# Gauss-Legendre Quadrature coefficients

• Grozny
In summary, The conversation is about finding the abscissas and weights for Gaussian integration based on Legendre polynomials. The person asking for help wrote a program for a 32-point quadrature in college but can't find it now. They are looking for information with 16 digits of accuracy and someone suggests checking Chapter 25 of Abramowitz and Stegun. The chapter lists formulas and tables with numerical values for the abscissas and weights. The person thanks the other for their help and mentions they will start typing in the numbers at home.
Grozny
When I was in college I found the abscissas and weights for a 32-point Gauss-Legendre Quadrature and wrote a program to do that. But I can't find that information now and need it again.

It doesn't have to be 32-point; anything from 30 to 50 would be fine, but 16 digits of accuracy please.

Chapter 25 of Abramowitz and Stegun has what you want. An online edition can be found here:

http://www.convertit.com/Go/Convertit/Reference/AMS55.ASP?Res=100&Page=875

The text at the beginning of the chapter list formulas, and the tables at the end of the chapter provide numerical values of the abscissas and weights. For Gaussian integration based upon Legendre polynomials, you get 24 digits for up to 96-point quadrature. Enjoy.

jason

Thanks!

I can't wait to hurry home and start typing 96 24-digit numbers into my computer!

## What is the Gauss-Legendre Quadrature method?

The Gauss-Legendre Quadrature method is a numerical integration technique used to approximate the definite integral of a function. It involves using a weighted sum of function values at specific points within the integration interval, known as Gauss-Legendre Quadrature coefficients, to estimate the integral.

## How are the Gauss-Legendre Quadrature coefficients calculated?

The Gauss-Legendre Quadrature coefficients are calculated using a mathematical formula that involves solving a system of equations. The number of coefficients used depends on the desired level of accuracy, with higher numbers of coefficients resulting in a more precise estimate of the integral.

## What is the advantage of using Gauss-Legendre Quadrature coefficients?

The use of Gauss-Legendre Quadrature coefficients allows for a more accurate estimation of definite integrals compared to other numerical integration methods. It also has the advantage of being able to handle integrands with singularities or discontinuities with relative ease.

## What is the relationship between Gauss-Legendre Quadrature coefficients and orthogonal polynomials?

The Gauss-Legendre Quadrature coefficients are closely related to orthogonal polynomials, specifically the Legendre polynomials. These polynomials are used to generate the coefficients and are also used as the basis functions for the integration.

## What are some applications of Gauss-Legendre Quadrature coefficients?

The Gauss-Legendre Quadrature method is commonly used in various fields of engineering and science to numerically solve integrals that arise in physical and mathematical models. It also has applications in numerical analysis, signal processing, and numerical weather prediction.

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