#ifndef lint
static char *RCSid() { return RCSid("$Id: interpol.c,v 1.39.2.2 2012/08/23 21:42:26 sfeam Exp $"); }
#endif

/* GNUPLOT - interpol.c */

/*[
 * Copyright 1986 - 1993, 1998, 2004   Thomas Williams, Colin Kelley
 *
 * Permission to use, copy, and distribute this software and its
 * documentation for any purpose with or without fee is hereby granted,
 * provided that the above copyright notice appear in all copies and
 * that both that copyright notice and this permission notice appear
 * in supporting documentation.
 *
 * Permission to modify the software is granted, but not the right to
 * distribute the complete modified source code.  Modifications are to
 * be distributed as patches to the released version.  Permission to
 * distribute binaries produced by compiling modified sources is granted,
 * provided you
 *   1. distribute the corresponding source modifications from the
 *    released version in the form of a patch file along with the binaries,
 *   2. add special version identification to distinguish your version
 *    in addition to the base release version number,
 *   3. provide your name and address as the primary contact for the
 *    support of your modified version, and
 *   4. retain our contact information in regard to use of the base
 *    software.
 * Permission to distribute the released version of the source code along
 * with corresponding source modifications in the form of a patch file is
 * granted with same provisions 2 through 4 for binary distributions.
 *
 * This software is provided "as is" without express or implied warranty
 * to the extent permitted by applicable law.
]*/


/*
 * C-Source file identification Header
 *
 * This file belongs to a project which is:
 *
 * done 1993 by MGR-Software, Asgard  (Lars Hanke)
 * written by Lars Hanke
 *
 * Contact me via:
 *
 *  InterNet: mgr@asgard.bo.open.de
 *      FIDO: Lars Hanke @ 2:243/4802.22   (as long as they keep addresses)
 *
 **************************************************************************
 *
 *   Project: gnuplot
 *    Module:
 *      File: interpol.c
 *
 *   Revisor: Lars Hanke
 *   Revised: 26/09/93
 *  Revision: 1.0
 *
 **************************************************************************
 *
 * LEGAL
 *  This module is part of gnuplot and distributed under whatever terms
 *  gnuplot is or will be published, unless exclusive rights are claimed.
 *
 * DESCRIPTION
 *  Supplies 2-D data interpolation and approximation routines
 *
 * IMPORTS
 *  plot.h
 *    - cp_extend()
 *    - structs: curve_points, coordval, coordinate
 *
 *  setshow.h
 *    - samples, axis array[] variables
 *    - plottypes
 *
 *  proto.h
 *    - solve_tri_diag()
 *    - typedef tri_diag
 *
 * EXPORTS
 *  gen_interp()
 *  sort_points()
 *  cp_implode()
 *
 * BUGS and TODO
 *  I would really have liked to use Gershon Elbers contouring code for
 *  all the stuff done here, but I failed. So I used my own code.
 *  If somebody is able to consolidate Gershon's code for this purpose
 *  a lot of gnuplot users would be very happy - due to memory problems.
 *
 **************************************************************************
 *
 * HISTORY
 * Changes:
 *  Nov 24, 1995  Markus Schuh (M.Schuh@meteo.uni-koeln.de):
 *      changed the algorithm for csplines
 *      added algorithm for approximation csplines
 *      copied point storage and range fix from plot2d.c
 *
 *  Dec 12, 1995 David Denholm
 *      oops - at the time this is called, stored co-ords are
 *      internal (ie maybe log of data) but min/max are in
 *      user co-ordinates.
 *      Work with min and max of internal co-ords, and
 *      check at the end whether external min and max need to
 *      be increased. (since samples_1 is typically 100 ; we
 *      dont want to take more logs than necessary)
 *      Also, need to take into account which axes are active
 *
 *  Jun 30, 1996 Jens Emmerich
 *      implemented handling of UNDEFINED points
 */

#include "interpol.h"

#include "alloc.h"
#include "axis.h"
#include "contour.h"
#include "graphics.h"
#include "misc.h"
#include "plot2d.h"
/*  #include "setshow.h" */
#include "util.h"


/* in order to support multiple axes, and to simplify ranging in
 * parametric plots, we use arrays to store some things. For 2d plots,
 * elements are z=0,y1=1,x1=2,z2=4,y2=5,x2=6 these are given symbolic
 * names in plot.h
 */


/*
 * IMHO, code is getting too cluttered with repeated chunks of
 * code. Some macros to simplify, I hope.
 */


/* store VALUE or log(VALUE) in STORE, set TYPE as appropriate Do
 * OUT_ACTION or UNDEF_ACTION as appropriate. Adjust range provided
 * type is INRANGE (ie dont adjust y if x is outrange). VALUE must not
 * be same as STORE */
/* FIXME 20010610: UNDEF_ACTION is completely unused ??? Furthermore,
 * this is so similar to STORE_WITH_LOG_AND_UPDATE_RANGE() from axis.h
 * that the two should probably be merged.  */
#define STORE_AND_FIXUP_RANGE(store, value, type, min, max, auto,	\
			      out_action, undef_action)			\
do {									\
    store=value;							\
    if (type != INRANGE)						\
	break;  /* don't set y range if x is outrange, for example */	\
    if ((value) < (min)) {						\
       if ((auto) & AUTOSCALE_MIN)					\
	   (min) = (value);						\
       else {								\
	   (type) = OUTRANGE;						\
	   out_action;							\
	   break;							\
       }								\
    }									\
    if ((value) > (max)) {						\
       if ((auto) & AUTOSCALE_MAX)					\
	   (max) = (value);						\
       else {								\
	   (type) = OUTRANGE;						\
	   out_action;							\
       }								\
    }									\
} while(0)

#define UPDATE_RANGE(TEST,OLD,NEW,AXIS)		\
do {						\
    if (TEST)					\
	(OLD) = AXIS_DE_LOG_VALUE(AXIS,NEW);	\
} while(0)

#define spline_coeff_size 4
typedef double spline_coeff[spline_coeff_size];
typedef double five_diag[5];

static int next_curve __PROTO((struct curve_points * plot, int *curve_start));
static int num_curves __PROTO((struct curve_points * plot));
static void eval_kdensity __PROTO((struct curve_points *cp, 
				   int first_point, int num_points, double sr,
				   coordval *px, coordval *py));
static void do_kdensity __PROTO((struct curve_points *cp, int first_point,
				 int num_points, struct coordinate *dest));
static double *cp_binomial __PROTO((int points));
static void eval_bezier __PROTO((struct curve_points * cp, int first_point,
				 int num_points, double sr, coordval * px,
				 coordval *py, double *c));
static void do_bezier __PROTO((struct curve_points * cp, double *bc, int first_point, int num_points, struct coordinate * dest));
static int solve_five_diag __PROTO((five_diag m[], double r[], double x[], int n));
static spline_coeff *cp_approx_spline __PROTO((struct curve_points * plot, int first_point, int num_points));
static spline_coeff *cp_tridiag __PROTO((struct curve_points * plot, int first_point, int num_points));
static void do_cubic __PROTO((struct curve_points * plot, spline_coeff * sc, int first_point, int num_points, struct coordinate * dest));
static void do_freq __PROTO((struct curve_points *plot,	int first_point, int num_points));
int compare_points __PROTO((SORTFUNC_ARGS p1, SORTFUNC_ARGS p2));


/*
 * position curve_start to index the next non-UNDEFINDED point,
 * start search at initial curve_start,
 * return number of non-UNDEFINDED points from there on,
 * if no more valid points are found, curve_start is set
 * to plot->p_count and 0 is returned
 */

static int
next_curve(struct curve_points *plot, int *curve_start)
{
    int curve_length;

    /* Skip undefined points */
    while (*curve_start < plot->p_count
	   && plot->points[*curve_start].type == UNDEFINED) {
	(*curve_start)++;
    };
    curve_length = 0;
    /* curve_length is first used as an offset, then the correct # points */
    while ((*curve_start) + curve_length < plot->p_count
	   && plot->points[(*curve_start) + curve_length].type != UNDEFINED) {
	curve_length++;
    };
    return (curve_length);
}


/*
 * determine the number of curves in plot->points, separated by
 * UNDEFINED points
 */

static int
num_curves(struct curve_points *plot)
{
    int curves;
    int first_point;
    int num_points;

    first_point = 0;
    curves = 0;
    while ((num_points = next_curve(plot, &first_point)) > 0) {
	curves++;
	first_point += num_points;
    }
    return (curves);
}


/* PKJ - May 2008 
   kdensity (short for Kernel Density) builds histograms using
   "Kernel Density Estimation" using Gaussian Kernels.
   Check: L. Wassermann: "All of Statistics" for example.

