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Krylov and core transformation algorithms for an inverse eigenvalue problem to compute recurrences of multiple orthogonal polynomials

Amin Faghih, Michele Rinelli, Marc Van Barel, Raf Vandebril, Robbe Vermeiren

TL;DR

Algorithms for computing the recurrence coefficients corresponding to multiple orthogonal polynomials on the step-line are developed and analyzed with numerical experiments on the ill-conditioned inverse eigenvalue problems related to Kravchuk and Hahn polynomials.

Abstract

In this paper, we develop algorithms for computing the recurrence coefficients corresponding to multiple orthogonal polynomials on the step-line. We reformulate the problem as an inverse eigenvalue problem, which can be solved using numerical linear algebra techniques. We consider two approaches: the first is based on the link with block Krylov subspaces and results in a biorthogonal Lanczos process with multiple starting vectors; the second consists of applying a sequence of Gaussian eliminations on a diagonal matrix to construct the banded Hessenberg matrix containing the recurrence coefficients. We analyze the accuracy and stability of the algorithms with numerical experiments on the ill-conditioned inverse eigenvalue problemshave related to Kravchuk and Hahn polynomials, as well as on other better conditioned examples.

Krylov and core transformation algorithms for an inverse eigenvalue problem to compute recurrences of multiple orthogonal polynomials

TL;DR

Algorithms for computing the recurrence coefficients corresponding to multiple orthogonal polynomials on the step-line are developed and analyzed with numerical experiments on the ill-conditioned inverse eigenvalue problems related to Kravchuk and Hahn polynomials.

Abstract

In this paper, we develop algorithms for computing the recurrence coefficients corresponding to multiple orthogonal polynomials on the step-line. We reformulate the problem as an inverse eigenvalue problem, which can be solved using numerical linear algebra techniques. We consider two approaches: the first is based on the link with block Krylov subspaces and results in a biorthogonal Lanczos process with multiple starting vectors; the second consists of applying a sequence of Gaussian eliminations on a diagonal matrix to construct the banded Hessenberg matrix containing the recurrence coefficients. We analyze the accuracy and stability of the algorithms with numerical experiments on the ill-conditioned inverse eigenvalue problemshave related to Kravchuk and Hahn polynomials, as well as on other better conditioned examples.

Paper Structure

This paper contains 17 sections, 7 theorems, 62 equations, 5 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Uniqueness of type I and type II MOPs
  3. AT-system
  4. Inverse eigenvalue problem
  5. The step-line recurrence relation
  6. The inverse eigenvalue problem
  7. Connection with (block) Krylov subspaces
  8. A biorthogonal Lanczos algorithm
  9. Numerical aspects
  10. An algorithm with normalized vectors and (re-)orthogonalization
  11. Moment matrix formulation
  12. Core transformation algorithm
  13. Numerical experiments
  14. Kravchuk and Hahn Multiple Orthogonal Polynomials
  15. Conditioning and stability
  16. ...and 2 more sections

Key Result

Theorem 2.3

Assume we have an AT-system with $r$ positive discrete measures on $\Delta$. Then for $|\vec{n}| \leq N$

Figures (5)

  • Figure 1: Visualization of different steps in \ref{['alg:coretransformation']} for $N=5$. To simplify the notation, we do not display the transformations for the biorthogonalization.
  • Figure 1: Accuracy and biorthogonalization loss of the various algorithms for the Kravchuk (left) and Hahn (right) MOPs.
  • Figure 2: Relative entry-wise accuracy $\left\lvert\frac{[H_{20}- \widehat{H}_{20}]_{i,j}}{[H_{20}]_{i,j}}\right\rvert$ of IEP_KRYLREORTH(full) for Kravchuk MOPs (left) and Hahn MOPs (right).
  • Figure 3: Comparison of backward error on nodes (left) and weights (right) for multiple Hahn polynomials. Note that the dashed lines in the right figure almost coincide with the solid lines.
  • Figure 6: Error comparison for equidistant nodes in [-1,1] and uniform random weights (left) and Chebyshev nodes in [-1,1] and uniform random weights (right) for larger values of $N$

Theorems & Definitions (14)

  • Definition 2.1: Chebyshev system
  • Definition 2.2: AT-system ArvCousVanAssche2003
  • Theorem 2.3: Uniqueness
  • Remark 3.2
  • Proposition 3.3: Chapter 23 of Ism05
  • Theorem 3.4
  • Proposition 3.7
  • Proof 1
  • Proposition 4.1
  • Proof 2
  • ...and 4 more