $n$, with $f(x)=\log(x)$. The timings are expressed in seconds. The values reported are the medians of 100 measurements.

Theorems & Definitions (8)

• Theorem 1
• Theorem 1
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof