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spline.cpp
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spline.cpp
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/* spline.c
Cubic interpolating spline. */
/************************************************/
/* */
/* CMATH. Copyright (c) 1989 Design Software */
/* */
/************************************************/
#include <math.h>
/*-----------------------------------------------------------------*/
int spline(int n, int end1, int end2,
double slope1, double slope2,
double x[], double y[],
double b[], double c[], double d[],
int *iflag)
{ /* begin procedure spline() */
int nm1, ib, i;
double t;
int ascend;
nm1 = n - 1;
*iflag = 0;
if (n < 2)
{ /* no possible interpolation */
*iflag = 1;
goto LeaveSpline;
}
ascend = 1;
for (i = 1; i < n; ++i) if (x[i] <= x[i - 1]) ascend = 0;
if (!ascend)
{
*iflag = 2;
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() */
/*-------------------------------------------------------------------*/
double seval(int n, double u,
double x[], double y[],
double b[], double c[], double d[],
int *last)
{ /* begin function seval() */
int i, j, k;
double w;
i = *last;
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);
}
*last = i;
/* ---- Evaluate the spline ---- */
w = u - x[i];
w = y[i] + w * (b[i] + w * (c[i] + w * d[i]));
return (w);
}
/*-------------------------------------------------------------------*/
double deriv(int n, double u,
double x[],
double b[], double c[], double d[],
int *last)
{ /* begin function deriv() */
int i, j, k;
double w;
i = *last;
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);
}
*last = i;
/* ---- 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() */
/*-------------------------------------------------------------------*/
double sinteg(int n, double u,
double x[], double y[],
double b[], double c[], double d[],
int *last)
/*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
Last : the segment that was last used to evaluate U
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
Last : the segment in which u lies
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.
*/
/*-------------------------------------------------------------------*/
{ /* begin function sinteg() */
int i, j, k;
double sum, dx;
i = *last;
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);
}
*last = i;
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);
}
/*-------------------------------------------------------------------*/