Wider contours and adaptive contours
Shev MacNamara, William McLean, Kevin Burrage

TL;DR
This paper introduces wider and adaptive contour strategies for numerical matrix function computations, especially for non-normal matrices, improving accuracy in applications like biology and finance.
Contribution
It proposes wider contours and a semi-analytic adaptive approach based on field of values estimation for better numerical stability.
Findings
Wider contours improve accuracy for non-normal matrices.
Adaptive contour method enhances numerical stability.
Applications demonstrated in biological reaction models.
Abstract
Contour integrals in the complex plane are the basis of effective numerical methods for computing matrix functions, such as the matrix exponential and the Mittag-Leffler function. These methods provide successful ways to solve partial differential equations, such as convection--diffusion models. Part of the success of these methods comes from exploiting the freedom to choose the contour, by appealing to Cauchy's theorem. However, the pseudospectra of non-normal matrices or operators present a challenge for these methods: if the contour is too close to regions where the norm of the resolvent matrix is large, then the accuracy suffers. Important applications that involve non-normal matrices or operators include the Black--Scholes equation of finance, and Fokker--Planck equations for stochastic models arising in biology. Consequently, it is crucial to choose the contour carefully. As a…
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