# NAG FL Interfaced02ucf (bvp_​ps_​lin_​cgl_​grid)

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## 1Purpose

d02ucf returns the Chebyshev Gauss–Lobatto grid points on $\left[a,b\right]$.

## 2Specification

Fortran Interface
 Subroutine d02ucf ( n, a, b, x,
 Integer, Intent (In) :: n Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: a, b Real (Kind=nag_wp), Intent (Out) :: x(n+1)
#include <nag.h>
 void d02ucf_ (const Integer *n, const double *a, const double *b, double x[], Integer *ifail)
The routine may be called by the names d02ucf or nagf_ode_bvp_ps_lin_cgl_grid.

## 3Description

d02ucf returns the Chebyshev Gauss–Lobatto grid points on $\left[a,b\right]$. The Chebyshev Gauss–Lobatto points on $\left[-1,1\right]$ are computed as ${t}_{\mathit{i}}=-\mathrm{cos}\left(\frac{\left(\mathit{i}-1\right)\pi }{n}\right)$, for $\mathit{i}=1,2,\dots ,n+1$. The Chebyshev Gauss–Lobatto points on an arbitrary domain $\left[a,b\right]$ are:
 $xi = b-a 2 ti + a+b 2 , i=1,2,…,n+1 .$

## 4References

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: $\mathbf{x}\left({\mathbf{n}}+1\right)$Real (Kind=nag_wp) array Output
On exit: the Chebyshev Gauss–Lobatto grid points, ${x}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n+1$, on $\left[a,b\right]$.
5: $\mathbf{ifail}$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: ${\mathbf{a}}<{\mathbf{b}}$.
${\mathbf{ifail}}=-99$
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

The Chebyshev Gauss–Lobatto grid points computed should be accurate to within a small multiple of machine precision.

## 8Parallelism and Performance

d02ucf is not threaded in any implementation.

The number of operations is of the order $n\mathrm{log}\left(n\right)$ and there are no internal memory requirements; thus the computation remains efficient and practical for very fine discretizations (very large values of $n$).