naginterfaces.library.sum.chebyshev

naginterfaces.library.sum.chebyshev(x, xmin, xmax, c, s)

chebyshev evaluates a polynomial from its Chebyshev series representation at a set of points.

For full information please refer to the NAG Library document for c06dc

https://www.nag.com/numeric/nl/nagdoc_29.3/flhtml/c06/c06dcf.html

Parameters

xfloat, array-like, shape $\left(\text{lx}\right)$

$x\in X$, the set of arguments of the series.

xminfloat

The lower end point of the interval $[x_{min},x_{max}]$.

xmaxfloat

The upper end point of the interval $[x_{min},x_{max}]$.

cfloat, array-like, shape $\left(n\right)$

$c[\text{j}-1]$ must contain the coefficient $c_{\text{j}}$ of the Chebyshev series, for $\text{j}=1,2,\dots ,n$.

sint

Determines the series (see Notes).

$s=1$

The series is general.

$s=2$

The series is even.

$s=3$

The series is odd.

Returns

resfloat, ndarray, shape $\left(\text{lx}\right)$

The Chebyshev series evaluated at the set of points $X$.

Raises

NagValueError

(errno $1$)

On entry, $\text{lx}=⟨value⟩$.

Constraint: $\text{lx}\ge 1$.

(errno $2$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 1$.

(errno $3$)

On entry, $s=⟨value⟩$.

Constraint: $s=1$, $2$ or $3$.

(errno $4$)

On entry, $xmax=⟨value⟩$ and $xmin=⟨value⟩$.

Constraint: $xmin<xmax$.

(errno $5$)

On entry, element $x[⟨value⟩]=⟨value⟩$, $xmin=⟨value⟩$ and $xmax=⟨value⟩$.

Constraint: $xmin\le x\left[i\right]\le xmax$, for all $i$.

Notes

chebyshev evaluates, at each point in a given set $X$, the sum of a Chebyshev series of one of three forms according to the value of the parameter $s$:

$s=1$:	$0.5c_1+{\sum }_2^n{c}_j{T}_{j-1}\left(\overline{x}\right)$
$s=2$:	$0.5c_1+{\sum }_2^n{c}_j{T}_{2j-2}\left(\overline{x}\right)$
$s=3$:	${\sum }_1^n{c}_j{T}_{2j-1}\left(\overline{x}\right)$

where $\overline{x}$ lies in the range $-1.0\le \overline{x}\le 1.0$. Here $T_r\left(x\right)$ is the Chebyshev polynomial of order $r$ in $\overline{x}$, defined by $\cos \left(ry\right)$ where $\cos \left(y\right)=\overline{x}$.

It is assumed that the independent variable $\overline{x}$ in the interval $[-1.0,+1.0]$ was obtained from your original variable $x\in X$, a set of real numbers in the interval $[x_{min},x_{max}]$, by the linear transformation

$$\overline{x}=\frac{2x-(x_{max}+x_{min})}{x_{max}-x_{min}}\text{.}$$

The method used is based upon a three-term recurrence relation; for details see Clenshaw (1962).

The coefficients $c_j$ are normally generated by other functions, for example they may be those returned by the interpolation function interp.dim1_cheb (in vector $\text{a}$), by a least squares fitting function in submodule fit, or as the solution of a boundary value problem by ode.bvp_coll_nth, ode.bvp_coll_sys or ode.bvp_ps_lin_solve.

References

Clenshaw, C W, 1962, Chebyshev Series for Mathematical Functions, Mathematical tables, HMSO