Line data    Source code

       1             : /* +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
       2             :    Copyright (c) 2016-2021 The VES code team
       3             :    (see the PEOPLE-VES file at the root of this folder for a list of names)
       4             : 
       5             :    See http://www.ves-code.org for more information.
       6             : 
       7             :    This file is part of VES code module.
       8             : 
       9             :    The VES code module is free software: you can redistribute it and/or modify
      10             :    it under the terms of the GNU Lesser General Public License as published by
      11             :    the Free Software Foundation, either version 3 of the License, or
      12             :    (at your option) any later version.
      13             : 
      14             :    The VES code module is distributed in the hope that it will be useful,
      15             :    but WITHOUT ANY WARRANTY; without even the implied warranty of
      16             :    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
      17             :    GNU Lesser General Public License for more details.
      18             : 
      19             :    You should have received a copy of the GNU Lesser General Public License
      20             :    along with the VES code module.  If not, see <http://www.gnu.org/licenses/>.
      21             : +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ */
      22             : 
      23             : #include "BasisFunctions.h"
      24             : 
      25             : #include "core/ActionRegister.h"
      26             : 
      27             : 
      28             : namespace PLMD {
      29             : namespace ves {
      30             : 
      31             : //+PLUMEDOC VES_BASISF BF_LEGENDRE
      32             : /*
      33             : Legendre polynomials basis functions.
      34             : 
      35             : Use as basis functions [Legendre polynomials](https://en.wikipedia.org/wiki/Legendre_polynomials)
      36             : $P_{n}(x)$ defined on a bounded interval.
      37             : You need to provide the interval $[a,b]$
      38             : on which the basis functions are to be used, and the order of the
      39             : expansion $N$ (i.e. the highest order polynomial used).
      40             : The total number of basis functions is $N+1$ as the constant $P_{0}(x)=1$
      41             : is also included.
      42             : These basis functions should not be used for periodic CVs.
      43             : 
      44             : Intrinsically the Legendre polynomials are defined on the interval $[-1,1]$.
      45             : A variable $t$ in the interval $[a,b]$ is transformed to a variable $x$
      46             : in the intrinsic interval $[-1,1]$ by using the transform function
      47             : 
      48             : $$
      49             : x(t) = \frac{t-(a+b)/2}
      50             : {(b-a)/2}
      51             : $$
      52             : 
      53             : The Legendre polynomials are given by the recurrence relation
      54             : 
      55             : $$
      56             : \begin{aligned}
      57             : P_{0}(x)    &= 1 \\
      58             : P_{1}(x)    &= x \\
      59             : P_{n+1}(x)  &= \frac{2n+1}{n+1} \, x \, P_{n}(x) -  \frac{n}{n+1} \, P_{n-1}(x)
      60             : \end{aligned}
      61             : $$
      62             : 
      63             : The first 6 polynomials are shown below
      64             : 
      65             : ![A graph showing the first 6 Legendre polynomials](figures/ves_basisf-legendre.png)
      66             : 
      67             : The Legendre polynomial are orthogonal over the interval $[-1,1]$
      68             : 
      69             : $$
      70             : \int_{-1}^{1} dx \, P_{n}(x)\, P_{m}(x)  =  \frac{2}{2n+1} \delta_{n,m}
      71             : $$
      72             : 
      73             : By using the SCALED keyword the polynomials are scaled by a factor of
      74             : $\sqrt{\frac{2n+1}{2}}$ such that they are orthonormal to 1.
      75             : 
      76             : 
      77             : From the above equation it follows that integral of the basis functions
      78             : over the uniform target distribution $p_{\mathrm{u}}(x)$ are given by
      79             : 
      80             : $$
      81             : \int_{-1}^{1} dx \, P_{n}(x) p_{\mathrm{u}}(x) =  \delta_{n,0},
      82             : $$
      83             : 
      84             : and thus always zero except for the constant $P_{0}(x)=1$.
      85             : 
      86             : 
      87             : For further mathematical properties of the Legendre polynomials see for example
      88             : the [Wikipedia page](https://en.wikipedia.org/wiki/Legendre_polynomials).
      89             : 
      90             : ## Examples
      91             : 
      92             : Here we employ a Legendre expansion of order 20 over the interval -4.0 to 8.0.
      93             : This results in a total number of 21 basis functions.
      94             : The label used to identify  the basis function action can then be
      95             : referenced later on in the input file.
