# NAG FL Interfaced02uaf (bvp_​ps_​lin_​coeffs)

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## 1Purpose

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.

## 2Specification

Fortran Interface
 Subroutine d02uaf ( n, f, c,
 Integer, Intent (In) :: n Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: f(n+1) Real (Kind=nag_wp), Intent (Out) :: c(n+1)
#include <nag.h>
 void d02uaf_ (const Integer *n, const double f[], double c[], Integer *ifail)
The routine may be called by the names d02uaf or nagf_ode_bvp_ps_lin_coeffs.

## 3Description

d02uaf computes the coefficients ${c}_{\mathit{j}}$, for $\mathit{j}=1,2,\dots ,n+1$, of the interpolating Chebyshev series
 $12 c1 T0 (x¯) + c2 T1 (x¯) + c3T2 (x¯) +⋯+ cn+1 Tn (x¯) ,$
which interpolates the function $f\left(x\right)$ evaluated at the Chebyshev Gauss–Lobatto points
 $x¯r = - cos((r-1)π/n) , r=1,2,…,n+1 .$
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.

## 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: $\mathbf{f}\left({\mathbf{n}}+1\right)$Real (Kind=nag_wp) array Input
On entry: the function values $f\left({x}_{\mathit{r}}\right)$, for $\mathit{r}=1,2,\dots ,n+1$.
3: $\mathbf{c}\left({\mathbf{n}}+1\right)$Real (Kind=nag_wp) array Output
On exit: the Chebyshev coefficients, ${c}_{\mathit{j}}$, for $\mathit{j}=1,2,\dots ,n+1$.
4: $\mathbf{ifail}$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: ${\mathbf{n}}>1$.
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: n is even.
${\mathbf{ifail}}=-99$
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.
${\mathbf{ifail}}=-399$
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

The Chebyshev coefficients computed should be accurate to within a small multiple of machine precision.

## 8Parallelism and Performance

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 $n\mathrm{log}\left(n\right)$ 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 $n$).