naginterfaces.library.ode.bvp_​ps_​lin_​cgl_​vals

naginterfaces.library.ode.bvp_ps_lin_cgl_vals(a, b, q, c)

bvp_ps_lin_cgl_vals evaluates a function, or one of its lower order derivatives, from its Chebyshev series representation at Chebyshev Gauss–Lobatto points on $[a,b]$. The coefficients of the Chebyshev series representation required are usually derived from those returned by bvp_ps_lin_coeffs() or bvp_ps_lin_solve().

For full information please refer to the NAG Library document for d02ub

https://www.nag.com/numeric/nl/nagdoc_29.3/flhtml/d02/d02ubf.html

Parameters

afloat

$a$, the lower bound of domain $[a,b]$.

bfloat

$b$, the upper bound of domain $[a,b]$.

qint

The order, $q$, of the derivative to evaluate.

cfloat, array-like, shape $(n+1)$

The Chebyshev coefficients, $c_{\text{i}}$, for $\text{i}=1,2,\dots ,n+1$.

Returns

ffloat, ndarray, shape $(n+1)$

The derivatives $S^{\left(q\right)}{x}_{\text{i}}$, for $\text{i}=1,2,\dots ,n+1$, of the Chebyshev series, $S$.

Raises

NagValueError

(errno $1$)

On entry, $n=⟨value⟩$.

Constraint: $n>0$.

(errno $1$)

On entry, $n=⟨value⟩$.

Constraint: $\text{n}$ is even.

(errno $2$)

On entry, $a=⟨value⟩$ and $b=⟨value⟩$.

Constraint: $a<b$.

(errno $3$)

On entry, $q=⟨value⟩$.

Constraint: $0\le q\le 4$.

Notes

bvp_ps_lin_cgl_vals evaluates the Chebyshev series

$$S\left(\overline{x}\right)=\frac{1}{2}{c}_1{T}_0\left(\overline{x}\right)+c_2{T}_1\left(\overline{x}\right)+c_3{T}_2\left(\overline{x}\right)+\cdots +c_{n+1}{T}_n\left(\overline{x}\right)\text{,}$$

or its derivative (up to fourth order) at the Chebyshev Gauss–Lobatto points on $[a,b]$. Here $T_j\left(\overline{x}\right)$ denotes the Chebyshev polynomial of the first kind of degree $j$ with argument $\overline{x}$ defined on $[-1,1]$. In terms of your original variable, $x$ say, the input values at which the function values are to be provided are

$$x_r=-\frac{1}{2}(b-a)\cos \left(\pi (r-1)/n\right)+\frac{1}{2}(b+a)\text{,}r=1,2,\dots ,n+1\text{,}$$

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).

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