d02 Chapter Contents
d02 Chapter Introduction
NAG Library Manual

# NAG Library Function Documentnag_ode_bvp_ps_lin_cgl_grid (d02ucc)

## 1  Purpose

nag_ode_bvp_ps_lin_cgl_grid (d02ucc) returns the Chebyshev Gauss–Lobatto grid points on $\left[a,b\right]$.

## 2  Specification

 #include #include
 void nag_ode_bvp_ps_lin_cgl_grid (Integer n, double a, double b, double x[], NagError *fail)

## 3  Description

nag_ode_bvp_ps_lin_cgl_grid (d02ucc) 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:
 $xi = b-a 2 ti + a+b 2 , i=1,2,…,n+1 .$

## 4  References

Trefethen L N (2000) Spectral Methods in MATLAB SIAM

## 5  Arguments

1:     nIntegerInput
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.
On entry: $a$, the lower bound of domain $\left[a,b\right]$.
Constraint: ${\mathbf{a}}<{\mathbf{b}}$.
3:     bdoubleInput
On entry: $b$, the upper bound of domain $\left[a,b\right]$.
Constraint: ${\mathbf{b}}>{\mathbf{a}}$.
4:     x[${\mathbf{n}}+1$]doubleOutput
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:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

On entry, argument $⟨\mathit{\text{value}}⟩$ had an illegal value.
NE_INT
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}>0$.
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: n is even.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
NE_REAL_2
On entry, ${\mathbf{a}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{b}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{a}}<{\mathbf{b}}$.

## 7  Accuracy

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

## 8  Parallelism and Performance

Not applicable.

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 $n$).