Action: BF_LEGENDRE
Module | ves |
---|---|
Description | Usage |
Legendre polynomials basis functions. |
Details and examples
Legendre polynomials basis functions.
Use as basis functions Legendre polynomials defined on a bounded interval. You need to provide the interval on which the basis functions are to be used, and the order of the expansion (i.e. the highest order polynomial used). The total number of basis functions is as the constant is also included. These basis functions should not be used for periodic CVs.
Intrinsically the Legendre polynomials are defined on the interval . A variable in the interval is transformed to a variable in the intrinsic interval by using the transform function
The Legendre polynomials are given by the recurrence relation
The first 6 polynomials are shown below
The Legendre polynomial are orthogonal over the interval
By using the SCALED keyword the polynomials are scaled by a factor of such that they are orthonormal to 1.
From the above equation it follows that integral of the basis functions over the uniform target distribution are given by
and thus always zero except for the constant .
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.
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
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 |