   The implementation is based closely on the implementation for Bezier
   curves, except for the way the actual interpolation is generated.
*/

/* eval_kdensity is a modification of eval_bezier */
static void 
eval_kdensity ( 
    struct curve_points *cp,
    int first_point,	/* where to start in plot->points (to find x-range) */
    int num_points,	/* to determine end in plot->points */
    double sr,		/* position inside curve, range [0:1] */
    coordval *px,	/* OUTPUT: x and y */
    coordval *py ) {

    unsigned int i;
    unsigned int n = num_points - 1;
    struct coordinate GPHUGE *this_points = (cp->points) + first_point;
  
    double x, y, tmp;
    double avg, sigma;
    double min =  DBL_MAX;
    double max = -DBL_MAX;
    double bandwidth, default_bandwidth;

    avg = 0.0;
    sigma = 0.0;
    for (i = 0; i <= n; i++) {
      avg   += this_points[i].x;
      sigma += this_points[i].x * this_points[i].x;

      /* Find min and max of x-range. Necessary since points not sorted! */
      min = this_points[i].x < min ? this_points[i].x : min;
      max = this_points[i].x > max ? this_points[i].x : max;
    }
    avg /= (double)num_points;
    sigma = sqrt( sigma/(double)num_points - avg*avg ); /* Standard Deviation */
    
    /* This is the optimal bandwidth if the point distribution is Gaussian.
       (Applied Smoothing Techniques for Data Analysis
       by Adrian W, Bowman & Adelchi Azzalini (1997)) */
    default_bandwidth = pow( 4.0/(3.0*num_points), 1.0/5.0 )*sigma;

    /* If the supplied bandwidth is zero of less, the default bandwidth
       is used. If only two columns have been specified in the using
       declaration, the value of the third column (z-value) is set to
       the value of '-1' in plot2d.c::get_data(). Therefore, a negative
       value of z indicates that the default bandwidth should be used.
       (I would prefer that a different magic value would be used to
       indicate a missing third column, so that we we could fail if 
       the user supplies a negative bandwidth, but the -1 seems too
       deeply ingrained.) */

    x = min + sr*(max-min); /* The current x-value */

    y = 0;
    for (i = 0; i <= n; i++) {
      if ( this_points[i].z <= 0 ) {
	bandwidth = default_bandwidth;
      } else {
	bandwidth = this_points[i].z;
      }
      
      tmp = ( x - this_points[i].x )/bandwidth;
      y += this_points[i].y * exp( - 0.5*tmp*tmp ) / bandwidth;
    }
    y /= sqrt(2.0*M_PI);

    *px = x;
    *py = y;
}

/* do_kdensity is based on do_bezier, except for the call to eval_bezier */
static void 
do_kdensity( 
    struct curve_points *cp,
    int first_point,		/* where to start in plot->points */
    int num_points,		/* to determine end in plot->points */
    struct coordinate *dest)	/* where to put the interpolated data */
{
    int i;
    coordval x, y;

    /* min and max in internal (eg logged) co-ordinates. We update
     * these, then update the external extrema in user co-ordinates
     * at the end.
     */

    double ixmin, ixmax, iymin, iymax;
    double sxmin, sxmax, symin, symax;	/* starting values of above */

    x_axis = cp->x_axis;
    y_axis = cp->y_axis;

    ixmin = sxmin = AXIS_LOG_VALUE(x_axis, X_AXIS.min);
    ixmax = sxmax = AXIS_LOG_VALUE(x_axis, X_AXIS.max);
    iymin = symin = AXIS_LOG_VALUE(y_axis, Y_AXIS.min);
    iymax = symax = AXIS_LOG_VALUE(y_axis, Y_AXIS.max);

    for (i = 0; i < samples_1; i++) {
        eval_kdensity( cp, first_point, num_points,
		       (double) i / (double) (samples_1 - 1),
		       &x, &y );

