D02UBF evaluates a function, or one of its lower order derivatives, from its Chebyshev series representation at Chebyshev Gauss–Lobatto points on
$\left[a,b\right]$. The coefficients of the Chebyshev series representation required are usually derived from those returned by
D02UAF or
D02UEF.
D02UBF evaluates the Chebyshev series
or its derivative (up to fourth order) at the Chebyshev Gauss–Lobatto points on
$\left[a,b\right]$. Here
${T}_{j}\left(\stackrel{}{x}\right)$ denotes the Chebyshev polynomial of the first kind of degree
$j$ with argument
$\stackrel{}{x}$ defined on
$\left[1,1\right]$. In terms of your original variable,
$x$ say, the input values at which the function values are to be provided are
where
$b$ and
$a$ are respectively the upper and lower ends of the range of
$x$ over which the function is required.
The calculation is implemented by a forward onedimensional discrete Fast Fourier Transform (DFT).
 1: $\mathrm{N}$ – INTEGERInput

On entry: $n$, where the number of grid points is $n+1$. This is also the largest order of Chebyshev polynomial in the Chebyshev series to be computed.
Constraint:
${\mathbf{N}}>0$ and
N is even.
 2: $\mathrm{A}$ – REAL (KIND=nag_wp)Input

On entry: $a$, the lower bound of domain $\left[a,b\right]$.
Constraint:
${\mathbf{A}}<{\mathbf{B}}$.
 3: $\mathrm{B}$ – REAL (KIND=nag_wp)Input

On entry: $b$, the upper bound of domain $\left[a,b\right]$.
Constraint:
${\mathbf{B}}>{\mathbf{A}}$.
 4: $\mathrm{Q}$ – INTEGERInput

On entry: the order, $q$, of the derivative to evaluate.
Constraint:
$0\le {\mathbf{Q}}\le 4$.
 5: $\mathrm{C}\left({\mathbf{N}}+1\right)$ – REAL (KIND=nag_wp) arrayInput

On entry: the Chebyshev coefficients,
${c}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n+1$.
 6: $\mathrm{F}\left({\mathbf{N}}+1\right)$ – REAL (KIND=nag_wp) arrayOutput

On exit: the derivatives
${S}^{\left(q\right)}{x}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n+1$, of the Chebyshev series, $S$.
 7: $\mathrm{IFAIL}$ – INTEGERInput/Output

On entry:
IFAIL must be set to
$0$,
$1\text{ or}1$. If you are unfamiliar with this argument you should refer to
Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value
$1\text{ or}1$ is recommended. If the output of error messages is undesirable, then the value
$1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is
$0$.
When the value $\mathbf{1}\text{ or}\mathbf{1}$ is used it is essential to test the value of IFAIL on exit.
On exit:
${\mathbf{IFAIL}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see
Section 6).
If on entry
${\mathbf{IFAIL}}=0$ or
$1$, explanatory error messages are output on the current error message unit (as defined by
X04AAF).
D02UBF is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
D02UBF 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 implementationspecific information.