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

Motohiro Otsuka, Fuminori Tatsuoka, Tomohiro Sogabe, Kota Takeda, Shao-Liang Zhang

Abstract

This study considers quadrature-based algorithms to compute $A^α\boldsymbol{b}$, 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 $A^α \boldsymbol{b}$ 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 $A^α\boldsymbol{b}$ below a prescribed error tolerance. Numerical results demonstrate that the proposed criterion enables the computation of $A^α\boldsymbol{b}$ within prescribed tolerance limits.

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

Abstract

This study considers quadrature-based algorithms to compute , the action of a real power of a Hermitian positive-definite matrix on a vector . In these algorithms, the computation of an integral representation of 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 below a prescribed error tolerance. Numerical results demonstrate that the proposed criterion enables the computation of within prescribed tolerance limits.

Paper Structure

This paper contains 6 sections, 2 theorems, 11 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Error Analysis and Control
  3. Estimating the integrand evaluation error
  4. Derivation of a stopping criterion
  5. Numerical Experiments
  6. Conclusion

Key Result

Proposition 1

Let $A \in \mathbb{C}^{n \times n}$ be Hermitian positive definite, and let $\lambda_{\text{max}}$ denote its largest eigenvalue. Then, for any real $\sigma \geq 0$ and any $\bm{x} \in \mathbb{C}^{n}$, the following bound holds: $\blacktriangleleft$$\blacktriangleleft$

Figures (3)

  • Figure 1: Illustrative example of the upper bound formulated by Proposition \ref{['prop:error_bound']}. The vertical axis represents the absolute error in the $2$-norm, and the horizontal axis the number of iterations $i$ of the Shifted CG method. The coefficient matrix $A \in \mathbb{R}^{1024 \times 1024}$ is the finite-difference matrix of the two-dimensional Laplacian. The right-hand-side vector in the linear systems is $\bm{b} = [1,\dots,1]^\top$.
  • Figure 2: Stopping criterion for the shifted systems at the quadrature nodes derived from Corollary \ref{['cor:convergence_criterion']} (DE formula, $\alpha = 0.2$, $\epsilon = 10^{-9}$). The vertical axis represents the admissible residual norm, and the horizontal axis the quadrature-node index $k$.
  • Figure 3: Absolute errors of $A^\alpha \bm{b}$ computed using quadrature-based algorithms. Left column: $\alpha = 0.2$; right column: $\alpha = 0.5$. Top row: one-dimensional Laplacian; bottom row: two-dimensional Laplacian. The horizontal axis represents the prescribed error tolerance $\epsilon$, and the vertical axis the absolute error.

Theorems & Definitions (4)

  • Proposition 1
  • proof
  • Corollary 1
  • proof