# NAG FL Interfaced02ubf (bvp_​ps_​lin_​cgl_​vals)

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## 1Purpose

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.

## 2Specification

Fortran Interface
 Subroutine d02ubf ( n, a, b, q, c, f,
 Integer, Intent (In) :: n, q Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: a, b, c(n+1) Real (Kind=nag_wp), Intent (Out) :: f(n+1)
#include <nag.h>
 void d02ubf_ (const Integer *n, const double *a, const double *b, const Integer *q, const double c[], double f[], Integer *ifail)
The routine may be called by the names d02ubf or nagf_ode_bvp_ps_lin_cgl_vals.

## 3Description

d02ubf evaluates the Chebyshev series
 $S (x¯) = 12 c1 T0 (x¯) + c2 T1 (x¯) + c3T2 (x¯) +⋯+ cn+1 Tn (x¯) ,$
or its derivative (up to fourth order) at the Chebyshev Gauss–Lobatto points on $\left[a,b\right]$. Here ${T}_{j}\left(\overline{x}\right)$ denotes the Chebyshev polynomial of the first kind of degree $j$ with argument $\overline{x}$ 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
 $xr = - 12 (b-a) cos(π(r-1)/n) + 1 2 (b+a) , r=1,2,…,n+1 , ​$
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 one-dimensional discrete Fast Fourier Transform (DFT).

## 4References

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

## 5Arguments

1: $\mathbf{n}$Integer Input
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: $\mathbf{a}$Real (Kind=nag_wp) Input
On entry: $a$, the lower bound of domain $\left[a,b\right]$.
Constraint: ${\mathbf{a}}<{\mathbf{b}}$.
3: $\mathbf{b}$Real (Kind=nag_wp) Input
On entry: $b$, the upper bound of domain $\left[a,b\right]$.
Constraint: ${\mathbf{b}}>{\mathbf{a}}$.
4: $\mathbf{q}$Integer Input
On entry: the order, $q$, of the derivative to evaluate.
Constraint: $0\le {\mathbf{q}}\le 4$.
5: $\mathbf{c}\left({\mathbf{n}}+1\right)$Real (Kind=nag_wp) array Input
On entry: the Chebyshev coefficients, ${c}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n+1$.
6: $\mathbf{f}\left({\mathbf{n}}+1\right)$Real (Kind=nag_wp) array Output
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: $\mathbf{ifail}$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 $-\mathbf{1}$ 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).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}>0$.
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: n is even.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{a}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{b}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{a}}<{\mathbf{b}}$.
${\mathbf{ifail}}=3$
On entry, ${\mathbf{q}}=⟨\mathit{\text{value}}⟩$.
Constraint: $0\le {\mathbf{q}}\le 4$.
${\mathbf{ifail}}=-99$
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
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.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

## 7Accuracy

Evaluations of DFT to obtain function or derivative values should be an order $n$ multiple of machine precision assuming full accuracy to machine precision in the given Chebyshev series representation.

## 8Parallelism and Performance

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 $n$).