      96             : 
      97             : ```plumed
      98             : bf_leg: BF_LEGENDRE MINIMUM=-4.0 MAXIMUM=8.0 ORDER=20
      99             : ```
     100             : 
     101             : */
     102             : //+ENDPLUMEDOC
     103             : 
     104             : class BF_Legendre : public BasisFunctions {
     105             :   bool scaled_;
     106             :   void setupUniformIntegrals() override;
     107             : public:
     108             :   static void registerKeywords(Keywords&);
     109             :   explicit BF_Legendre(const ActionOptions&);
     110             :   void getAllValues(const double, double&, bool&, std::vector<double>&, std::vector<double>&) const override;
     111             : };
     112             : 
     113             : 
     114             : PLUMED_REGISTER_ACTION(BF_Legendre,"BF_LEGENDRE")
     115             : 
     116             : 
     117          66 : void BF_Legendre::registerKeywords(Keywords& keys) {
     118          66 :   BasisFunctions::registerKeywords(keys);
     119          66 :   keys.addFlag("SCALED",false,"Scale the polynomials such that they are orthonormal to 1.");
     120          66 : }
     121             : 
     122          62 : BF_Legendre::BF_Legendre(const ActionOptions&ao):
     123             :   PLUMED_VES_BASISFUNCTIONS_INIT(ao),
     124          62 :   scaled_(false) {
     125          62 :   parseFlag("SCALED",scaled_);
     126         124 :   addKeywordToList("SCALED",scaled_);
     127          62 :   setNumberOfBasisFunctions(getOrder()+1);
     128         124 :   setIntrinsicInterval("-1.0","+1.0");
     129             :   setNonPeriodic();
     130             :   setIntervalBounded();
     131          62 :   setType("Legendre");
     132          62 :   setDescription("Legendre polynomials");
     133          62 :   setLabelPrefix("L");
     134          62 :   setupBF();
     135          62 :   checkRead();
     136          62 : }
     137             : 
     138             : 
     139     1625168 : void BF_Legendre::getAllValues(const double arg, double& argT, bool& inside_range, std::vector<double>& values, std::vector<double>& derivs) const {
     140     1625168 :   inside_range=true;
     141     1625168 :   argT=translateArgument(arg, inside_range);
     142     1625168 :   std::vector<double> derivsT(derivs.size());
     143             :   //
     144     1625168 :   values[0]=1.0;
     145     1625168 :   derivsT[0]=0.0;
     146     1625168 :   derivs[0]=0.0;
     147     1625168 :   values[1]=argT;
     148     1625168 :   derivsT[1]=1.0;
     149     1625168 :   derivs[1]=intervalDerivf();
     150    15470653 :   for(unsigned int i=1; i < getOrder(); i++) {
     151    13845485 :     double io = static_cast<double>(i);
     152    13845485 :     values[i+1]  = ((2.0*io+1.0)/(io+1.0))*argT*values[i] - (io/(io+1.0))*values[i-1];
     153    13845485 :     derivsT[i+1] = ((2.0*io+1.0)/(io+1.0))*(values[i]+argT*derivsT[i])-(io/(io+1.0))*derivsT[i-1];
     154    13845485 :     derivs[i+1]  = intervalDerivf()*derivsT[i+1];
     155             :   }
     156     1625168 :   if(scaled_) {
     157             :     // L0 is also scaled!
     158     5088852 :     for(unsigned int i=0; i<values.size(); i++) {
     159     4632062 :       double io = static_cast<double>(i);
     160     4632062 :       double sf = sqrt(io+0.5);
     161     4632062 :       values[i] *= sf;
     162     4632062 :       derivs[i] *= sf;
     163             :     }
     164             :   }
     165     1625168 :   if(!inside_range) {
     166      131604 :     for(unsigned int i=0; i<derivs.size(); i++) {
     167      120046 :       derivs[i]=0.0;
     168             :     }
     169             :   }
     170     1625168 : }
     171             : 
     172             : 
     173          60 : void BF_Legendre::setupUniformIntegrals() {
     174          60 :   setAllUniformIntegralsToZero();
     175             :   double L0_int = 1.0;
     176          60 :   if(scaled_) {
     177             :     L0_int = sqrt(0.5);
     178             :   }
     179             :   setUniformIntegral(0,L0_int);
     180          60 : }
     181             : 
     182             : 
     183             : }
     184             : }