Error control technique of quadrature-based algorithms for the action of real powers of a Hermitian positive-definite matrix

TL;DR

This paper develops an error control method for quadrature algorithms computing matrix fractional powers, incorporating iterative solver residuals to ensure prescribed accuracy.

Contribution

It introduces a residual-based error bound and stopping criterion for iterative solvers in quadrature algorithms for matrix powers.

Findings

The proposed error control method effectively maintains prescribed accuracy.

Numerical tests confirm the stopping criterion's practical utility.

The approach accounts for additional solver errors beyond quadrature discretization.

Abstract

This study considers quadrature-based algorithms to compute $A^\alpha \boldsymbol{b}$, the action of a real power of a Hermitian positive-definite matrix $A$ on a vector $ \boldsymbol{b}$. In these algorithms, the computation of an integral representation of $A^{\alpha} \boldsymbol{b}$ is reduced to solving several tens or hundreds of shifted linear systems. Current approaches usually analyze the quadrature discretization error, but rarely take into account the additional error introduced by solving these shifted linear systems with iterative solvers. Here, we bound this error with the residual of the approximated solution of these linear systems. This allows the derivation of a stopping criterion for iterative solvers to keep the error of $A^\alpha \boldsymbol{b}$ below a prescribed error tolerance. Numerical results demonstrate that the proposed criterion enables the computation of $A^\alpha \boldsymbol{b}$ within prescribed tolerance limits.