thirdparty_interpolate.cpp

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/* This file is part of Cloudy and is copyright (C)1978-2017 by Gary J. Ferland and
 * others.  For conditions of distribution and use see copyright notice in license.txt */
#include "cddefines.h"
#include "thirdparty.h"

STATIC double *d3_np_fs( long n, double a[], const double b[], double x[] );

//****************************************************************************80
//
// The routines d3_np_fs, spline_cubic_set, spline_cubic_val where written
// by John Burkardt (Computer Science Department, Florida State University)
// and have been slightly modified and adapted for use in Cloudy by Peter
// van Hoof (Royal Observatory of Belgium).
//
// The original sources can be found at
//    http://www.scs.fsu.edu/~burkardt/cpp_src/spline/spline.html
//
//****************************************************************************80

STATIC double *d3_np_fs( long n, double a[], const double b[], double x[] )

//****************************************************************************80
//
//  Purpose:
//
//    D3_NP_FS factors and solves a D3 system.
//
//  Discussion:
//
//    The D3 storage format is used for a tridiagonal matrix.
//    The superdiagonal is stored in entries (1,2:N), the diagonal in
//    entries (2,1:N), and the subdiagonal in (3,1:N-1).  Thus, the
//    original matrix is "collapsed" vertically into the array.
//
//    This algorithm requires that each diagonal entry be nonzero.
//    It does not use pivoting, and so can fail on systems that
//    are actually nonsingular.
//
//  Example:
//
//    Here is how a D3 matrix of order 5 would be stored:
//
//       *  A12 A23 A34 A45
//      A11 A22 A33 A44 A55
//      A21 A32 A43 A54  *
//
//  Modified:
//
//    15 November 2003
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, long N, the order of the linear system.
//
//    Input/output, double A[3*N].
//    On input, the nonzero diagonals of the linear system.
//    On output, the data in these vectors has been overwritten
//    by factorization information.
//
//    Input, double B[N], the right hand side.
//
//    Output, double X[N] and D3_NP_FS[N], the solution of the linear system.
//    D3_NP_FS returns NULL if there was an error because one of the diagonal
//    entries was zero.
//
{
        long i;
        double xmult;

        DEBUG_ENTRY( "d3_np_fs()" );
//
//  Check.
//
        for( i = 0; i < n; i++ )
        {
                if( a[1+i*3] == 0.0 )
                {
                        return NULL;
                }
        }

        x[0] = b[0];

        for( i = 1; i < n; i++ )
        {
                xmult = a[2+(i-1)*3] / a[1+(i-1)*3];
                a[1+i*3] = a[1+i*3] - xmult * a[0+i*3];
                x[i] = b[i] - xmult * x[i-1];
        }

        x[n-1] = x[n-1] / a[1+(n-1)*3];
        for( i = n-2; 0 <= i; i-- )
        {
                x[i] = ( x[i] - a[0+(i+1)*3] * x[i+1] ) / a[1+i*3];
        }
        return x;
}

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

void spline_cubic_set( long n, const double t[], const double y[], double ypp[],
                       int ibcbeg, double ybcbeg, int ibcend, double ybcend )

