Action: BF_LEGENDRE

Module	ves
Description	Usage
Legendre polynomials basis functions.

Details and examples

Legendre polynomials basis functions.

Use as basis functions Legendre polynomials $P_{n}(x)$ 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 $P_{0}(x)=1$ is also included. These basis functions should not be used for periodic CVs.

Intrinsically the Legendre 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 Legendre polynomials are given by the recurrence relation

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

The first 6 polynomials are shown below

The Legendre polynomial are orthogonal over the interval $[-1,1]$

$$\int_{-1}^{1} dx \, P_{n}(x)\, P_{m}(x) = \frac{2}{2n+1} \delta_{n,m}$$

By using the SCALED keyword the polynomials are scaled by a factor of $\sqrt{\frac{2n+1}{2}}$ such that they are orthonormal to 1.

From the above equation it follows that integral of the basis functions over the uniform target distribution $p_{\mathrm{u}}(x)$ are given by

$$\int_{-1}^{1} dx \, P_{n}(x) p_{\mathrm{u}}(x) = \delta_{n,0},$$

and thus always zero except for the constant $P_{0}(x)=1$.

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

Examples

Here we employ a Legendre expansion of order 20 over the interval -4.0 to 8.0. 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

bf_leg: BF_LEGENDRELegendre polynomials basis functions. More details MINIMUMThe minimum of the interval on which the basis functions are defined=-4.0 MAXIMUMThe maximum of the interval on which the basis functions are defined=8.0 ORDERThe order of the basis function expansion=20
The BF_LEGENDRE action with label bf_leg 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_INFO	optional	false	Print out more detailed information about the basis set
NUMERICAL_INTEGRALS	optional	false	Calculate basis function integral for the uniform distribution numerically
SCALED	optional	false	Scale the polynomials such that they are orthonormal to 1