Conditioning of Matrix Functions at Quasi-Triangular Matrices

SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 2, Page 954-966, June 2024.
Abstract. The area of matrix functions has received growing interest for a long period of time due to their growing applications. Having a numerical algorithm for a matrix function, the ideal situation is to have an estimate or bound for the error returned alongside the solution. Condition numbers serve this purpose; they measure the first-order sensitivity of matrix functions to perturbations in the input data. We have observed that the existing unstructured condition number leads most of the time to inferior bounds of relative forward errors for some matrix functions at triangular and quasi-triangular matrices. We propose a condition number of matrix functions exploiting the structure of triangular and quasi-triangular matrices. We then adapt an existing algorithm for exact computation of the unstructured condition number to an algorithm for exact evaluation of the structured condition number. Although these algorithms are direct rather than iterative and useful for testing the numerical stability of numerical algorithms, they are less practical for relatively large problems. Therefore, we use an implicit power method approach to estimate the structured condition number. Our numerical experiments show that the structured condition number captures the behavior of the numerical algorithms and provides sharp bounds for the relative forward errors. In addition, the experiment indicates that the power method algorithm is reliable to estimate the structured condition number.