In this method, some known singular nonpolynomial functions in the approximation space of the conventional Jacobi-Galerkin method were first introduced and then the Gauss-Jacobi
quadrature rules to approximate the integral term in the resulting equation were used to obtain high-order accuracy for the approximation.
In this section, we illustrate the accuracy and efficiency of the Clenshaw-Curtis, Fejer's first- and second-type rules for the functions tan [absolute value of (x)], 1/(1 + 16[x.sup.2]) and [[absolute value of (x - 0.5)].sup.0.6] by the algorithms presented in this paper, which are compared with the Gauss-Jacobi
quadrature used [x,w] = jacpts(n, [alpha], [beta]) in CHEBFUN v4.2  (see Figure 1).
The following Gauss-Jacobi quadrature, suggested by Trefethen , was employed to take care of the singularities at the endpoints
The singularities other than the integration limits, however, are not taken care of by Gauss-Jacobi quadrature.
To overcome this difficulty, an adaptive compound Gauss-Jacobi quadrature was used, which is based on the following criterion, again suggested by Trefethen .
Having done with the subdividing, (2.11) or pure Gauss-Jacobi quadrature then applied to each of the resulting subpaths.
DSCPACK, the software package to solve accessory parameter problem described in the last section, contains 21 subroutines and uses two library routines, HYBRD, a nonlinear system solver based on MINPACK subroutines, and GAUSSJ, a routine to generate Gauss-Jacobi weights and nodes.
Hence, the main purpose of this section is going to approximate the integral operator and the inner product based on the Gauss-Jacobi quadrature rule.
A direct computation using the Gauss-Jacobi quadrature rule (40) yields that
It follows from the Gauss-Jacobi quadrature rule (40) that
To this end, suppose N [greater than or equal to] n + 1; replacing the operator K in (33) by the operator [mathematical expression not reproducible] given in (54) and then using Gauss-Jacobi quadrature (40) produce that
Here, we compute the Gauss-Jacobi quadrature rule nodes and weights by Theorems 3.4 and 3.6 discussed in .