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A note on Laguerre truncated polynomials and quadrature formula

Juan C. García-Ardila, Francisco Marcellán

TL;DR

The paper addresses the construction of Gaussian quadrature formulas for the truncated Gamma distribution with weight $w(x)=x^{\alpha} e^{-x}$ on $(0,z)$ and analyzes how the quadrature data depend on the truncation parameter $z$. It advances a stable computational methodology based on the modified Chebyshev algorithm with shifted Jacobi polynomials, deriving modified moments in closed form via incomplete gamma and hypergeometric functions and establishing asymptotic limits for the recurrence coefficients. Comprehensive numerical experiments in MATLAB/OPQ quantify conditioning effects as $z$ grows, propose interpolation strategies for the recurrence data across $z$, and provide explicit quadrature data for representative cases. The work offers a practical route for accurate and robust Gaussian-type quadrature on truncated Laguerre measures, with implications for implementations that require stable handling of the associated Jacobi matrices and their spectra.

Abstract

In this contribution we deal with Gaussian quadrature rules based on orthogonal polynomials associated with a weight function $w(x)= x^α e^{-x}$ supported on an interval $(0,z)$, $z>0.$ The modified Chebyshev algorithm is used in order to test the accuracy in the computation of the coefficients of the three-term recurrence relation, the zeros and weights, as well as the dependence on the parameter $z.$

A note on Laguerre truncated polynomials and quadrature formula

TL;DR

The paper addresses the construction of Gaussian quadrature formulas for the truncated Gamma distribution with weight on and analyzes how the quadrature data depend on the truncation parameter . It advances a stable computational methodology based on the modified Chebyshev algorithm with shifted Jacobi polynomials, deriving modified moments in closed form via incomplete gamma and hypergeometric functions and establishing asymptotic limits for the recurrence coefficients. Comprehensive numerical experiments in MATLAB/OPQ quantify conditioning effects as grows, propose interpolation strategies for the recurrence data across , and provide explicit quadrature data for representative cases. The work offers a practical route for accurate and robust Gaussian-type quadrature on truncated Laguerre measures, with implications for implementations that require stable handling of the associated Jacobi matrices and their spectra.

Abstract

In this contribution we deal with Gaussian quadrature rules based on orthogonal polynomials associated with a weight function supported on an interval , The modified Chebyshev algorithm is used in order to test the accuracy in the computation of the coefficients of the three-term recurrence relation, the zeros and weights, as well as the dependence on the parameter
Paper Structure (6 sections, 4 theorems, 64 equations, 2 figures, 5 tables)

This paper contains 6 sections, 4 theorems, 64 equations, 2 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Basic background
  3. Gauss quadrature formula
  4. Quadrature Formula for the truncated Laguerre polynomials
  5. Numerical experiments
  6. Concluding remarks

Key Result

Theorem 2.3

Let $\mathbf{u}$ be a positive-definite linear functional and let $(P_n)_{n\ge 0}$ be its corresponding SMOP. There exist positive real numbers $A_{N,1},\ldots, A_{N,N}$ such that for every polynomial $q(x)$ of degree at most $2N-1$, we have where $(x_{N,k})_{k=1}^N$ are the zeros of de $P_N(x)$.

Figures (2)

  • Figure 1: The coefficients $b_k(z)$ for $z\in[0,30]$ in the case $\alpha=1$.
  • Figure 2: The coefficients $a_k(z)$ for $z\in[0,30]$ in the case $\alpha=1$.

Theorems & Definitions (7)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3: Gauss Quadrature Rule
  • Corollary 2.4
  • Proposition 3.1
  • Proposition 3.2
  • proof