The R Project SVN R

/*
 *  R : A Computer Language for Statistical Data Analysis
 *  Copyright (C) 1995, 1996  Robert Gentleman and Ross Ihaka
 *  Copyright (C) 1998--2001  Robert Gentleman, Ross Ihaka and the
 *                            R Development Core Team
 *
 *  This program is free software; you can redistribute it and/or modify
 *  it under the terms of the GNU General Public License as published by
 *  the Free Software Foundation; either version 2 of the License, or
 *  (at your option) any later version.
 *
 *  This program is distributed in the hope that it will be useful,
 *  but WITHOUT ANY WARRANTY; without even the implied warranty of
 *  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
 *  GNU General Public License for more details.
 *
 *  You should have received a copy of the GNU General Public License
 *  along with this program; if not, write to the Free Software
 *  Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA
 */

/*  Spline Interpolation
 *  --------------------
 *  C code to perform spline fitting and interpolation.
 *  There is code here for:
 *
 *  1. Natural splines.
 *
 *  2. Periodic splines
 *
 *  3. Splines with end-conditions determined by fitting
 *     cubics in the start and end intervals (Forsythe et al).
 *
 *
 *  Computational Techniques
 *  ------------------------
 *  A special LU decomposition for symmetric tridiagonal matrices
 *  is used for all computations, except for periodic splines where
 *  Choleski is more efficient.
 */

#ifdef HAVE_CONFIG_H
#include <config.h>
#endif

#include "R_ext/Arith.h"
#include "R_ext/Applic.h"
#include <stdio.h>
#include <errno.h>

/*
 *  Natural Splines
 *  ---------------
 *  Here the end-conditions are determined by setting the second
 *  derivative of the spline at the end-points to equal to zero.
 *
 *  There are n-2 unknowns (y[i]'' at x[2], ..., x[n-1]) and n-2
 *  equations to determine them.  Either Choleski or Gaussian
 *  elimination could be used.
 */

void natural_spline(int n, double *x, double *y, double *b, double *c, double *d)
{
    int nm1, i;
    double t;

    x--; y--; b--; c--; d--;

    if(n < 2) {
    errno = EDOM;
    return;
    }

    if(n < 3) {
    t = (y[2] - y[1]);
    b[1] = t / (x[2]-x[1]);
    b[2] = b[1];
    c[1] = c[2] = d[1] = d[2] = 0.0;
    return;
    }

    nm1 = n - 1;

    /* Set up the tridiagonal system */
    /* b = diagonal, d = offdiagonal, c = right hand side */

    d[1] = x[2] - x[1];
    c[2] = (y[2] - y[1])/d[1];
    for( i=2 ; i<n ; i++) {
    d[i] = x[i+1] - x[i];
    b[i] = 2.0 * (d[i-1] + d[i]);
    c[i+1] = (y[i+1] - y[i])/d[i];
    c[i] = c[i+1] - c[i];
    }

    /* Gaussian elimination */

    for(i=3 ; i<n ; i++) {
    t = d[i-1]/b[i-1];
    b[i] = b[i] - t*d[i-1];
    c[i] = c[i] - t*c[i-1];
    }

    /* Backward substitution */

    c[nm1] = c[nm1]/b[nm1];
    for(i=n-2 ; i>1 ; i--)
    c[i] = (c[i]-d[i]*c[i+1])/b[i];

    /* End conditions */

    c[1] = c[n] = 0.0;

    /* Get cubic coefficients */

    b[1] = (y[2] - y[1])/d[1] - d[i] * c[2];
    c[1] = 0.0;
    d[1] = c[2]/d[1];
    b[n] = (y[n] - y[nm1])/d[nm1] + d[nm1] * c[nm1];
    for(i=2 ; i<n ; i++) {
    b[i] = (y[i+1]-y[i])/d[i] - d[i]*(c[i+1]+2.0*c[i]);
    d[i] = (c[i+1]-c[i])/d[i];
    c[i] = 3.0*c[i];
    }
    c[n] = 0.0;
    d[n] = 0.0;

    return;
}

/*
 *  Splines a la Forsythe Malcolm and Moler
 *  ---------------------------------------
 *  In this case the end-conditions are determined by fitting
 *  cubic polynomials to the first and last 4 points and matching
 *  the third derivitives of the spline at the end-points to the
 *  third derivatives of these cubics at the end-points.
 */

void fmm_spline(int n, double *x, double *y, double *b, double *c, double *d)
{
    int nm1, i;
    double t;

