/************************************************/
/*  adapted from                                */
/*  CMATH.  Copyright (c) 1989 Design Software  */
/*                                              */
/************************************************/


/* Purpose ...
   -------
   Evaluate the coefficients b[i], c[i], d[i], i = 0, 1, .. n-1 for
   a cubic interpolating spline

   S(xx) = Y[i] + b[i] * w + c[i] * w**2 + d[i] * w**3
   where w = xx - x[i]
   and   x[i] <= xx <= x[i+1]

   The n supplied data points are x[i], y[i], i = 0 ... n-1.

   Input :
   -------
   n       : The number of data points or knots (n >= 2)
   end1,
   end2    : = 1 to specify the slopes at the end points
             = 0 to obtain the default conditions
   slope1,
   slope2  : the slopes at the end points x[0] and x[n-1]
             respectively
   x[]     : the abscissas of the knots in strictly
             increasing order
   y[]     : the ordinates of the knots

   Output :
   --------
   b, c, d : arrays of spline coefficients as defined above
             (See note 2 for a definition.)
            = 0 normal return
            = 1 less than two data points; cannot interpolate
            = 2 x[] are not in ascending order

   This C code written by ...  Peter & Nigel,
   ----------------------      Design Software,
                               42 Gubberley St,
                               Kenmore, 4069,
                               Australia.

   Version ... 1.1, 30 September 1987
   -------     2.0, 6 April 1989    (start with zero subscript)
                                     remove ndim from parameter list
               2.1, 28 April 1989   (check on x[])
               2.2, 10 Oct   1989   change number order of matrix

   Notes ...
   -----
   (1) The accompanying function seval() may be used to evaluate the
       spline while deriv will provide the first derivative.
   (2) Using p to denote differentiation
       y[i] = S(X[i])
       b[i] = Sp(X[i])
       c[i] = Spp(X[i])/2
       d[i] = Sppp(X[i])/6  ( Derivative from the right )
   (3) Since the zero elements of the arrays ARE NOW used here,
       all arrays to be passed from the main program should be
       dimensioned at least [n].  These routines will use elements
       [0 .. n-1].
   (4) Adapted from the text
       Forsythe, G.E., Malcolm, M.A. and Moler, C.B. (1977)
       "Computer Methods for Mathematical Computations"
       Prentice Hall
   (5) Note that although there are only n-1 polynomial segments,
       n elements are requird in b, c, d.  The elements b[n-1],
       c[n-1] and d[n-1] are set to continue the last segment
       past x[n-1].
*/

int spline (int n, int end1, int end2,
            double slope1, double slope2,
            double x[], double y[],
            double b[], double c[], double d[])
{  /* begin procedure spline() */

int    nm1, ib, i;
double t;
int    ascend;

nm1    = n - 1;

if (n < 2)
  {  /* no possible interpolation */
  goto LeaveSpline;
  }

ascend = 1;
for (i = 1; i < n; ++i) if (x[i] <= x[i-1]) ascend = 0;
if (!ascend)
   {
   goto LeaveSpline;
   }

if (n >= 3)
   {    /* ---- At least quadratic ---- */

   /* ---- Set up the symmetric tri-diagonal system
           b = diagonal
           d = offdiagonal
           c = right-hand-side  */
   d[0] = x[1] - x[0];
   c[1] = (y[1] - y[0]) / d[0];
   for (i = 1; i < nm1; ++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];
      }

   /* ---- Default End conditions
           Third derivatives at x[0] and x[n-1] obtained
           from divided differences  */
   b[0]   = -d[0];
   b[nm1] = -d[n-2];
   c[0]   = 0.0;
   c[nm1] = 0.0;
   if (n != 3)
      {
      c[0]   = c[2] / (x[3] - x[1]) - c[1] / (x[2] - x[0]);
      c[nm1] = c[n-2] / (x[nm1] - x[n-3]) - c[n-3] / (x[n-2] - x[n-4]);
      c[0]   = c[0] * d[0] * d[0] / (x[3] - x[0]);
      c[nm1] = -c[nm1] * d[n-2] * d[n-2] / (x[nm1] - x[n-4]);
      }

   /* Alternative end conditions -- known slopes */
   if (end1 == 1)
      {
      b[0] = 2.0 * (x[1] - x[0]);
      c[0] = (y[1] - y[0]) / (x[1] - x[0]) - slope1;
      }
   if (end2 == 1)
      {
      b[nm1] = 2.0 * (x[nm1] - x[n-2]);
      c[nm1] = slope2 - (y[nm1] - y[n-2]) / (x[nm1] - x[n-2]);
      }

   /* Forward elimination */
   for (i = 1; 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];
     }

   /* Back substitution */
   c[nm1] = c[nm1] / b[nm1];
   for (ib = 0; ib < nm1; ++ib)
      {
      i    = n - ib - 2;
      c[i] = (c[i] - d[i] * c[i+1]) / b[i];
      }

   /* c[i] is now the sigma[i] of the text */

   /* Compute the polynomial coefficients */
   b[nm1] = (y[nm1] - y[n-2]) / d[n-2] + d[n-2] * (c[n-2] + 2.0 * c[nm1]);
   for (i = 0; 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[nm1] = 3.0 * c[nm1];
   d[nm1] = d[n-2];

   }  /* at least quadratic */

else  /* if n >= 3 */
   {  /* linear segment only  */
   b[0] = (y[1] - y[0]) / (x[1] - x[0]);
   c[0] = 0.0;
   d[0] = 0.0;
   b[1] = b[0];
   c[1] = 0.0;
   d[1] = 0.0;
   }

LeaveSpline:
return 0;
}  /* end of spline() */


/*Purpose ...
  -------
  Evaluate the cubic spline function

  S(xx) = y[i] + b[i] * w + c[i] * w**2 + d[i] * w**3
  where w = u - x[i]
  and   x[i] <= u <= x[i+1]
  Note that Horner's rule is used.
  If u < x[0]   then i = 0 is used.
  If u > x[n-1] then i = n-1 is used.

