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The Akhiezer iteration

Cade Ballew, Thomas Trogdon

TL;DR

The Akhiezer iteration extends the Chebyshev iteration to indefinite linear systems whose spectrum lies on multiple disjoint intervals by leveraging orthogonal polynomials with respect to a measure on $\Sigma$. It provides $\mathcal{O}(k)$-complexity in computing the first $k$ recurrence coefficients and Cauchy transforms via a numerical Riemann–Hilbert method, with a dramatic speedup in the special Akhiezer two-interval case. The framework is further extended to matrix-function evaluation $f(\mathbf{A})\mathbf{b}$ through contour quadrature and resolvents, with proven convergence bounds and practical adaptive band selection for symmetric $\mathbf{A}$. Empirical results on two-interval spectra and preconditioned BVPs demonstrate rapid, sometimes superexponential, convergence, illustrating the method’s potential for efficient indefinite-system solves and reliable matrix-function approximations in applications requiring spectral information on multiple intervals.

Abstract

We develop the Akhiezer iteration, a generalization of the classical Chebyshev iteration, for the inner product-free, iterative solution of indefinite linear systems using orthogonal polynomials for measures supported on multiple, disjoint intervals. The iteration applies to shifted linear solves and can then be used for efficient matrix function approximation. Using the asymptotics of orthogonal polynomials, error bounds are provided. A key component in the efficiency of the method is the ability to compute the first $k$ orthogonal polynomial recurrence coefficients and the first $k$ weighted Stieltjes transforms of these orthogonal polynomials in $\mathrm{O}(k)$ complexity using a numerical Riemann--Hilbert approach. For a special class of orthogonal polynomials, the Akhiezer polynomials, the method can be sped up significantly, with the greatest speedup occurring in the two interval case where important formulae of Akhiezer are employed and the Riemann--Hilbert approach is bypassed.

The Akhiezer iteration

TL;DR

The Akhiezer iteration extends the Chebyshev iteration to indefinite linear systems whose spectrum lies on multiple disjoint intervals by leveraging orthogonal polynomials with respect to a measure on . It provides -complexity in computing the first recurrence coefficients and Cauchy transforms via a numerical Riemann–Hilbert method, with a dramatic speedup in the special Akhiezer two-interval case. The framework is further extended to matrix-function evaluation through contour quadrature and resolvents, with proven convergence bounds and practical adaptive band selection for symmetric . Empirical results on two-interval spectra and preconditioned BVPs demonstrate rapid, sometimes superexponential, convergence, illustrating the method’s potential for efficient indefinite-system solves and reliable matrix-function approximations in applications requiring spectral information on multiple intervals.

Abstract

We develop the Akhiezer iteration, a generalization of the classical Chebyshev iteration, for the inner product-free, iterative solution of indefinite linear systems using orthogonal polynomials for measures supported on multiple, disjoint intervals. The iteration applies to shifted linear solves and can then be used for efficient matrix function approximation. Using the asymptotics of orthogonal polynomials, error bounds are provided. A key component in the efficiency of the method is the ability to compute the first orthogonal polynomial recurrence coefficients and the first weighted Stieltjes transforms of these orthogonal polynomials in complexity using a numerical Riemann--Hilbert approach. For a special class of orthogonal polynomials, the Akhiezer polynomials, the method can be sped up significantly, with the greatest speedup occurring in the two interval case where important formulae of Akhiezer are employed and the Riemann--Hilbert approach is bypassed.
Paper Structure (21 sections, 5 theorems, 129 equations, 8 figures, 4 algorithms)

This paper contains 21 sections, 5 theorems, 129 equations, 8 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. Orthogonal polynomials and their Cauchy integrals
  3. Orthogonal polynomials
  4. Cauchy integrals
  5. Akhiezer polynomials
  6. Akhiezer's formulae and notation
  7. The Akhiezer iteration and matrix function evaluation
  8. The Akhiezer iteration for solving linear systems
  9. Convergence
  10. The Akhiezer iteration for matrix function approximation
  11. Convergence
  12. Complexity
  13. Adaptivity
  14. Examples and Applications
  15. Discussion and open questions
  16. ...and 6 more sections

Key Result

Lemma 4.2

Fix $\epsilon>0$ and the set $V=\{z\in\mathbb{C}~:~|z-a_j|\geq\epsilon,~|z-b_j|\geq\epsilon,~j=1,\ldots,g+1\}$. Then, where $\delta_n(z)$ and $\hat{\delta}_n(z)$ are uniformly bounded in both $n$ and $z\in V$ and $\mathfrak g$ is the exterior Green's function with pole at infinitySee Appendix ap:rhp for the construction of $\mathfrak g$. We point out that it is determined by $\Sigma$ alone. corre

Figures (8)

  • Figure 1: Histogram of the eigenvalues of $\mathbf{A}\in\mathbb{R}^{200\times200}$ (left) and the convergence of the Akhiezer iteration applied to $\mathbf{A}\mathbf{x}=\mathbf{b}$ where $\mathbf{b}\in\mathbb{R}^{200}$ is a random Gaussian vector. The reference line that approximates the convergence rate is derived in Section \ref{['sect:lin_conv']}.
  • Figure 2: Relative residuals at each iteration of the Akhiezer iteration applied to a diagonal matrix $\mathbf{A}\in\mathbb{R}^{200\times200}$ with eigenvalues spaced uniformly in $[-2,-0.5]\cup[0.5,6]$ and $\mathbf{b}=\mathbf{A}\mathbf{e}$. The rate of convergence is well-approximated by the reference line $C\mathrm{e}^{-k\mathop{\mathrm{Re}}\nolimits \mathfrak{g}(0)}.$
  • Figure 3: Eigenvalues of a matrix $\mathbf{A}$ superimposed on a contour plot of $\mathop{\mathrm{Re}}\nolimits\mathfrak g(z)$ (left) and relative residuals at each iteration for the Akhiezer iteration applied to $\mathbf{A}\mathbf{x}=\mathbf{b}:=\mathbf{A}\mathbf{e}$. The asymptotic rate of convergence is well-approximated by the reference line $C\mathrm{e}^{k\nu(0;\mathbf{A})}.$
  • Figure 4: 2-norm error at each iteration for the Akhiezer iteration approximation to $f(\mathbf{A})\mathbf{b}$ where $f(x)=\exp(x)$, $\mathbf{A}\in\mathbb{R}^{200\times200}$ is as in Figure \ref{['error_and_contour']}, and $\mathbf{b}\in\mathbb{R}^{200}$ is a random Gaussian vector. The iteration converges superexponentially, saturating within 20 iterations.
  • Figure 5: 2-norm error at each iteration for the Akhiezer iteration approximation to $f(\mathbf{A})\mathbf{b}$ where $f(x)=\tanh(x)$ (left) and $f(x)=\exp(x)/x$ (right), $\mathbf{A}\in\mathbb{R}^{200\times200}$ is has randomly generated eigenvalues in $[-2,-0.5]\cup[0.5,6]$, and $\mathbf{b}\in\mathbb{R}^{200}$ is a random Gaussian vector. The convergence rate bounds $C_1\mathrm{e}^{k\nu(\mathrm{i}\pi/2;\mathbf{A})}$ (left) and $C_2\mathrm{e}^{k\nu(0;\mathbf{A})}$ (right) approximate the true convergence rates.
  • ...and 3 more figures

Theorems & Definitions (17)

  • Remark 2.1
  • Remark 4.1
  • Lemma 4.2
  • proof
  • Definition 4.3
  • Definition 4.4
  • Theorem 4.5
  • proof
  • Remark 4.6
  • Remark 4.7
  • ...and 7 more