NAG FL Interface
d02uyf (bvp_​ps_​lin_​quad_​weights)

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1 Purpose

d02uyf obtains the weights for Clenshaw–Curtis quadrature at Chebyshev points. This allows for fast approximations of integrals for functions specified on Chebyshev Gauss–Lobatto points on [−1,1].

2 Specification

Fortran Interface
Subroutine d02uyf ( n, w, ifail)
Integer, Intent (In) :: n
Integer, Intent (Inout) :: ifail
Real (Kind=nag_wp), Intent (Out) :: w(n+1)
C Header Interface
#include <nag.h>
void  d02uyf_ (const Integer *n, double w[], Integer *ifail)
The routine may be called by the names d02uyf or nagf_ode_bvp_ps_lin_quad_weights.

3 Description

d02uyf obtains the weights for Clenshaw–Curtis quadrature at Chebyshev points.
Given the (Clenshaw–Curtis) weights wi, for i=0,1,,n, and function values fi=f(ti) (where ti=-cos(i×π/n), for i=0,1,,n, are the Chebyshev Gauss–Lobatto points), then −1 1 f(x) dx i=0 n wi fi .
For a function discretized on a Chebyshev Gauss–Lobatto grid on [a,b] the resultant summation must be multiplied by the factor (b-a)/2.

4 References

Trefethen L N (2000) Spectral Methods in MATLAB SIAM

5 Arguments

1: n Integer Input
On entry: n, where the number of grid points is n+1.
Constraint: n>0 and n is even.
2: w(n+1) Real (Kind=nag_wp) array Output
On exit: the Clenshaw–Curtis quadrature weights, wi, for i=0,1,,n.
3: ifail Integer Input/Output
On entry: ifail must be set to 0, −1 or 1 to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of 0 causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of −1 means that an error message is printed while a value of 1 means that it is not.
If halting is not appropriate, the value −1 or 1 is recommended. If message printing is undesirable, then the value 1 is recommended. Otherwise, the value 0 is recommended. When the value -1 or 1 is used it is essential to test the value of ifail on exit.
On exit: ifail=0 unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry ifail=0 or −1, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
ifail=1
On entry, n=value.
Constraint: n>0.
On entry, n=value.
Constraint: n is even.
ifail=-99
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
ifail=-399
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
ifail=-999
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

The accuracy should be close to machine precision.

8 Parallelism and Performance

d02uyf is not threaded in any implementation.

9 Further Comments

A real array of length 2n is internally allocated.

10 Example

This example approximates the integral −1 3 3 x2 dx using 65 Clenshaw–Curtis weights and a 65-point Chebyshev Gauss–Lobatto grid on [−1,3].

10.1 Program Text

Program Text (d02uyfe.f90)

10.2 Program Data

Program Data (d02uyfe.d)

10.3 Program Results

Program Results (d02uyfe.r)