naginterfaces.library.ode.bvp_​ps_​lin_​coeffs

naginterfaces.library.ode.bvp_ps_lin_coeffs(f)

bvp_ps_lin_coeffs 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 bvp_ps_lin_cgl_grid().

For full information please refer to the NAG Library document for d02ua

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/d02/d02uaf.html

Parameters

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

The function values $f\left(x_{\text{r}}\right)$, for $\text{r}=1,2,\dots ,n+1$.

Returns

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

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

Raises

NagValueError

(errno $1$)

On entry, $n=⟨value⟩$.

Constraint: $n>1$.

(errno $1$)

On entry, $n=⟨value⟩$.

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

Notes

bvp_ps_lin_coeffs computes the coefficients $c_{\text{j}}$, for $\text{j}=1,2,\dots ,n+1$, of the interpolating Chebyshev series

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

which interpolates the function $f\left(x\right)$ evaluated at the Chebyshev Gauss–Lobatto points

$${\overline{x}}_r=-\cos \left((r-1)\pi /n\right)\text{,}r=1,2,\dots ,n+1\text{.}$$

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.

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