NAG C Library Function Document

1Purpose

nag_ode_bvp_ps_lin_cgl_deriv (d02udc) differentiates a function discretized on Chebyshev Gauss–Lobatto points. The grid points on which the function values are to be provided are normally returned by a previous call to nag_ode_bvp_ps_lin_cgl_grid (d02ucc).

2Specification

 #include #include
 void nag_ode_bvp_ps_lin_cgl_deriv (Integer n, const double f[], double fd[], NagError *fail)

3Description

nag_ode_bvp_ps_lin_cgl_deriv (d02udc) differentiates a function discretized on Chebyshev Gauss–Lobatto points on $\left[-1,1\right]$. The polynomial interpolation on Chebyshev points is equivalent to trigonometric interpolation on equally spaced points. Hence the differentiation on the Chebyshev points can be implemented by the Fast Fourier transform (FFT).
Given the function values $f\left({x}_{i}\right)$ on Chebyshev Gauss–Lobatto points ${x}_{\mathit{i}}=-\mathrm{cos}\left(\left(\mathit{i}-1\right)\pi /n\right)$, for $\mathit{i}=1,2,\dots ,n+1$, $f$ is differentiated with respect to $x$ by means of forward and backward FFTs on the function values $f\left({x}_{i}\right)$. nag_ode_bvp_ps_lin_cgl_deriv (d02udc) returns the computed derivative values ${f}^{\prime }\left({x}_{\mathit{i}}\right)$, for $\mathit{i}=1,2,\dots ,n+1$. The derivatives are computed with respect to the standard Chebyshev Gauss–Lobatto points on $\left[-1,1\right]$; for derivatives of a function on $\left[a,b\right]$ the returned values have to be scaled by a factor $2/\left(b-a\right)$.

4References

Canuto C, Hussaini M Y, Quarteroni A and Zang T A (2006) Spectral Methods: Fundamentals in Single Domains Springer
Greengard L (1991) Spectral integration and two-point boundary value problems SIAM J. Numer. Anal. 28(4) 1071–80
Trefethen L N (2000) Spectral Methods in MATLAB SIAM

5Arguments

1:    $\mathbf{n}$IntegerInput
On entry: $n$, where the number of grid points is $n+1$.
Constraint: ${\mathbf{n}}>0$ and n is even.
2:    $\mathbf{f}\left[{\mathbf{n}}+1\right]$const doubleInput
On entry: the function values $f\left({x}_{\mathit{i}}\right)$, for $\mathit{i}=1,2,\dots ,n+1$
3:    $\mathbf{fd}\left[{\mathbf{n}}+1\right]$doubleOutput
On exit: the approximations to the derivatives of the function evaluated at the Chebyshev Gauss–Lobatto points. For functions defined on $\left[a,b\right]$, the returned derivative values (corresponding to the domain $\left[-1,1\right]$) must be multiplied by the factor $2/\left(b-a\right)$ to obtain the correct values on $\left[a,b\right]$.
4:    $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}>0$.
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: n is even.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.

7Accuracy

The accuracy is close to machine precision for small numbers of grid points, typically less than $100$. For larger numbers of grid points, the error in differentiation grows with the number of grid points. See Greengard (1991) for more details.

8Parallelism and Performance

nag_ode_bvp_ps_lin_cgl_deriv (d02udc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_ode_bvp_ps_lin_cgl_deriv (d02udc) makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the x06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

The number of operations is of the order $n\mathrm{log}\left(n\right)$ and the memory requirements are $\mathit{O}\left(n\right)$; thus the computation remains efficient and practical for very fine discretizations (very large values of $n$).

10Example

The function $2x+\mathrm{exp}\left(-x\right)$, defined on $\left[0,1.5\right]$, is supplied and then differentiated on a grid.

10.1Program Text

Program Text (d02udce.c)

10.2Program Data

Program Data (d02udce.d)

10.3Program Results

Program Results (d02udce.r)