Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

From an Elementary Proof of Error Representation for Hermite Quadrature to a Rediscovery of Legendre Polynomials and Rodrigues Formula

Tan Bui-Thanh, Giancarlo Villatoro, C. G. Krishnanunni

Abstract

We generalize two-point interpolatory Hermite quadrature to functions with available values and the first (n-1) derivatives at both end points. Armed with integration by parts in the reverse form we provide an elementary derivation of an exact error represenation of Hermite quadrature rule. This approach possesses several advantages over the classical approaches: i) Only integration by parts is needed for the derivation; ii) the error representation requires much milder regularity, namely the existence of nth-order derivative rather than a (2n)th-order derivative of the function under consideration. As a result, our error formula is valid for less regular functions for which the classical ones are not valid; iii) our approach rediscovers Legendre polynomials and more interestingly it provides a surprisingly elegant relation between Legendre polynomial and Hermite interpolation. In particular, Legendre polynomials are precisely the error kernels for interpolatory Hermite quadrature rules; and iv) We also rediscover the Rodrigues formula for Legendre polynomials as part of our findings. For those who are interested in a different proof of the exact error representation for Hermite quadrature rule, we provide an alternative proof using the Peano kernel theorem. We also provide a composite interpolatory Hermite quadrature rule for practical applications.

From an Elementary Proof of Error Representation for Hermite Quadrature to a Rediscovery of Legendre Polynomials and Rodrigues Formula

Abstract

We generalize two-point interpolatory Hermite quadrature to functions with available values and the first (n-1) derivatives at both end points. Armed with integration by parts in the reverse form we provide an elementary derivation of an exact error represenation of Hermite quadrature rule. This approach possesses several advantages over the classical approaches: i) Only integration by parts is needed for the derivation; ii) the error representation requires much milder regularity, namely the existence of nth-order derivative rather than a (2n)th-order derivative of the function under consideration. As a result, our error formula is valid for less regular functions for which the classical ones are not valid; iii) our approach rediscovers Legendre polynomials and more interestingly it provides a surprisingly elegant relation between Legendre polynomial and Hermite interpolation. In particular, Legendre polynomials are precisely the error kernels for interpolatory Hermite quadrature rules; and iv) We also rediscover the Rodrigues formula for Legendre polynomials as part of our findings. For those who are interested in a different proof of the exact error representation for Hermite quadrature rule, we provide an alternative proof using the Peano kernel theorem. We also provide a composite interpolatory Hermite quadrature rule for practical applications.
Paper Structure (21 sections, 14 theorems, 93 equations, 1 figure)

This paper contains 21 sections, 14 theorems, 93 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Hermite interpolating function and its integral
  3. Hermite quadrature from Hermite interpolation
  4. An elementary approach to derive Hermite quadrature error using milder conditions
  5. $n=2$: Exact expression for $E_2$ and its bounds
  6. Exact expression and characterizations of $E_2$
  7. General bounds for $E_2$ with uniform and $L_2$ norms
  8. Improved Bounds for $E_2$ when $f^{(3)}$ exists
  9. An enhance of the classical representation for $E_2$ when $f^{(4)}$ exists
  10. General $n$ case
  11. Exact error representation via coefficient matching
  12. Redundancy in matching conditions and properties of the error kernel
  13. Polynomial kernels $K_n\left( x,\boldsymbol{\theta}^* \right)$ are shifted Legendre polynomials
  14. Rediscovery of Rodrigues formula when $f^{(2n)}$ is available
  15. Uniqueness of the parameters from matching conditions
  16. ...and 6 more sections

Key Result

Proposition 3.1

Consider the Hermite interpolating polynomial in her_def for a given $n$. Then the weights $w_j^a$ and $w_j^b$ in her_rewrite are given by: where $\binom{k}{j}=\frac{k!}{j!(k-j)!}$. Consequently,

Figures (1)

  • Figure 1: Plot of the function $f(x)=x^2\sin{(x)}$ with the shaded region denoting the area under the curve between $a=0$ and $b=\pi$. Note that $\int_a^b f(x)\, dx$ equals the area under the curve. The curve corresponding to the trapezoidal rule is identically $0$ and therefore does not appear in the plot.

Theorems & Definitions (33)

  • Definition 2.1: Two-Point Hermite Interpolating Polynomial
  • Proposition 3.1
  • Proof 1
  • Proposition 4.1: $n$th-time reverse integration by parts
  • Proof 2
  • Remark 4.2
  • Theorem 5.1: Exact expression for $E_2$
  • Proof 3
  • Lemma 5.2: Kernel Properties
  • Proof 4
  • ...and 23 more