Calculate the weights and nodes used in integrating a function by Gauss-Legendre quadrature.
Usage
call PreGLQ (lower, upper, n, zero, w, exitstatus)
Parameters
lower: input, real(dp)- The lower bound of the integration.
upper: input, real(dp)- The upper bound of the integration.
n: input, integer(int32)- The number of integration points to use. This will integrate exactly a polynomial of degree
2n-1. zero: output, real(dp), dimension (n)- The zeros used in the Gauss-Legendre quadrature.
w: output, real(dp), dimension (n)- The weights used in the Gauss-Legendre quadrature.
exitstatus: output, optional, integer(int32)- If present, instead of executing a STOP when an error is encountered, the variable exitstatus will be returned describing the error. 0 = No errors; 1 = Improper dimensions of input array; 2 = Improper bounds for input variable; 3 = Error allocating memory; 4 = File IO error.
Description
PreGLQ will calculate the weights and zeros used to integrate a function using Gauss-Legendre quadrature. For n quadrature points, the integration will be exact if the function is a polynomial of degree 2n-1, or less. The quadrature nodes correspond to the zeros of the Legendre polynomial of degree n. The number of quadrature points required to integrate a polynomial of degree L is ceiling((L+1)/2).
To integrate a function between the bounds lower and upper it is only necessary to calculate the sum of the function evaluated at the nodes zero multiplied by the weights.
This is a slightly modified version of the algorithm that was published in NUMERICAL RECIPES.
References
Press, W.H., S.A. Teukolsky, W.T. Vetterling, and B.P. Flannery, Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed., Cambridge Univ. Press, Cambridge, UK, 1992.