Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Polynomial and rational measure modifications of orthogonal polynomials via infinite-dimensional banded matrix factorizations

Timon S. Gutleb, Sheehan Olver, Richard Mikael Slevinsky

TL;DR

A family of orthogonal polynomials with modified classical weights is constructed that support banded differentiation matrices, enabling sparse spectral methods with modified Classical orthogonaial methods.

Abstract

We describe fast algorithms for approximating the connection coefficients between a family of orthogonal polynomials and another family with a polynomially or rationally modified measure. The connection coefficients are computed via infinite-dimensional banded matrix factorizations and may be used to compute the modified Jacobi matrices all in linear complexity with respect to the truncation degree. A family of orthogonal polynomials with modified classical weights is constructed that support banded differentiation matrices, enabling sparse spectral methods with modified classical orthogonal polynomials.

Polynomial and rational measure modifications of orthogonal polynomials via infinite-dimensional banded matrix factorizations

TL;DR

A family of orthogonal polynomials with modified classical weights is constructed that support banded differentiation matrices, enabling sparse spectral methods with modified Classical orthogonaial methods.

Abstract

We describe fast algorithms for approximating the connection coefficients between a family of orthogonal polynomials and another family with a polynomially or rationally modified measure. The connection coefficients are computed via infinite-dimensional banded matrix factorizations and may be used to compute the modified Jacobi matrices all in linear complexity with respect to the truncation degree. A family of orthogonal polynomials with modified classical weights is constructed that support banded differentiation matrices, enabling sparse spectral methods with modified classical orthogonal polynomials.
Paper Structure (21 sections, 26 theorems, 159 equations, 3 figures, 1 table)

This paper contains 21 sections, 26 theorems, 159 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Previous work
  3. Infinite-dimensional matrix factorizations
  4. Contemporary Applications
  5. Rational measure modification via infinite-dimensional matrix factorizations
  6. $L^\infty$ measure modifications
  7. Measure modifications with $M$-matrix connection coefficients
  8. Banded derivatives of modified classical orthogonal polynomials
  9. Algorithms for infinite-dimensional matrix factorizations
  10. Approximating the $QL$ factorization
  11. Approximating the reverse Cholesky factorization
  12. A model for infinite-dimensional matrix factorizations
  13. $QL$ factorizations
  14. Reverse Cholesky factorizations
  15. Applications and numerical experiments
  16. ...and 6 more sections

Key Result

Theorem 1.3

\newlabelthm:connectioncoeffresult0 Let $X_P$ and $X_Q$ be the Jacobi matrices for the original and modified orthonormal polynomials, ${\bf P}(x)$ and ${\bf Q}(x)$, respectively. Then there exists a unique infinite invertible upper triangular operator $R : \ell^0 \rightarrow \ell^0$ such that:

Figures (3)

  • Figure 1: Left: synthesis of $q_{500}^{(-0.25,-0.75)}(x;\gamma=0.0001)$ on a Chebyshev grid with the Szegő envelope. Right: the nodes of the $30$-point modified Gaussian quadrature rule with $(\alpha,\beta) = (-0.25, -0.75)$.
  • Figure 2: Conversion of a degree-$n$ expansion in modified Jacobi polynomials $q_n^{(-0.25,-0.75)}(x;\gamma=0.01)$ with standard normally distributed pseudorandom coefficients to Jacobi polynomials with the same parameters. Left: $2$-norm and $\infty$-norm relative error in the forward and backward transformation. Right: precomputation and execution time as well as a complexity estimate based on Theorem \ref{['theorem:QLconvergence']}.
  • Figure 3: Conversion of a degree-$n$ expansion in modified generalized Laguerre polynomials $q_n^{(0.25)}(x;\gamma=0.1)$ with standard normally distributed pseudorandom coefficients to generalized Laguerre polynomials with the same parameter. Left: $2$-norm and $\infty$-norm relative error in the forward and backward transformation. Right: precomputation and execution time as well as a complexity estimate inspired by Theorem \ref{['theorem:QLconvergence']}.

Theorems & Definitions (52)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3: Gautschi Gautschi-24-245-70
  • Proof 1
  • Remark 1.4
  • Proposition 2.1: Deift-Li-Tomei-64-358-85Colbrook-Hansen-143-17-19Hansen-254-2092-08
  • Proposition 2.2: Webb-Thesis-17
  • Definition 2.3
  • Proposition 2.4: Chui-Ward-Smith-5-1-82Goodman-et-al-35-233-95Goodman-et-al-18-331-98
  • Corollary 2.5
  • ...and 42 more