  Input :
  -------
  n       : The number of data points or knots (n >= 2)
  u       : the abscissa at which the spline is to be evaluated
  x[]     : the abscissas of the knots in strictly increasing order
  y[]     : the ordinates of the knots
  b, c, d : arrays of spline coefficients computed by spline().

  Output :
  --------
  seval   : the value of the spline function at u

  Notes ...
  -----
  (1) If u is not in the same interval as the previous call then a
      binary search is performed to determine the proper interval.

*/

double seval (int n, double u,
              double x[], double y[],
              double b[], double c[], double d[])

{  /* begin function seval() */

int    i, j, k;
double w;

if (i >= n-1) i = 0;
if (i < 0)  i = 0;

if ((x[i] > u) || (x[i+1] < u))
  {  /* ---- perform a binary search ---- */
  i = 0;
  j = n;
  do
    {
    k = (i + j) / 2;         /* split the domain to search */
    if (u < x[k])  j = k;    /* move the upper bound */
    if (u >= x[k]) i = k;    /* move the lower bound */
    }                        /* there are no more segments to search */
  while (j > i+1);
  }

/* ---- Evaluate the spline ---- */
w = u - x[i];
w = y[i] + w * (b[i] + w * (c[i] + w * d[i]));
return (w);
}

/* Purpose ...
   -------
   Evaluate the derivative of the cubic spline function

   S(x) = B[i] + 2.0 * C[i] * w + 3.0 * D[i] * w**2
   where w = u - X[i]
   and   X[i] <= u <= X[i+1]
   Note that Horner's rule is used.
   If U < X[0] then i = 0 is used.
   If U > X[n-1] then i = n-1 is used.

   Input :
   -------
   n       : The number of data points or knots (n >= 2)
   u       : the abscissa at which the derivative is to be evaluated
   x       : the abscissas of the knots in strictly increasing order
   b, c, d : arrays of spline coefficients computed by spline()

   Output :
   --------
   deriv : the value of the derivative of the spline
           function at u

   Notes ...
   -----
   (1) If u is not in the same interval as the previous call then a
       binary search is performed to determine the proper interval.

*/

double deriv (int n, double u,
              double x[],
              double b[], double c[], double d[])
{  /* begin function deriv() */

int    i, j, k;
double w;

if (i >= n-1) i = 0;
if (i < 0) i = 0;

if ((x[i] > u) || (x[i+1] < u))
  {  /* ---- perform a binary search ---- */
  i = 0;
  j = n;
  do
    {
    k = (i + j) / 2;          /* split the domain to search */
    if (u < x[k])  j = k;     /* move the upper bound */
    if (u >= x[k]) i = k;     /* move the lower bound */
    }                         /* there are no more segments to search */
  while (j > i+1);
  }

/* ---- Evaluate the derivative ---- */
w = u - x[i];
w = b[i] + w * (2.0 * c[i] + w * 3.0 * d[i]);
return (w);

} /* end of deriv() */


/*Purpose ...
  -------
  Integrate the cubic spline function

  S(xx) = y[i] + b[i] * w + c[i] * w**2 + d[i] * w**3
  where w = u - x[i]
  and   x[i] <= u <= x[i+1]

  The integral is zero at u = x[0].

  If u < x[0]   then i = 0 segment is extrapolated.
  If u > x[n-1] then i = n-1 segment is extrapolated.

  Input :
  -------
  n       : The number of data points or knots (n >= 2)
  u       : the abscissa at which the spline is to be evaluated
  x[]     : the abscissas of the knots in strictly increasing order
  y[]     : the ordinates of the knots
  b, c, d : arrays of spline coefficients computed by spline().

  Output :
  --------
  sinteg  : the value of the spline function at u

  Notes ...
  -----
  (1) If u is not in the same interval as the previous call then a
      binary search is performed to determine the proper interval.

*/

double sinteg (int n, double u,
              double x[], double y[],
              double b[], double c[], double d[])
{  /* begin function sinteg() */

int    i, j, k;
double sum, dx;

if (i >= n-1) i = 0;
if (i < 0)  i = 0;

if ((x[i] > u) || (x[i+1] < u))
  {  /* ---- perform a binary search ---- */
  i = 0;
  j = n;
  do
    {
    k = (i + j) / 2;         /* split the domain to search */
    if (u < x[k])  j = k;    /* move the upper bound */
    if (u >= x[k]) i = k;    /* move the lower bound */
    }                        /* there are no more segments to search */
  while (j > i+1);
  }

sum = 0.0;
/* ---- Evaluate the integral for segments x < u ---- */
for (j = 0; j < i; ++j)
   {
   dx = x[j+1] - x[j];
   sum += dx *
          (y[j] + dx *
          (0.5 * b[j] + dx *
          (c[j] / 3.0 + dx * 0.25 * d[j])));
   }

/* ---- Evaluate the integral fot this segment ---- */
dx = u - x[i];
sum += dx *
       (y[i] + dx *
       (0.5 * b[i] + dx *
       (c[i] / 3.0 + dx * 0.25 * d[i])));

return (sum);
}