    /* Adjustment for 1-based arrays */

    x--; y--; b--; c--; d--;

    if(n < 2) {
    errno = EDOM;
    return;
    }

    if(n < 3) {
    t = (y[2] - y[1]);
    b[1] = t / (x[2]-x[1]);
    b[2] = b[1];
    c[1] = c[2] = d[1] = d[2] = 0.0;
    return;
    }

    nm1 = n - 1;

    /* Set up tridiagonal system */
    /* b = diagonal, d = offdiagonal, c = right hand side */

    d[1] = x[2] - x[1];
    c[2] = (y[2] - y[1])/d[1];/* = +/- Inf  for x[1]=x[2] -- problem? */
    for(i=2 ; i<n ; i++) {
    d[i] = x[i+1] - x[i];
    b[i] = 2.0 * (d[i-1] + d[i]);
    c[i+1] = (y[i+1] - y[i])/d[i];
    c[i] = c[i+1] - c[i];
    }

    /* End conditions. */
    /* Third derivatives at x[0] and x[n-1] obtained */
    /* from divided differences */

    b[1] = -d[1];
    b[n] = -d[nm1];
    c[1] = c[n] = 0.0;
    if(n > 3) {
    c[1] = c[3]/(x[4]-x[2]) - c[2]/(x[3]-x[1]);
    c[n] = c[nm1]/(x[n] - x[n-2]) - c[n-2]/(x[nm1]-x[n-3]);
    c[1] = c[1]*d[1]*d[1]/(x[4]-x[1]);
    c[n] = -c[n]*d[nm1]*d[nm1]/(x[n]-x[n-3]);
    }

    /* Gaussian elimination */

    for(i=2 ; i<=n ; i++) {
    t = d[i-1]/b[i-1];
    b[i] = b[i] - t*d[i-1];
    c[i] = c[i] - t*c[i-1];
    }

    /* Backward substitution */

    c[n] = c[n]/b[n];
    for(i=nm1 ; i>=1 ; i--)
    c[i] = (c[i]-d[i]*c[i+1])/b[i];

    /* c[i] is now the sigma[i-1] of the text */
    /* Compute polynomial coefficients */

    b[n] = (y[n] - y[n-1])/d[n-1] + d[n-1]*(c[n-1]+ 2.0*c[n]);
    for(i=1 ; i<=nm1 ; i++) {
    b[i] = (y[i+1]-y[i])/d[i] - d[i]*(c[i+1]+2.0*c[i]);
    d[i] = (c[i+1]-c[i])/d[i];
    c[i] = 3.0*c[i];
    }
    c[n] = 3.0*c[n];
    d[n] = d[nm1];
    return;
}


/*
 *  Periodic Spline
 *  ---------------
 *  The end conditions here match spline (and its derivatives)
 *  at x[1] and x[n].
 *
 *  Note: There is an explicit check that the user has supplied
 *  data with y[1] equal to y[n].
 */

void periodic_spline(int n, double *x, double *y,
             double *b, double *c, double *d, double *e)
{
    double s;
    int i, nm1;

    /* Adjustment for 1-based arrays */

    x--; y--; b--; c--; d--; e--;

    if(n < 2 || y[1] != y[n]) {
    errno = EDOM;
    return;
    }

    nm1 = n-1;

    /* Set up the matrix system */
    /* A = diagonal  B = off-diagonal  C = rhs */

#define A   b
#define B   d
#define C   c

    B[1]  = x[2] - x[1];
    B[nm1]= x[n] - x[nm1];
    A[1] = 2.0 * (B[1] + (x[nm1] - x[n-2]));
    C[1] = (y[2] - y[1])/B[1] - (y[n] - y[nm1])/B[nm1];

    for(i=2 ; i<n ; i++) {
    B[i] = x[i+1] - x[i];
    A[i] = 2.0 * (B[i] + B[i-1]);
    C[i] = (y[i+1] - y[i])/B[i] - (y[i] - y[i-1])/B[i-1];
    }

