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Error formulas for block rational Krylov approximations of matrix functions

Stefano Massei, Leonardo Robol

Abstract

This paper investigates explicit expressions for the error associated with the block rational Krylov approximation of matrix functions. Two formulas are proposed, both derived from characterizations of the block FOM residual. The first formula employs a block generalization of the residual polynomial, while the second leverages the block collinearity of the residuals. A posteriori error bounds based on the knowledge of spectral information of the argument are derived and tested on a set of examples. Notably, both error formulas and their corresponding upper bounds do not require the use of quadratures for their practical evaluation.

Error formulas for block rational Krylov approximations of matrix functions

Abstract

This paper investigates explicit expressions for the error associated with the block rational Krylov approximation of matrix functions. Two formulas are proposed, both derived from characterizations of the block FOM residual. The first formula employs a block generalization of the residual polynomial, while the second leverages the block collinearity of the residuals. A posteriori error bounds based on the knowledge of spectral information of the argument are derived and tested on a set of examples. Notably, both error formulas and their corresponding upper bounds do not require the use of quadratures for their practical evaluation.

Paper Structure

This paper contains 24 sections, 18 theorems, 114 equations, 5 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Literature review
  3. Contribution
  4. Outline
  5. Petrov-Galerkin approximation of shifted linear systems with block Krylov subspaces
  6. Matrix polynomials and their action on block vectors
  7. Residual and error block polynomials
  8. Collinearity of residuals
  9. Block rational Krylov methods
  10. Collinearity of residuals in the rational case
  11. A formula for the moment matching approximation error of transfer functions
  12. Matrix functions approximation
  13. Interpolation error formulas for matrix functions
  14. Numerical experiments
  15. Galerkin approximation of the matrix exponential
  16. ...and 9 more sections

Key Result

Lemma 2.1

Let $P(\lambda) = \lambda^kP_k + \ldots + \lambda P_1+'P_0$ be an $s \times s$ matrix polynomial of degree $k$. Under Assumptions 1, where $\Upsilon_j^U = \Pi_{UZ}U_{j+1} \Gamma_{j+1}^U \ldots \Gamma_{2}^U$ has full rank.

Figures (5)

  • Figure 1: Galerkin approximation error for the action of the matrix exponential onto a random block vector, and a posteriori bounds. The results are averaged over $10$ runs.
  • Figure 2: Galerkin approximation error for the action of the inverse square root onto a random block vector, and a posteriori bounds. The results are averaged over $10$ runs. The plot on the left refers to the matrix $A_1$, while the plot on the right refers to the matrix $A_2$.
  • Figure 3: Petrov-Galerkin approximation error for the action of the matrix exponential onto a random block vector, and a posteriori bounds. The matrix $A$ is diagonal with eigenvalues as in the left part of the Figure. The results are averaged over $10$ runs.
  • Figure 4: Petrov-Galerkin approximation error for the inverse square root of $A$ applied to a random block vector, and a posteriori bounds. The matrix $A$ is diagonal with eigenvalues as in the left part of the Figure. The results are averaged over $10$ runs.
  • Figure 5: Galerkin approximation error for the action of the matrix exponential (left) and the inverse square root (right) for a diagonal matrix $A$ with logarithmically spaced eigenvalues, onto a block vector $B$ containing the two lowest frequency eigenvectors of the 1D discrete Laplacian.

Theorems & Definitions (43)

  • Lemma 2.1
  • Proof 1
  • Definition 2.2: Definition 2.23 in lund2018new
  • Remark 2.3
  • Lemma 2.4
  • Proof 2
  • Lemma 2.5
  • Proof 3
  • Definition 2.6
  • Lemma 2.7
  • ...and 33 more