lagrange_approx_1d.cpp

# include <cstdlib>
# include <iostream>
# include <iomanip>
# include <cmath>

using namespace std;

# include "lagrange_approx_1d.hpp"
# include "qr_solve.hpp"
# include "r8lib.hpp"

//****************************************************************************80

double *lagrange_approx_1d ( int m, int nd, double xd[], double yd[], 
  int ni, double xi[] )

//****************************************************************************80
//
//  Purpose:
//
//    LAGRANGE_APPROX_1D evaluates the Lagrange approximant of degree M.
//
//  Discussion:
//
//    The Lagrange approximant L(M,ND,XD,YD)(X) is a polynomial of
//    degree M which approximates the data (XD(I),YD(I)) for I = 1 to ND.
//
//    We can represent any polynomial of degree M+1 as the sum of the Lagrange 
//    basis functions at the M+1 Chebyshev points.
//
//      L(M)(X) = sum ( 1 <= I <= M+1 ) C(I) LB(M,XC)(X)
//
//    Given our data, we can seek the M+1 unknown coefficients C which minimize
//    the norm of || L(M)(XD(1:ND)) - YD(1:ND) ||.
//
//    Given the coefficients, we can then evaluate the polynomial at the
//    points XI.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    09 October 2012
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int M, the polynomial degree.
//
//    Input, int ND, the number of data points.
//    ND must be at least 1.
//
//    Input, double XD[ND], the data points.
//
//    Input, double YD[ND], the data values.
//
//    Input, int NI, the number of interpolation points.
//
//    Input, double XI[NI], the interpolation points.
//
//    Output, double LAGRANGE_APPROX_1D[NI], the interpolated values.
//
{
  double a;
  double b;
  double *ld;
  double *li;
  int nc;
  double *xc;
  double *yc;
  double *yi;

  nc = m + 1;
//
//  Evaluate the Chebyshev points.
//
  a = -1.0;
  b = +1.0;
  xc = r8vec_cheby_extreme_new ( nc, a, b );
//
//  Evaluate the Lagrange basis functions for the Chebyshev points 
//  at the data points.
//
  ld = lagrange_basis_1d ( nc, xc, nd, xd );
//
//  The value of the Lagrange approximant at each data point should
//  approximate the data value: LD * YC = YD, where YC are the unknown
//  coefficients.
//
  yc = qr_solve ( nd, nc, ld, yd );
//
//  Now we want to evaluate the Lagrange approximant at the "interpolant
//  points": LI * YC = YI
//
  li = lagrange_basis_1d ( nc, xc, ni, xi );

  yi = r8mat_mv_new ( ni, nc, li, yc );

  delete [] ld;
  delete [] li;
  delete [] xc;
  delete [] yc;

  return yi;
}
//****************************************************************************80

double *lagrange_basis_1d ( int nd, double xd[], int ni, double xi[] )

//****************************************************************************80
//
//  Purpose:
//
//    LAGRANGE_BASIS_1D evaluates the Lagrange basis polynomials.
//
//  Discussion:
//
//    Given ND distinct abscissas, XD(1:ND),
//    the I-th Lagrange basis polynomial LB(I)(T) is defined as the polynomial of
//    degree ND - 1 which is 1 at XD(I) and 0 at the ND - 1
//    other abscissas.
//
//    A formal representation is:
//
//      LB(I)(T) = Product ( 1 <= J <= ND, I /= J )
//       ( T - T(J) ) / ( T(I) - T(J) )
//
//    This routine accepts a set of NI values at which all the Lagrange
//    basis polynomials should be evaluated.
//
//    Given data values YD at each of the abscissas, the value of the
//    Lagrange interpolating polynomial at each of the interpolation points
//    is then simple to compute by matrix multiplication:
//
//      YI(1:NI) = LB(1:NI,1:ND) * YD(1:ND)
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    09 October 2012
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int ND, the number of data points.
//    ND must be at least 1.
//
//    Input, double XD[ND], the data points.
//
//    Input, int NI, the number of interpolation points.
//
//    Input, double XI[NI], the interpolation points.
//
//    Output, double LAGRANGE_BASIS[NI*ND], the values
//    of the Lagrange basis polynomials at the interpolation points.
//
{
  int i;
  int j;
  int k;
  double *lb;
//
//  Evaluate the polynomial.
//
  lb = new double[ni*nd];

  for ( j = 0; j < nd; j++ )
  {
    for ( i = 0; i < ni; i++ )
    {
      lb[i+j*ni] = 1.0;
    }
  }

  for ( i = 0; i < nd; i++ )
  {
    for ( j = 0; j < nd; j++ )
    {
      if ( j != i )
      {
        for ( k = 0; k < ni; k++ )
        {
          lb[k+i*ni] = lb[k+i*ni] * ( xi[k] - xd[j] ) / ( xd[i] - xd[j] );
        }
      }
    }
  }

  return lb;
}