    /* Choleski decomposition */

#define L   b
#define M   d
#define E   e

    L[1] = sqrt(A[1]);
    E[1] = (x[n] - x[nm1])/L[1];
    s = 0.0;
    for(i=1 ; i<=nm1-2; i++) {
    M[i] = B[i]/L[i];
    if(i != 1) E[i] = -E[i-1] * M[i-1] / L[i];
    L[i+1] = sqrt(A[i+1]-M[i]*M[i]);
    s = s + E[i]*E[i];
    }
    M[nm1-1] = (B[nm1-1] - E[nm1-2] * M[nm1-2])/L[nm1-1];
    L[nm1] = sqrt(A[nm1] - M[nm1-1]*M[nm1-1] - s);

    /* Forward Elimination */

#define Y   c
#define D   c

    Y[1] = D[1]/L[1];
    s = 0.0;
    for(i=2 ; i<=nm1-1 ; i++) {
    Y[i] = (D[i] - M[i-1]*Y[i-1])/L[i];
    s = s + E[i-1] * Y[i-1];
    }
    Y[nm1] = (D[nm1] - M[nm1-1] * Y[nm1-1] - s) / L[nm1];

#define X   c

    /*
      X[nm1] = -Y[nm1]/L[nm1];
      X[nm1-1] = -(Y[nm1-1] + M[nm1-1] * X[nm1])/L[nm1-1];
      for(i=nm1-2 ; i>=1 ; i--)
      X[i] = -(Y[i] + M[i] * X[i+1] + E[i] * X[nm1])/L[i];
    */

    X[nm1] = Y[nm1]/L[nm1];
    X[nm1-1] = (Y[nm1-1] - M[nm1-1] * X[nm1])/L[nm1-1];
    for(i=nm1-2 ; i>=1 ; i--)
    X[i] = (Y[i] - M[i] * X[i+1] - E[i] * X[nm1])/L[i];

    /* Compute polynomial coefficients */

    for(i=1 ; i<=nm1 ; i++) {
    s = x[i+1] - x[i];
    b[i] = (y[i+1]-y[i])/s - s*(c[i+1]+2.0*c[i]);
    d[i] = (c[i+1]-c[i])/s;
    c[i] = 3.0*c[i];
    }
    b[n] = b[1];
    c[n] = c[1];
    d[n] = d[1];
    return;
}
#undef A
#undef B
#undef C
#undef L
#undef M
#undef E
#undef Y
#undef D
#undef X

void spline_coef(int *method, int *n, double *x, double *y,
         double *b, double *c, double *d, double *e)
{
    switch(*method) {
    case 1:
    periodic_spline(*n, x, y, b, c, d, e);
    break;

    case 2:
    natural_spline(*n, x, y, b, c, d);
    break;

    case 3:
    fmm_spline(*n, x, y, b, c, d);
    break;
    }
}

void spline_eval(int *method, int *nu, double *u, double *v,
         int *n, double *x, double *y, double *b, double *c, double *d)
{
    int i, j, k, l;
    double ul, dx, tmp;

    u--; v--;
    x--; y--;
    b--; c--; d--;

    if(*method == 1) {
    dx = x[*n] - x[1];
    for( l=1 ; l<=*nu ; l++) {
        v[l] = fmod(u[l]-x[1], dx);
        if(v[l] < 0.0) v[l] += dx;
        v[l] = v[l] + x[1];
    }
    }
    else {
    for( l=1 ; l<=*nu ; l++)
        v[l] = u[l];
    }

    i = 1;
    for( l=1 ; l<=*nu ; l++) {
    ul = v[l];
    if(ul < x[i] || x[i+1] < ul) {
        i = 1;
        j = *n + 1;
        do {
        k = (i+j)/2;
        if(ul < x[k]) j = k;
        else i = k;
        }
        while(j > i+1);
    }
    dx = ul - x[i];
    /* for natural splines extrapolate linearly left */
    tmp = d[i];
    if(*method == 2 && ul < x[1]) tmp = 0.0;
    v[l] = y[i] + dx*(b[i] + dx*(c[i] + dx*tmp));
    }
}