//****************************************************************************80
//
//  Purpose:
//
//    SPLINE_CUBIC_SET computes the second derivatives of a piecewise cubic spline.
//
//  Discussion:
//
//    For data interpolation, the user must call SPLINE_SET to determine
//    the second derivative data, passing in the data to be interpolated,
//    and the desired boundary conditions.
//
//    The data to be interpolated, plus the SPLINE_SET output, defines
//    the spline.  The user may then call SPLINE_VAL to evaluate the
//    spline at any point.
//
//    The cubic spline is a piecewise cubic polynomial.  The intervals
//    are determined by the "knots" or abscissas of the data to be
//    interpolated.  The cubic spline has continous first and second
//    derivatives over the entire interval of interpolation.
//
//    For any point T in the interval T(IVAL), T(IVAL+1), the form of
//    the spline is
//
//      SPL(T) = A(IVAL)
//             + B(IVAL) * ( T - T(IVAL) )
//             + C(IVAL) * ( T - T(IVAL) )**2
//             + D(IVAL) * ( T - T(IVAL) )**3
//
//    If we assume that we know the values Y(*) and YPP(*), which represent
//    the values and second derivatives of the spline at each knot, then
//    the coefficients can be computed as:
//
//      A(IVAL) = Y(IVAL)
//      B(IVAL) = ( Y(IVAL+1) - Y(IVAL) ) / ( T(IVAL+1) - T(IVAL) )
//        - ( YPP(IVAL+1) + 2 * YPP(IVAL) ) * ( T(IVAL+1) - T(IVAL) ) / 6
//      C(IVAL) = YPP(IVAL) / 2
//      D(IVAL) = ( YPP(IVAL+1) - YPP(IVAL) ) / ( 6 * ( T(IVAL+1) - T(IVAL) ) )
//
//    Since the first derivative of the spline is
//
//      SPL'(T) =     B(IVAL)
//              + 2 * C(IVAL) * ( T - T(IVAL) )
//              + 3 * D(IVAL) * ( T - T(IVAL) )**2,
//
//    the requirement that the first derivative be continuous at interior
//    knot I results in a total of N-2 equations, of the form:
//
//      B(IVAL-1) + 2 C(IVAL-1) * (T(IVAL)-T(IVAL-1))
//      + 3 * D(IVAL-1) * (T(IVAL) - T(IVAL-1))**2 = B(IVAL)
//
//    or, setting H(IVAL) = T(IVAL+1) - T(IVAL)
//
//      ( Y(IVAL) - Y(IVAL-1) ) / H(IVAL-1)
//      - ( YPP(IVAL) + 2 * YPP(IVAL-1) ) * H(IVAL-1) / 6
//      + YPP(IVAL-1) * H(IVAL-1)
//      + ( YPP(IVAL) - YPP(IVAL-1) ) * H(IVAL-1) / 2
//      =
//      ( Y(IVAL+1) - Y(IVAL) ) / H(IVAL)
//      - ( YPP(IVAL+1) + 2 * YPP(IVAL) ) * H(IVAL) / 6
//
//    or
//
//      YPP(IVAL-1) * H(IVAL-1) + 2 * YPP(IVAL) * ( H(IVAL-1) + H(IVAL) )
//      + YPP(IVAL) * H(IVAL)
//      =
//      6 * ( Y(IVAL+1) - Y(IVAL) ) / H(IVAL)
//      - 6 * ( Y(IVAL) - Y(IVAL-1) ) / H(IVAL-1)
//
//    Boundary conditions must be applied at the first and last knots.  
//    The resulting tridiagonal system can be solved for the YPP values.
//
//  Modified:
//
//    06 February 2004
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, long N, the number of data points.  N must be at least 2.
//    In the special case where N = 2 and IBCBEG = IBCEND = 0, the
//    spline will actually be linear.
//
//    Input, double T[N], the knot values, that is, the points were data is
//    specified.  The knot values should be distinct, and increasing.
//
//    Input, double Y[N], the data values to be interpolated.
//
//    Input, int IBCBEG, left boundary condition flag:
//      0: the cubic spline should be a quadratic over the first interval;
//      1: the first derivative at the left endpoint should be YBCBEG;
//      2: the second derivative at the left endpoint should be YBCBEG.
//
//    Input, double YBCBEG, the values to be used in the boundary
//    conditions if IBCBEG is equal to 1 or 2.
//
//    Input, int IBCEND, right boundary condition flag:
//      0: the cubic spline should be a quadratic over the last interval;
//      1: the first derivative at the right endpoint should be YBCEND;
//      2: the second derivative at the right endpoint should be YBCEND.
//
//    Input, double YBCEND, the values to be used in the boundary
//    conditions if IBCEND is equal to 1 or 2.
//
//    Output, double YPP[N] and SPLINE_CUBIC_SET[N], the second derivatives
//    of the cubic spline.
//
{
        long i;

        DEBUG_ENTRY( "spline_cubic_set()" );
//
//  Check.
//
        ASSERT( n >= 2 );

