splines.hpp

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namespace tmbutils {

// Copyright (C) 2013-2015 Kasper Kristensen
// License: GPL-2

/*
 *  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, a copy is available at
 *  http://www.r-project.org/Licenses/
 */

/*      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.
 */

template <class Type>
class splinefun
{
private:
  /* Input to R's "spline_coef" */
  int method[1];
  int n[1];
  Type *x;
  Type *y;
  Type *b;
  Type *c;
  Type *d;
  Type *e;

  /* Not used */
  //int errno, EDOM;

  /* Memory management helpers */
  void clear() {
    if (*n != 0) {
      delete [] x;
      delete [] y;
      delete [] b;
      delete [] c;
      delete [] d;
      delete [] e;
    }
  }
  void alloc(int n_) {
    *n = n_;
    x = new Type[*n];
    y = new Type[*n];
    b = new Type[*n];
    c = new Type[*n];
    d = new Type[*n];
    e = new Type[*n];
  }
  void copy_from(const splinefun &fun) {
    *method = fun.method[0];
    *n = fun.n[0];
    alloc(*n);
    for(int i = 0; i < *n; i++) {
      x[i] = fun.x[i];
      y[i] = fun.y[i];
      b[i] = fun.b[i];
      c[i] = fun.c[i];
      d[i] = fun.d[i];
      e[i] = fun.e[i];
    }
  }

public:
  /* Construct spline */
  splinefun() {
    x = y = b = c = d = e = NULL;
    *method = *n = 0;
  };
  splinefun(const splinefun &fun){
    copy_from(fun);
  }
  splinefun& operator=(const splinefun &x) {
    if (this != &x) {
      (*this).clear();
      (*this).copy_from(x);
    }
    return *this;
  }
  ~splinefun() {
    clear();
  };
  splinefun(const vector<Type> &x_,
            const vector<Type> &y_,
            int method_ = 3) {
    method[0] = method_;
    n[0] = x_.size();
    alloc( x_.size() );
    for(int i=0; i < *n; i++) {
      x[i] = x_[i];
      y[i] = y_[i];
    }
    spline_coef(method, n, x, y, b, c, d, e);
  }

  Type operator()(const Type &x_) {
    Type u[1];
    Type v[1];
    int nu[1];
    u[0] = x_;
    nu[0] = 1;
    spline_eval(method, nu, u, v,
                n, x, y, b, c, d);
    return v[0];
  }
  vector<Type> operator() (const vector<Type> &x) {
    vector<Type> y(x.size());
    for (int i=0; i<x.size(); i++) y[i] = (*this)(x[i]);
    return y;
  }

  /* ------------------------------------------------------------------ 
   * The following is copy-pasted from R-package "stats" where "double"
   * has been replaced by "Type".
   *-------------------------------------------------------------------
   */

  /*
   *    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, Type *x, Type *y, Type *b, Type *c, Type *d)
  {
    int nm1, i;
    Type 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, Type *x, Type *y, Type *b, Type *c, Type *d)
  {
    int nm1, i;
    Type 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, Type *x, Type *y,
                       Type *b, Type *c, Type *d, Type *e)
  {
    Type 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;
    }
    
    if(n == 2) {
      b[1] = b[2] = c[1] = c[2] = d[1] = d[2] = 0.0;
      return;
    } else if (n == 3) {
      b[1] = b[2] = b[3] = -(y[1] - y[2])*(x[1] - 2*x[2] + x[3])/(x[3]-x[2])/(x[2]-x[1]);
      c[1] = -3*(y[1]-y[2])/(x[3]-x[2])/(x[2]-x[1]);
      c[2] = -c[1];
      c[3] = c[1];
      d[1] = -2*c[1]/3/(x[2]-x[1]);
      d[2] = -d[1]*(x[2]-x[1])/(x[3]-x[2]);
      d[3] = d[1];
      return;
    }
    
    /* else --------- n >= 4 --------- */
    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] + B[nm1]);
    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];
    
    /* Wrap around */
    
    X[n] = X[1];
    
    /* 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, Type *x, Type *y,
                   Type *b, Type *c, Type *d, Type *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, Type *u, Type *v,
                   int *n, Type *x, Type *y, Type *b, Type *c, Type *d)
  {
    /* Evaluate  v[l] := spline(u[l], ...),         l = 1,..,nu, i.e. 0:(nu-1)
     * Nodes x[i], coef (y[i]; b[i],c[i],d[i]); i = 1,..,n , i.e. 0:(*n-1)
     */
    const int n_1 = *n - 1;
    int i, j, k, l;
    Type ul, dx, tmp;
    
    if(*method == 1 && *n > 1) { /* periodic */
      dx = x[n_1] - x[0];
      for(l = 0; l < *nu; l++) {

        /* WARNING - "fmod(AD<double>,AD<double>)" is not defined */
        v[l] = fmod(asDouble(u[l]-x[0]), asDouble(dx));
        if(v[l] < 0.0) v[l] += dx;
        v[l] += x[0];
      }
    }
    else {
      for(l = 0; l < *nu; l++)
        v[l] = u[l];
    }
    
    i = 0;
    for(l = 0; l < *nu; l++) {
      ul = v[l];
      if(ul < x[i] || (i < n_1 && x[i+1] < ul)) {
        /* reset i  such that  x[i] <= ul <= x[i+1] : */
        i = 0;
        j = *n;
        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 = (*method == 2 && ul < x[0]) ? 0.0 : d[i];
      
      v[l] = y[i] + dx*(b[i] + dx*(c[i] + dx*tmp));
    }
  }
  
};

}