	/* now we have to store the points and adjust the ranges */
	dest[i].type = INRANGE;
	STORE_AND_FIXUP_RANGE( dest[i].x, x, dest[i].type, ixmin, ixmax, 
			       X_AXIS.autoscale, NOOP, continue);
	STORE_AND_FIXUP_RANGE( dest[i].y, y, dest[i].type, iymin, iymax, 
			       Y_AXIS.autoscale, NOOP, NOOP);

	dest[i].xlow = dest[i].xhigh = dest[i].x;
	dest[i].ylow = dest[i].yhigh = dest[i].y;

	dest[i].z = -1;
    }

    UPDATE_RANGE(ixmax > sxmax, X_AXIS.max, ixmax, x_axis);
    UPDATE_RANGE(ixmin < sxmin, X_AXIS.min, ixmin, x_axis);
    UPDATE_RANGE(iymax > symax, Y_AXIS.max, iymax, y_axis);
    UPDATE_RANGE(iymin < symin, Y_AXIS.min, iymin, y_axis);
}


/*
 * build up a cntr_struct list from curve_points
 * this funtion is only used for the alternate entry point to
 * Gershon's code and thus commented out
 ***deleted***
 */


/* HBB 990205: rewrote the 'bezier' interpolation routine,
 * to prevent numerical overflow and other undesirable things happening
 * for large data files (num_data about 1000 or so), where binomial
 * coefficients would explode, and powers of 'sr' (0 < sr < 1) become
 * extremely small. Method used: compute logarithms of these
 * extremely large and small numbers, and only go back to the
 * real numbers once they've cancelled out each other, leaving
 * a reasonable-sized one. */

/*
 * cp_binomial() computes the binomial coefficients needed for BEZIER stuff
 *   and stores them into an array which is hooked to sdat.
 * (MGR 1992)
 */
static double *
cp_binomial(int points)
{
    double *coeff;
    int n, k;
    int e;

    e = points;			/* well we're going from k=0 to k=p_count-1 */
    coeff = gp_alloc(e * sizeof(double), "bezier coefficients");

    n = points - 1;
    e = n / 2;
    /* HBB 990205: calculate these in 'logarithmic space',
     * as they become _very_ large, with growing n (4^n) */
    coeff[0] = 0.0;

    for (k = 0; k < e; k++) {
	coeff[k + 1] = coeff[k] + log(((double) (n - k)) / ((double) (k + 1)));
    }

    for (k = n; k >= e; k--)
	coeff[k] = coeff[n - k];

    return (coeff);
}


/* This is a subfunction of do_bezier() for BEZIER style computations.
 * It is passed the stepration (STEP/MAXSTEPS) and the addresses of
 * the double values holding the next x and y coordinates.
 * (MGR 1992)
 */

static void
eval_bezier(
    struct curve_points *cp,
    int first_point,		/* where to start in plot->points (to find x-range) */
    int num_points,		/* to determine end in plot->points */
    double sr,			/* position inside curve, range [0:1] */
    coordval *px,		/* OUTPUT: x and y */
    coordval *py,
    double *c)			/* Bezier coefficient array */
{
    unsigned int n = num_points - 1;
    struct coordinate GPHUGE *this_points;

    this_points = (cp->points) + first_point;

    if (sr == 0.0) {
	*px = this_points[0].x;
	*py = this_points[0].y;
    } else if (sr == 1.0) {
	*px = this_points[n].x;
	*py = this_points[n].y;
    } else {
	/* HBB 990205: do calculation in 'logarithmic space',
	 * to avoid over/underflow errors, which would exactly cancel
	 * out each other, anyway, in an exact calculation
	 */
	unsigned int i;
	double lx = 0.0, ly = 0.0;
	double log_dsr_to_the_n = n * log(1 - sr);
	double log_sr_over_dsr = log(sr) - log(1 - sr);

	for (i = 0; i <= n; i++) {
	    double u = exp(c[i] + log_dsr_to_the_n + i * log_sr_over_dsr);

	    lx += this_points[i].x * u;
	    ly += this_points[i].y * u;
	}

	*px = lx;
	*py = ly;
    }
}

/*
 * generate a new set of coordinates representing the bezier curve and
 * set it to the plot
 */

static void
do_bezier(
    struct curve_points *cp,
    double *bc,			/* Bezier coefficient array */
    int first_point,		/* where to start in plot->points */
    int num_points,		/* to determine end in plot->points */
    struct coordinate *dest)	/* where to put the interpolated data */
{
    int i;
    coordval x, y;