#       ifndef NDEBUG
        for( i = 0; i < n - 1; i++ )
        {
                if( t[i+1] <= t[i] )
                {
                        fprintf( ioQQQ, "SPLINE_CUBIC_SET - Fatal error!\n" );
                        fprintf( ioQQQ,  "  The knots must be strictly increasing\n" );
                        cdEXIT(EXIT_FAILURE);
                }
        }
#       endif

        vector<double> a(3*n), b(n);
//
//  Set up the first equation.
//
        if( ibcbeg == 0 )
        {
                b[0] = 0.0;
                a[1+0*3] = 1.0;
                a[0+1*3] = -1.0;
        }
        else if( ibcbeg == 1 )
        {
                b[0] = ( y[1] - y[0] ) / ( t[1] - t[0] ) - ybcbeg;
                a[1+0*3] = ( t[1] - t[0] ) / 3.0;
                a[0+1*3] = ( t[1] - t[0] ) / 6.0;
        }
        else if( ibcbeg == 2 )
        {
                b[0] = ybcbeg;
                a[1+0*3] = 1.0;
                a[0+1*3] = 0.0;
        }
        else
        {
                fprintf( ioQQQ, "SPLINE_CUBIC_SET - Fatal error!\n" );
                fprintf( ioQQQ, "  IBCBEG must be 0, 1 or 2, but I found %d.\n", ibcbeg );
                cdEXIT(EXIT_FAILURE);
        }
//
//  Set up the intermediate equations.
//
        for( i = 1; i < n-1; i++ )
        {
                b[i] = ( y[i+1] - y[i] ) / ( t[i+1] - t[i] )
                        - ( y[i] - y[i-1] ) / ( t[i] - t[i-1] );
                a[2+(i-1)*3] = ( t[i] - t[i-1] ) / 6.0;
                a[1+ i   *3] = ( t[i+1] - t[i-1] ) / 3.0;
                a[0+(i+1)*3] = ( t[i+1] - t[i] ) / 6.0;
        }
//
//  Set up the last equation.
//
        if( ibcend == 0 )
        {
                b[n-1] = 0.0;
                a[2+(n-2)*3] = -1.0;
                a[1+(n-1)*3] = 1.0;
        }
        else if( ibcend == 1 )
        {
                b[n-1] = ybcend - ( y[n-1] - y[n-2] ) / ( t[n-1] - t[n-2] );
                a[2+(n-2)*3] = ( t[n-1] - t[n-2] ) / 6.0;
                a[1+(n-1)*3] = ( t[n-1] - t[n-2] ) / 3.0;
        }
        else if( ibcend == 2 )
        {
                b[n-1] = ybcend;
                a[2+(n-2)*3] = 0.0;
                a[1+(n-1)*3] = 1.0;
        }
        else
        {
                fprintf( ioQQQ, "SPLINE_CUBIC_SET - Fatal error!\n" );
                fprintf( ioQQQ, "  IBCEND must be 0, 1 or 2, but I found %d.\n", ibcend );
                cdEXIT(EXIT_FAILURE);
        }
//
//  Solve the linear system.
//
        if( n == 2 && ibcbeg == 0 && ibcend == 0 )
        {
                ypp[0] = 0.0;
                ypp[1] = 0.0;
        }
        else
        {
                if( d3_np_fs( n, &a[0], &b[0], ypp ) == NULL )
                {
                        fprintf( ioQQQ, "SPLINE_CUBIC_SET - Fatal error!\n" );
                        fprintf( ioQQQ, "  The linear system could not be solved.\n" );
                        cdEXIT(EXIT_FAILURE);
                }
        }

        return;
}

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

void spline_cubic_val( long n, const double t[], double tval, const double y[], const double ypp[],
                       double *yval, double *ypval, double *yppval )

