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             : \f$P_{n}(x)\f$ defined on a bounded interval.
      37             : You need to provide the interval \f$[a,b]\f$
      38             : on which the basis functions are to be used, and the order of the
      39             : expansion \f$N\f$ (i.e. the highest order polynomial used).
      40             : The total number of basis functions is \f$N+1\f$ as the constant \f$P_{0}(x)=1\f$
      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 \f$[-1,1]\f$.
      45             : A variable \f$t\f$ in the interval \f$[a,b]\f$ is transformed to a variable \f$x\f$
      46             : in the intrinsic interval \f$[-1,1]\f$ by using the transform function
      47             : \f[
      48             : x(t) = \frac{t-(a+b)/2}
      49             : {(b-a)/2}
      50             : \f]
      51             : 
      52             : The Legendre polynomials are given by the recurrence relation
      53             : \f{align}{
      54             : P_{0}(x)    &= 1 \\
      55             : P_{1}(x)    &= x \\
      56             : P_{n+1}(x)  &= \frac{2n+1}{n+1} \, x \, P_{n}(x) -  \frac{n}{n+1} \, P_{n-1}(x)
      57             : \f}
      58             : 
      59             : The first 6 polynomials are shown below
      60             : \image html ves_basisf-legendre.png
      61             : 
      62             : The Legendre polynomial are orthogonal over the interval \f$[-1,1]\f$
      63             : \f[
      64             : \int_{-1}^{1} dx \, P_{n}(x)\, P_{m}(x)  =  \frac{2}{2n+1} \delta_{n,m}
      65             : \f]
      66             : By using the SCALED keyword the polynomials are scaled by a factor of
      67             : \f$ \sqrt{\frac{2n+1}{2}}\f$ such that they are orthonormal to 1.
      68             : 
      69             : 
      70             : From the above equation it follows that integral of the basis functions
      71             : over the uniform target distribution \f$p_{\mathrm{u}}(x)\f$ are given by
      72             : \f[
      73             : \int_{-1}^{1} dx \, P_{n}(x) p_{\mathrm{u}}(x) =  \delta_{n,0},
      74             : \f]
      75             : and thus always zero except for the constant \f$P_{0}(x)=1\f$.
      76             : 
      77             : 
      78             : For further mathematical properties of the Legendre polynomials see for example
      79             : the [Wikipedia page](https://en.wikipedia.org/wiki/Legendre_polynomials).
      80             : 
      81             : \par Examples
      82             : 
      83             : Here we employ a Legendre expansion of order 20 over the interval -4.0 to 8.0.
      84             : This results in a total number of 21 basis functions.
      85             : The label used to identify  the basis function action can then be
      86             : referenced later on in the input file.
      87             : \plumedfile
      88             : bf_leg: BF_LEGENDRE MINIMUM=-4.0 MAXIMUM=8.0 ORDER=20
      89             : \endplumedfile
      90             : 
      91             : \par Examples
      92             : 
      93             : */
      94             : //+ENDPLUMEDOC
      95             : 
      96             : class BF_Legendre : public BasisFunctions {
      97             :   bool scaled_;
      98             :   void setupUniformIntegrals() override;
      99             : public:
     100             :   static void registerKeywords(Keywords&);
     101             :   explicit BF_Legendre(const ActionOptions&);
     102             :   void getAllValues(const double, double&, bool&, std::vector<double>&, std::vector<double>&) const override;
     103             : };
     104             : 
     105             : 
     106             : PLUMED_REGISTER_ACTION(BF_Legendre,"BF_LEGENDRE")
     107             : 
     108             : 
     109          63 : void BF_Legendre::registerKeywords(Keywords& keys) {
     110          63 :   BasisFunctions::registerKeywords(keys);
     111         126 :   keys.addFlag("SCALED",false,"Scale the polynomials such that they are orthonormal to 1.");
     112          63 : }
     113             : 
     114          61 : BF_Legendre::BF_Legendre(const ActionOptions&ao):
     115             :   PLUMED_VES_BASISFUNCTIONS_INIT(ao),
     116          61 :   scaled_(false)
     117             : {
     118         183 :   parseFlag("SCALED",scaled_); addKeywordToList("SCALED",scaled_);
     119          61 :   setNumberOfBasisFunctions(getOrder()+1);
     120         122 :   setIntrinsicInterval("-1.0","+1.0");
     121             :   setNonPeriodic();
     122             :   setIntervalBounded();
     123          61 :   setType("Legendre");
     124          61 :   setDescription("Legendre polynomials");
     125          61 :   setLabelPrefix("L");
     126          61 :   setupBF();
     127          61 :   checkRead();
     128          61 : }
     129             : 
     130             : 
     131     1625057 : void BF_Legendre::getAllValues(const double arg, double& argT, bool& inside_range, std::vector<double>& values, std::vector<double>& derivs) const {
     132             :   // plumed_assert(values.size()==numberOfBasisFunctions());
     133             :   // plumed_assert(derivs.size()==numberOfBasisFunctions());
     134     1625057 :   inside_range=true;
     135     1625057 :   argT=translateArgument(arg, inside_range);
     136     1625057 :   std::vector<double> derivsT(derivs.size());
     137             :   //
     138     1625057 :   values[0]=1.0;
     139     1625057 :   derivsT[0]=0.0;
     140     1625057 :   derivs[0]=0.0;
     141     1625057 :   values[1]=argT;
     142     1625057 :   derivsT[1]=1.0;
     143     1625057 :   derivs[1]=intervalDerivf();
     144    15470320 :   for(unsigned int i=1; i < getOrder(); i++) {
     145    13845263 :     double io = static_cast<double>(i);
     146    13845263 :     values[i+1]  = ((2.0*io+1.0)/(io+1.0))*argT*values[i] - (io/(io+1.0))*values[i-1];
     147    13845263 :     derivsT[i+1] = ((2.0*io+1.0)/(io+1.0))*(values[i]+argT*derivsT[i])-(io/(io+1.0))*derivsT[i-1];
     148    13845263 :     derivs[i+1]  = intervalDerivf()*derivsT[i+1];
     149             :   }
     150     1625057 :   if(scaled_) {
     151             :     // L0 is also scaled!
     152     5088852 :     for(unsigned int i=0; i<values.size(); i++) {
     153     4632062 :       double io = static_cast<double>(i);
     154     4632062 :       double sf = sqrt(io+0.5);
     155     4632062 :       values[i] *= sf;
     156     4632062 :       derivs[i] *= sf;
     157             :     }
     158             :   }
     159     1756541 :   if(!inside_range) {for(unsigned int i=0; i<derivs.size(); i++) {derivs[i]=0.0;}}
     160     1625057 : }
     161             : 
     162             : 
     163          59 : void BF_Legendre::setupUniformIntegrals() {
     164          59 :   setAllUniformIntegralsToZero();
     165             :   double L0_int = 1.0;
     166          59 :   if(scaled_) {L0_int = sqrt(0.5);}
     167             :   setUniformIntegral(0,L0_int);
     168          59 : }
     169             : 
     170             : 
     171             : }
     172             : }