d02 Chapter Contents
d02 Chapter Introduction
NAG C Library Manual

# NAG Library Function Documentnag_ode_bvp_ps_lin_coeffs (d02uac)

## 1  Purpose

nag_ode_bvp_ps_lin_coeffs (d02uac) 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 nag_ode_bvp_ps_lin_cgl_grid (d02ucc).

## 2  Specification

 #include #include
 void nag_ode_bvp_ps_lin_coeffs (Integer n, const double f[], double c[], NagError *fail)

## 3  Description

nag_ode_bvp_ps_lin_coeffs (d02uac) 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 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 ${T}_{j}\left(\stackrel{-}{x}\right)$ 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
 $xr = - 12 b - a cos πr-1 /n + 1 2 b + a , r=1,2,…,n+1 , ​$
where $b$ and $a$ are respectively the upper and lower ends of the range of $x$ over which the function is required.

## 4  References

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

## 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.
2:     f[${\mathbf{n}}+1$]const doubleInput
On entry: the function values $f\left({x}_{\mathit{r}}\right)$, for $\mathit{r}=1,2,\dots ,n+1$.
3:     c[${\mathbf{n}}+1$]doubleOutput
On exit: the Chebyshev coefficients, ${c}_{\mathit{j}}$, for $\mathit{j}=1,2,\dots ,n+1$.
4:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}>1$.
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.

## 7  Accuracy

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

The number of operations is of the order $n\mathrm{log}n$ 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 $n$).