//****************************************************************************80
//
//  Purpose:
//
//    SPLINE_CUBIC_VAL evaluates a piecewise cubic spline at a point.
//
//  Discussion:
//
//    SPLINE_CUBIC_SET must have already been called to define the values of YPP.
//
//    For any point T in the interval T(IVAL), T(IVAL+1), the form of
//    the spline is
//
//      SPL(T) = A
//             + B * ( T - T(IVAL) )
//             + C * ( T - T(IVAL) )**2
//             + D * ( T - T(IVAL) )**3
//
//    Here:
//      A = Y(IVAL)
//      B = ( Y(IVAL+1) - Y(IVAL) ) / ( T(IVAL+1) - T(IVAL) )
//        - ( YPP(IVAL+1) + 2 * YPP(IVAL) ) * ( T(IVAL+1) - T(IVAL) ) / 6
//      C = YPP(IVAL) / 2
//      D = ( YPP(IVAL+1) - YPP(IVAL) ) / ( 6 * ( T(IVAL+1) - T(IVAL) ) )
//
//  Modified:
//
//    04 February 1999
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, long N, the number of knots.
//
//    Input, double Y[N], the data values at the knots.
//
//    Input, double T[N], the knot values.
//
//    Input, double TVAL, a point, typically between T[0] and T[N-1], at
//    which the spline is to be evalulated.  If TVAL lies outside
//    this range, extrapolation is used.
//
//    Input, double Y[N], the data values at the knots.
//
//    Input, double YPP[N], the second derivatives of the spline at
//    the knots.
//
//    Output, double *YPVAL, the derivative of the spline at TVAL.
//
//    Output, double *YPPVAL, the second derivative of the spline at TVAL.
//
//    Output, double SPLINE_VAL, the value of the spline at TVAL.
//
{
        DEBUG_ENTRY( "spline_cubic_val()" );
//
//  Determine the interval [ T(I), T(I+1) ] that contains TVAL.
//  Values below T[0] or above T[N-1] use extrapolation.
//
        long ival = hunt_bisect( t, n, tval );
//
//  In the interval I, the polynomial is in terms of a normalized
//  coordinate between 0 and 1.
//
        double dt = tval - t[ival];
        double h = t[ival+1] - t[ival];

        if( yval != NULL )
        {
                *yval = y[ival]
                        + dt * ( ( y[ival+1] - y[ival] ) / h
                        - ( ypp[ival+1] / 6.0 + ypp[ival] / 3.0 ) * h
                        + dt * ( 0.5 * ypp[ival]
                        + dt * ( ( ypp[ival+1] - ypp[ival] ) / ( 6.0 * h ) ) ) );
        }
        if( ypval != NULL )
        {
                *ypval = ( y[ival+1] - y[ival] ) / h
                        - ( ypp[ival+1] / 6.0 + ypp[ival] / 3.0 ) * h
                        + dt * ( ypp[ival]
                        + dt * ( 0.5 * ( ypp[ival+1] - ypp[ival] ) / h ) );
        }
        if( yppval != NULL )
        {
                *yppval = ypp[ival] + dt * ( ypp[ival+1] - ypp[ival] ) / h;
        }
        return;
}

//****************************************************************************80
//
// the routines lagrange and linint where written by Peter van Hoof (ROB)
//
//****************************************************************************80

double lagrange(const double x[], /* x[n] */
                const double y[], /* y[n] */
                long n,
                double xval)
{
        double yval = 0.;

        DEBUG_ENTRY( "lagrange()" );

        for( long i=0; i < n; i++ )
        {
                double l = 1.;
                for( long j=0; j < n; j++ )
                {
                        if( i != j )
                                l *= (xval-x[j])/(x[i]-x[j]);
                }
                yval += y[i]*l;
        }
        return yval;
}

double linint(const double x[], /* x[n] */
              const double y[], /* y[n] */
              long n,
              double xval)
{
        double yval;

        DEBUG_ENTRY( "linint()" );

        ASSERT( n >= 2 );

        if( xval <= x[0] )
                yval = y[0];
        else if( xval >= x[n-1] )
                yval = y[n-1];
        else
        {
                long ilo = hunt_bisect( x, n, xval );
                double deriv = (y[ilo+1]-y[ilo])/(x[ilo+1]-x[ilo]);
                yval = y[ilo] + deriv*(xval-x[ilo]);
        }
        return yval;
}