Action: BF_CHEBYSHEV

Module	ves
Description	Usage
Chebyshev polynomial basis functions.

Details and examples

Chebyshev polynomial basis functions.

Use as basis functions Chebyshev polynomials of the first kind $T_{n}(x)$ defined on a bounded interval. You need to provide the interval $[a,b]$ on which the basis functions are to be used, and the order of the expansion $N$ (i.e. the highest order polynomial used). The total number of basis functions is $N+1$ as the constant $T_{0}(x)=1$ is also included. These basis functions should not be used for periodic CVs.

Intrinsically the Chebyshev polynomials are defined on the interval $[-1,1]$. A variable $t$ in the interval $[a,b]$ is transformed to a variable $x$ in the intrinsic interval $[-1,1]$ by using the transform function

$$x(t) = \frac{t-(a+b)/2} {(b-a)/2}$$

The Chebyshev polynomials are given by the recurrence relation

$$\begin{aligned} T_{0}(x) &= 1 \\ T_{1}(x) &= x \\ T_{n+1}(x) &= 2 \, x \, T_{n}(x) - T_{n-1}(x) \end{aligned}$$

The first 6 polynomials are shown below

The Chebyshev polynomial are orthogonal over the interval $[-1,1]$ with respect to the weight $\frac{1}{\sqrt{1-x^2}}$

$$\int_{-1}^{1} dx \, T_{n}(x)\, T_{m}(x) \, \frac{1}{\sqrt{1-x^2}} = \begin{cases} 0 & n \neq m \\ \pi & n = m = 0 \\ \pi/2 & n = m \neq 0 \end{cases}$$

For further mathematical properties of the Chebyshev polynomials see for example the Wikipedia page.

Examples

Here we employ a Chebyshev expansion of order 20 over the interval 0.0 to 10.0. This results in a total number of 21 basis functions. The label used to identify the basis function action can then be referenced later on in the input file.

Click on the labels of the actions for more information on what each action computes

bfC: BF_CHEBYSHEVChebyshev polynomial basis functions. More details MINIMUMThe minimum of the interval on which the basis functions are defined=0.0 MAXIMUMThe maximum of the interval on which the basis functions are defined=10.0 ORDERThe order of the basis function expansion=20
The BF_CHEBYSHEV action with label bfC calculates something

Full list of keywords

The following table describes the keywords and options that can be used with this action

Keyword	Type	Default	Description
ORDER	compulsory	none	The order of the basis function expansion
MINIMUM	compulsory	none	The minimum of the interval on which the basis functions are defined
MAXIMUM	compulsory	none	The maximum of the interval on which the basis functions are defined
DEBUG_INFOThis keyword do not have examples	optional	false	Print out more detailed information about the basis set
NUMERICAL_INTEGRALSThis keyword do not have examples	optional	false	Calculate basis function integral for the uniform distribution numerically