Gauss-Legendre Quadrature | Lab Report |

**Purpose
**The purpose of this experiment is to calculate the area
under a curve using Gauss-Legendre quadrature.

**Materials and Equipment**

Software RequirementsWindows 95

Delphi 3 or 4 (to recompile)

Hardware RequirementsVGA display

**Procedure**

- If desired, change the possible Integrand functions that are selectable in the
*ComboBoxFunction*ComboBox. Be sure to modify the*TFormQuadrature.ComboBoxFunctionChange*procedure. - Compile the
*GaussLegendreTest*project in Delphi 3 or 4. - Run the
*GaussLegendreTest*program. - Experiment with various Integrand functions and definite integral limits.

**Discussion
**The following data were taken using the

Integrand |
A |
B |
Observed Value |
Exact Value |

1/x | 1.0 | 2.0 | 0.69314 71806 27782 | 0.69314 71805 59945 |

exp(x) | 0.0 | 1.0 | 1.71828 18286 2713 | 1.71828 18284 59045 = e - 1 |

sin(x) | 0.0 | 3.141593 | 2.00000 00002 5578 | 2 |

This technique of integrand evaluation at n+1 functional values should yield the exact integral when the Integrand is a polynomial of degree 2n+1 or less. This is because of the use of orthogonal Legendre polynomials.

A function can be passed as a parameter using a construct like the following:

TYPE TRealFunction = FUNCTION(CONST X:
DOUBLE): DOUBLE;

FUNCTION GaussLegendreQuadrature(CONST A, B: DOUBLE; CONST N: INTEGER; CONST F:
TRealFunction): DOUBLE;

The function "F" is passed to *GaussLegendreQuadrature*
as the integrand of the definite integral evaluated from A to B.

For a commerical package, take a look at Quadrature by Engineering Objects International.

**Conclusions
**Gauss-Lengendre is an efficient technique for numeric
evaluation of definite integrals. The use of 15-point quadrature can result in area
calculations that are accurate to about 10 significant digits for some integrands.

**Keywords**

quadrature, numeric integration, Gauss-Legendre, function parameter

**Files
**Delphi 3/4 Source (6 KB): GaussLegendre.ZIP

**References
**

Quadrature

www.engineeringobjects.com/Quadrature.htm

*Eric Weisstein's World of Mathematics
*http://mathworld.wolfram.com/topics/NumericalIntegration.html

Updated 18 Feb 2002

since 1 Nov 1998