d02uaf obtains the Chebyshev coefficients of a function discretized on Chebyshev Gauss–Lobatto points. The set of discretization points on which the function is evaluated is usually obtained by a previous call to d02ucf.
The routine may be called by the names d02uaf or nagf_ode_bvp_ps_lin_coeffs.
d02uaf computes the coefficients
, for , of the interpolating Chebyshev series
which interpolates the function evaluated at the Chebyshev Gauss–Lobatto points
Here denotes the Chebyshev polynomial of the first kind of degree with argument defined on . In terms of your original variable, say, the input values at which the function values are to be provided are
where and are respectively the upper and lower ends of the range of over which the function is required.
Canuto C (1988) Spectral Methods in Fluid Dynamics 502 Springer
Canuto C, Hussaini M Y, Quarteroni A and Zang T A (2006) Spectral Methods: Fundamentals in Single Domains Springer
Trefethen L N (2000) Spectral Methods in MATLAB SIAM
1: – IntegerInput
On entry: , where the number of grid points is . This is also the largest order of Chebyshev polynomial in the Chebyshev series to be computed.
On entry: ifail must be set to , or to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of means that an error message is printed while a value of means that it is not.
If halting is not appropriate, the value or is recommended. If message printing is undesirable, then the value is recommended. Otherwise, the value is recommended. When the value or is used it is essential to test the value of ifail on exit.
On exit: unless the routine detects an error or a warning has been flagged (see Section 6).
6Error Indicators and Warnings
If on entry or , explanatory error messages are output on the current error message unit (as defined by x04aaf).
An unexpected error has been triggered by this routine. Please
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
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.
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.
The Chebyshev coefficients computed should be accurate to within a small multiple of machine precision.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
d02uaf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
d02uaf 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 routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.
The number of operations is of the order and the memory requirements are ; thus the computation remains efficient and practical for very fine discretizations (very large values of ).