Line data    Source code

       1             : /* +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
       2             :    Copyright (c) 2016-2018 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_CHEBYSHEV
      32             : /*
      33             : Chebyshev polynomial basis functions.
      34             : 
      35             : Use as basis functions [Chebyshev polynomials](https://en.wikipedia.org/wiki/Chebyshev_polynomials)
      36             : of the first kind \f$T_{n}(x)\f$ 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$T_{0}(x)=1\f$
      41             : is also included.
      42             : These basis functions should not be used for periodic CVs.
      43             : 
      44             : Intrinsically the Chebyshev 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 Chebyshev polynomials are given by the recurrence relation
      53             : \f{align}{
      54             : T_{0}(x)    &= 1 \\
      55             : T_{1}(x)    &= x \\
      56             : T_{n+1}(x)  &= 2 \, x \, T_{n}(x) - T_{n-1}(x)
      57             : \f}
      58             : 
      59             : The first 6 polynomials are shown below
      60             : \image html ves_basisf-chebyshev.png
      61             : 
      62             : The Chebyshev polynomial are orthogonal over the interval \f$[-1,1]\f$
      63             : with respect to the weight \f$\frac{1}{\sqrt{1-x^2}}\f$
      64             : \f[
      65             : \int_{-1}^{1} dx \, T_{n}(x)\, T_{m}(x) \, \frac{1}{\sqrt{1-x^2}} =
      66             : \begin{cases}
      67             : 0 & n \neq m \\
      68             : \pi & n = m = 0 \\
      69             : \pi/2 & n = m \neq 0
      70             : \end{cases}
      71             : \f]
      72             : 
      73             : For further mathematical properties of the Chebyshev polynomials see for example
      74             : the [Wikipedia page](https://en.wikipedia.org/wiki/Chebyshev_polynomials).
      75             : 
      76             : \par Examples
      77             : 
      78             : Here we employ a Chebyshev expansion of order 20 over the interval 0.0 to 10.0.
      79             : This results in a total number of 21 basis functions.
      80             : The label used to identify  the basis function action can then be
      81             : referenced later on in the input file.
      82             : \plumedfile
      83             : bfC: BF_CHEBYSHEV MINIMUM=0.0 MAXIMUM=10.0 ORDER=20
      84             : \endplumedfile
      85             : 
      86             : */
      87             : //+ENDPLUMEDOC
      88             : 
      89             : 
      90           4 : class BF_Chebyshev : public BasisFunctions {
      91             :   virtual void setupUniformIntegrals();
      92             : public:
      93             :   static void registerKeywords(Keywords&);
      94             :   explicit BF_Chebyshev(const ActionOptions&);
      95             :   void getAllValues(const double, double&, bool&, std::vector<double>&, std::vector<double>&) const;
      96             :   double getInnerProductWeight(const double arg) const;
      97          14 :   std::string getInnerProductWeightStr() const {return "1/sqrt(1-x^2)";}
      98             : };
      99             : 
     100             : 
     101        7840 : PLUMED_REGISTER_ACTION(BF_Chebyshev,"BF_CHEBYSHEV")
     102             : 
     103             : 
     104           5 : void BF_Chebyshev::registerKeywords(Keywords& keys) {
     105           5 :   BasisFunctions::registerKeywords(keys);
     106           5 : }
     107             : 
     108           4 : BF_Chebyshev::BF_Chebyshev(const ActionOptions&ao):
     109           4 :   PLUMED_VES_BASISFUNCTIONS_INIT(ao)
     110             : {
     111           4 :   setNumberOfBasisFunctions(getOrder()+1);
     112          12 :   setIntrinsicInterval("-1.0","+1.0");
     113             :   setNonPeriodic();
     114             :   setOrthogonal();
     115             :   setIntervalBounded();
     116           8 :   setType("chebyshev-1st-kind");
     117           8 :   setDescription("Chebyshev polynomials of the first kind");
     118           8 :   setLabelPrefix("T");
     119           4 :   setupBF();
     120           4 :   checkRead();
     121           4 : }
     122             : 
     123             : 
     124     1397256 : void BF_Chebyshev::getAllValues(const double arg, double& argT, bool& inside_range, std::vector<double>& values, std::vector<double>& derivs) const {
     125             :   // plumed_assert(values.size()==numberOfBasisFunctions());
     126             :   // plumed_assert(derivs.size()==numberOfBasisFunctions());
     127     1397256 :   inside_range=true;
     128     1397256 :   argT=translateArgument(arg, inside_range);
     129     1397256 :   std::vector<double> derivsT(derivs.size());
     130             :   //
     131     1397256 :   values[0]=1.0;
     132     1397256 :   derivsT[0]=0.0;
     133     1397256 :   derivs[0]=0.0;
     134     1397256 :   values[1]=argT;
     135     1397256 :   derivsT[1]=1.0;
     136     1397256 :   derivs[1]=intervalDerivf();
     137    29319836 :   for(unsigned int i=1; i < getOrder(); i++) {
     138   106101296 :     values[i+1]  = 2.0*argT*values[i]-values[i-1];
     139    79575972 :     derivsT[i+1] = 2.0*values[i]+2.0*argT*derivsT[i]-derivsT[i-1];
     140    26525324 :     derivs[i+1]  = intervalDerivf()*derivsT[i+1];
     141             :   }
     142     1401202 :   if(!inside_range) {for(unsigned int i=0; i<derivs.size(); i++) {derivs[i]=0.0;}}
     143     1397256 : }
     144             : 
     145             : 
     146           3 : void BF_Chebyshev::setupUniformIntegrals() {
     147         112 :   for(unsigned int i=0; i<numberOfBasisFunctions(); i++) {
     148          53 :     double io = i;
     149             :     double value = 0.0;
     150          53 :     if(i % 2 == 0) {
     151          28 :       value = -2.0/( pow(io,2.0)-1.0)*0.5;
     152             :     }
     153             :     setUniformIntegral(i,value);
     154             :   }
     155           3 : }
     156             : 
     157             : 
     158           0 : double BF_Chebyshev::getInnerProductWeight(const double arg) const {
     159             :   bool inside_range=true;
     160             :   double argT=translateArgument(arg, inside_range);
     161           0 :   double value = 1/sqrt(1-argT*argT);
     162           0 :   if(!inside_range) {value = 0.0;}
     163           0 :   return value;
     164             : }
     165             : 
     166             : 
     167             : }
     168        5874 : }