Module for Gauss-Legendre Quadrature

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Background. Gauss-Legendre Quadrature.  To approximate the integral by sampling    at the  n  unequally spaced abscissas   , where the corresponding weights are   .  The abscissas and weights are obtained from a table of values.  For convenience we will illustrate three cases and simplify the notation by using single subscripted values. The method is attributed to Johann Carl Friedrich Gauss (1777-1855) and Adrien-Marie Legendre (1752-1833).  

The Gauss-Legendre quadrature rule for n = 2 points.

The Gauss-Legendre quadrature rule for n = 3 points.

The Gauss-Legendre quadrature rule for n = 4 points.

Animations (Gauss-Legendre Quadrature  Gauss-Legendre Quadrature).  Internet hyperlinks to animations.

Example 1.  Use the  Gauss-Legendre quadrature rules for n = 2, 3, and 4 points to compute numerical approximations for .  

Solution 1.

Example 2.  Compare the accuracy of the Gauss-Legendre quadrature rules for n = 2, 3, and 4 points
with the Trapezoidal rule, Simpson's Rule and Simpson's rule for n = 2, 3, and 4 points respectively.
Illustrate the comparisons for the integral   .  

Solution 2.

More Background. The shifted Gauss-Legendre rule for [a,b].   To approximate the integral use the change of variable
. Then use and apply the Gauss-Legendre rules for .

Example 3.  Use the shifted Gauss-Legendre rules for n = 3 points to approximate the integrals
Illustrate the comparisons for the integral   .  

Solution 3.

Example 4.  Investigate the truncation error bound formulas for the Gauss-Legendre quadrature rules of n = 2, 3, and 4 points.
Use the integral   for the investigation.

Solution 4.

Old Lab Project (Gauss-Legendre Quadrature  Gauss-Legendre Quadrature).  Internet hyperlinks to an old lab project.  

Research Experience for Undergraduates

Gauss-Legendre Quadrature  Gauss-Legendre Quadrature  

Internet hyperlinks to web sites and a bibliography of articles.  
  

Downloads (Gauss-Legendre Quadrature Gauss-Legendre Quadrature).  

Download this Mathematica notebook.  

(c) John H. Mathews 2003