Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Polynomial Preconditioning for Indefinite Matrices

Hayden Henson, Ronald B. Morgan

TL;DR

The paper tackles the challenge of solving large linear systems and interior eigenvalue problems with indefinite matrices using polynomial preconditioning derived from the GMRES residual polynomial. It introduces balancing techniques to render the preconditioned spectrum definite, along with a specialized stability-control framework that addresses outlier eigenvalues and complex spectra, supported by convergence estimates. It demonstrates practical efficacy through extensive examples (e.g., Hatano–Nelson, CZ, QCD, Af23560) and interior eigenvalue tests, achieving substantial speedups and improved eigenvalue targeting. The findings highlight the potential for parallelizable polynomial preconditioners to outperform traditional methods on difficult problems and point to future work on broader problem classes and GPU/parallel architectures.

Abstract

Polynomial preconditioning is an important tool in solving large linear systems and eigenvalue problems. A polynomial from GMRES can be used to precondition restarted GMRES and restarted Arnoldi. Here we give methods for indefinite matrices that make polynomial preconditioning more generally applicable. The new techniques include balancing the polynomial so that it produces a definite spectrum. Then a stability approach is given that is specialized for the indefinite case. Also, very complex spectra are examined. Then convergence estimates are given for polynomial preconditioning of real, indefinite spectra. Finally, tests are preformed of finding interior eigenvalues.

Polynomial Preconditioning for Indefinite Matrices

TL;DR

The paper tackles the challenge of solving large linear systems and interior eigenvalue problems with indefinite matrices using polynomial preconditioning derived from the GMRES residual polynomial. It introduces balancing techniques to render the preconditioned spectrum definite, along with a specialized stability-control framework that addresses outlier eigenvalues and complex spectra, supported by convergence estimates. It demonstrates practical efficacy through extensive examples (e.g., Hatano–Nelson, CZ, QCD, Af23560) and interior eigenvalue tests, achieving substantial speedups and improved eigenvalue targeting. The findings highlight the potential for parallelizable polynomial preconditioners to outperform traditional methods on difficult problems and point to future work on broader problem classes and GPU/parallel architectures.

Abstract

Polynomial preconditioning is an important tool in solving large linear systems and eigenvalue problems. A polynomial from GMRES can be used to precondition restarted GMRES and restarted Arnoldi. Here we give methods for indefinite matrices that make polynomial preconditioning more generally applicable. The new techniques include balancing the polynomial so that it produces a definite spectrum. Then a stability approach is given that is specialized for the indefinite case. Also, very complex spectra are examined. Then convergence estimates are given for polynomial preconditioning of real, indefinite spectra. Finally, tests are preformed of finding interior eigenvalues.
Paper Structure (16 sections, 1 theorem, 16 equations, 11 figures, 9 tables, 7 algorithms)

This paper contains 16 sections, 1 theorem, 16 equations, 11 figures, 9 tables, 7 algorithms.

Table of Contents

  1. Introduction
  2. Review
  3. Polynomial Preconditioning with the GMRES Polynomial
  4. Stability control
  5. Range Restricted GMRES
  6. Balancing the Polynomial
  7. Balance Method 1: Add a root
  8. Balance Method 2: Remove root(s), then add a root
  9. Using Cubic Splines
  10. Balancing with a polynomial from Range Restricted GMRES
  11. Choice of balancing
  12. Complex Spectra
  13. Stability Control
  14. Convergence Estimates
  15. Interior Eigenvalue Problems
  16. ...and 1 more sections

Key Result

Theorem 5.1

Assume a GMRES residual polynomial $\pi$ is generated with $d$ iterations of solving $Ax=b$, where $A$ is diagonalizable and $b$ is norm one. Let the roots of $\pi$ (the harmonic Ritz values) be $\theta_i$, for $i=1,\ldots,d.$ Let the eigenvectors of $A$ be $z_i$'s and let $Z$ be the matrix with the where $pof(\theta_j) = \prod_{i\neq j} \left(1-\frac{\theta_j}{\theta_i}\right).$

Figures (11)

  • Figure 3.1: Bidiagonal matrix of size $n = 5000$. Top has $\varphi$ polynomials of degree $d=5$, $d=25$ and $d=50$. Bottom has a closeup of the same polynomials.
  • Figure 3.2: Bidiagonal matrix of size $n = 5000$. Top has $\varphi$ polynomials of degree $d=150$ and $d= 151$. Bottom has a closeup of the same polynomials plotted at the eigenvalues.
  • Figure 3.3: Bidiagonal matrix of size $n = 5000$ with 100 negative eigenvalues. Top has $\varphi$ polynomials of original degree $d=5$ with different balancing methods. Bottom has a closeup of the same polynomials near the origin.
  • Figure 3.4: Bidiagonal matrix of size $n = 5000$ with 100 negative eigenvalues. Top has $\varphi$ polynomials of degree $9$ with different balancing. Bottom has a closeup of the same polynomials near the origin.
  • Figure 3.5: \newlabelfig:Bal2_40 Bidiagonal matrix of size $n=10{,}000$. The bottom graph has the $\varphi$ polynomial of degree 40 with no balance, balance method 1 and balance method 2. Top right has the same polynomials zoomed in at the origin. Top left demonstrates the intervals where the polynomial from Balance Method 1 is negative.
  • ...and 6 more figures

Theorems & Definitions (2)

  • Theorem 5.1
  • proof