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_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 $T_{n}(x)$ 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 $T_{0}(x)=1$
      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 $[-1,1]$.
      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 Chebyshev polynomials are given by the recurrence relation
      54             : 
      55             : $$
      56             : \begin{aligned}
      57             : T_{0}(x)    &= 1 \\
      58             : T_{1}(x)    &= x \\
      59             : T_{n+1}(x)  &= 2 \, x \, T_{n}(x) - T_{n-1}(x)
      60             : \end{aligned}
      61             : $$
      62             : 
      63             : The first 6 polynomials are shown below
      64             : 
      65             : ![Graph showing first 6 Chebyshev polynomials](figures/ves_basisf-chebyshev.png)
      66             : 
      67             : The Chebyshev polynomial are orthogonal over the interval $[-1,1]$
      68             : with respect to the weight $\frac{1}{\sqrt{1-x^2}}$
      69             : 
      70             : $$
      71             : \int_{-1}^{1} dx \, T_{n}(x)\, T_{m}(x) \, \frac{1}{\sqrt{1-x^2}} =
      72             : \begin{cases}
      73             : 0 & n \neq m \\
      74             : \pi & n = m = 0 \\
      75             : \pi/2 & n = m \neq 0
      76             : \end{cases}
      77             : $$
      78             : 
      79             : For further mathematical properties of the Chebyshev polynomials see for example
      80             : the [Wikipedia page](https://en.wikipedia.org/wiki/Chebyshev_polynomials).
      81             : 
      82             : ## Examples
      83             : 
      84             : Here we employ a Chebyshev expansion of order 20 over the interval 0.0 to 10.0.
      85             : This results in a total number of 21 basis functions.
      86             : The label used to identify  the basis function action can then be
      87             : referenced later on in the input file.
      88             : 
      89             : ```plumed
      90             : bfC: BF_CHEBYSHEV MINIMUM=0.0 MAXIMUM=10.0 ORDER=20
      91             : ```
      92             : 
      93             : */
      94             : //+ENDPLUMEDOC
      95             : 
      96             : 
      97             : class BF_Chebyshev : public BasisFunctions {
      98             :   void setupUniformIntegrals() override;
      99             : public:
     100             :   static void registerKeywords(Keywords&);
     101             :   explicit BF_Chebyshev(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_Chebyshev,"BF_CHEBYSHEV")
     107             : 
     108             : 
     109           6 : void BF_Chebyshev::registerKeywords(Keywords& keys) {
     110           6 :   BasisFunctions::registerKeywords(keys);
     111           6 : }
     112             : 
     113           4 : BF_Chebyshev::BF_Chebyshev(const ActionOptions&ao):
     114           4 :   PLUMED_VES_BASISFUNCTIONS_INIT(ao) {
     115           4 :   setNumberOfBasisFunctions(getOrder()+1);
     116           8 :   setIntrinsicInterval("-1.0","+1.0");
     117             :   setNonPeriodic();
     118             :   setIntervalBounded();
     119           4 :   setType("chebyshev-1st-kind");
     120           4 :   setDescription("Chebyshev polynomials of the first kind");
     121           4 :   setLabelPrefix("T");
     122           4 :   setupBF();
     123           4 :   checkRead();
     124           4 : }
     125             : 
     126             : 
     127        7882 : void BF_Chebyshev::getAllValues(const double arg, double& argT, bool& inside_range, std::vector<double>& values, std::vector<double>& derivs) const {
     128             :   // plumed_assert(values.size()==numberOfBasisFunctions());
     129             :   // plumed_assert(derivs.size()==numberOfBasisFunctions());
     130        7882 :   inside_range=true;
     131        7882 :   argT=translateArgument(arg, inside_range);
     132        7882 :   std::vector<double> derivsT(derivs.size());
     133             :   //
     134        7882 :   values[0]=1.0;
     135        7882 :   derivsT[0]=0.0;
     136        7882 :   derivs[0]=0.0;
     137        7882 :   values[1]=argT;
     138        7882 :   derivsT[1]=1.0;
     139        7882 :   derivs[1]=intervalDerivf();
     140      135100 :   for(unsigned int i=1; i < getOrder(); i++) {
     141      127218 :     values[i+1]  = 2.0*argT*values[i]-values[i-1];
     142      127218 :     derivsT[i+1] = 2.0*values[i]+2.0*argT*derivsT[i]-derivsT[i-1];
     143      127218 :     derivs[i+1]  = intervalDerivf()*derivsT[i+1];
     144             :   }
     145        7882 :   if(!inside_range) {
     146        2024 :     for(unsigned int i=0; i<derivs.size(); i++) {
     147        1922 :       derivs[i]=0.0;
     148             :     }
     149             :   }
     150        7882 : }
     151             : 
     152             : 
     153           3 : void BF_Chebyshev::setupUniformIntegrals() {
     154          56 :   for(unsigned int i=0; i<numberOfBasisFunctions(); i++) {
     155          53 :     double io = i;
     156             :     double value = 0.0;
     157          53 :     if(i % 2 == 0) {
     158          28 :       value = -2.0/( pow(io,2.0)-1.0)*0.5;
     159             :     }
     160             :     setUniformIntegral(i,value);
     161             :   }
     162           3 : }
     163             : 
     164             : 
     165             : }
     166             : }