A simple, efficient and more reliable scheme for automatic numerical integration

Abstract

We have recently proposed a simple, reliable and efficient scheme for automatic numerical integration that uses the change of variable x(z) = 1+(2/Π){[1+ 2 3 (1−z 2)]z√1−z 2−arccos z} to transform the integral to be computed, ∫ 1-1 f( x) dx, into (16/3 Π) ∫ 1-1 f( x( z)) (1− z 2)√1− z 2 dz,, which is approximated by successive n-point Gauss-Chebyshev quadrature formulas of the second kind ( I n ). The following sequence of formulas was generated: I 1, I 3, I 7,..., I ( n−1)/2 , I n , I 2 n+1 . In the present work we generate the same sequence, together with another one, I 2, I 5,..., I m , I 2 m +1. Both sequences are generated in an alternate way, I 2, I 1, I 5, 3,..., I ( n−1)/2 , I m , I n , I 2 m+1 , where m + 1 = 2 3 (n + 1) . The main advantage of the new scheme is that, unlike the previous one, the total number of points increases more moderately than doubling. This is possible because all the abscissas of I ( n−1)/2 are also abscissas of I n , all the abscissas of I m are also abscissas of I 2 m+1 , and, especially, all the abscissas of I n are also abscissas of I 2 m+1 . These allows us to implement a simple and efficient algorithm to generate both sequences alternatively (a FORTRAN 77 version is included). We propose an error estimate for the approximate integral that is more conservative than the one used in our previous work, which results in a more reliable automatic numerical integrator, very suitable for multiple integrals.