    /* min and max in internal (eg logged) co-ordinates. We update
     * these, then update the external extrema in user co-ordinates
     * at the end.
     */

    double ixmin, ixmax, iymin, iymax;
    double sxmin, sxmax, symin, symax;	/* starting values of above */

    x_axis = cp->x_axis;
    y_axis = cp->y_axis;

    ixmin = sxmin = AXIS_LOG_VALUE(x_axis, X_AXIS.min);
    ixmax = sxmax = AXIS_LOG_VALUE(x_axis, X_AXIS.max);
    iymin = symin = AXIS_LOG_VALUE(y_axis, Y_AXIS.min);
    iymax = symax = AXIS_LOG_VALUE(y_axis, Y_AXIS.max);

    for (i = 0; i < samples_1; i++) {
	eval_bezier(cp, first_point, num_points,
		    (double) i / (double) (samples_1 - 1),
		    &x, &y, bc);

	/* now we have to store the points and adjust the ranges */

	dest[i].type = INRANGE;
	STORE_AND_FIXUP_RANGE(dest[i].x, x, dest[i].type, ixmin, ixmax, X_AXIS.autoscale, NOOP, continue);
	STORE_AND_FIXUP_RANGE(dest[i].y, y, dest[i].type, iymin, iymax, Y_AXIS.autoscale, NOOP, NOOP);

	dest[i].xlow = dest[i].xhigh = dest[i].x;
	dest[i].ylow = dest[i].yhigh = dest[i].y;

	dest[i].z = -1;
    }

    UPDATE_RANGE(ixmax > sxmax, X_AXIS.max, ixmax, x_axis);
    UPDATE_RANGE(ixmin < sxmin, X_AXIS.min, ixmin, x_axis);
    UPDATE_RANGE(iymax > symax, Y_AXIS.max, iymax, y_axis);
    UPDATE_RANGE(iymin < symin, Y_AXIS.min, iymin, y_axis);
}

/*
 * call contouring routines -- main entry
 */

/*
 * it should be like this, but it doesn't run. If you find out why,
 * contact me: mgr@asgard.bo.open.de or Lars Hanke 2:243/4802.22@fidonet
 *
 * Well, all this had originally been inside contour.c, so maybe links
 * to functions and of contour.c are broken.
 * ***deleted***
 * end of unused entry point to Gershon's code
 *
 */

/*
 * Solve five diagonal linear system equation. The five diagonal matrix is
 * defined via matrix M, right side is r, and solution X i.e. M * X = R.
 * Size of system given in n. Return TRUE if solution exist.
 *  G. Engeln-Muellges/ F.Reutter:
 *  "Formelsammlung zur Numerischen Mathematik mit Standard-FORTRAN-Programmen"
 *  ISBN 3-411-01677-9
 *
 * /  m02 m03 m04   0   0   0   0    .       .       . \   /  x0  \    / r0  \
 * I  m11 m12 m13 m14   0   0   0    .       .       . I   I  x1  I   I  r1  I
 * I  m20 m21 m22 m23 m24   0   0    .       .       . I * I  x2  I = I  r2  I
 * I    0 m30 m31 m32 m33 m34   0    .       .       . I   I  x3  I   I  r3  I
 *      .   .   .   .   .   .   .    .       .       .        .        .
 * \                           m(n-3)0 m(n-2)1 m(n-1)2 /   \x(n-1)/   \r(n-1)/
 *
 */
static int
solve_five_diag(five_diag m[], double r[], double x[], int n)
{
    int i;
    five_diag *hv;

    hv = gp_alloc((n + 1) * sizeof(five_diag), "five_diag help vars");

    hv[0][0] = m[0][2];
    if (hv[0][0] == 0) {
	free(hv);
	return FALSE;
    }
    hv[0][1] = m[0][3] / hv[0][0];
    hv[0][2] = m[0][4] / hv[0][0];

    hv[1][3] = m[1][1];
    hv[1][0] = m[1][2] - hv[1][3] * hv[0][1];
    if (hv[1][0] == 0) {
	free(hv);
	return FALSE;
    }
    hv[1][1] = (m[1][3] - hv[1][3] * hv[0][2]) / hv[1][0];
    hv[1][2] = m[1][4] / hv[1][0];

    for (i = 2; i < n; i++) {
	hv[i][3] = m[i][1] - m[i][0] * hv[i - 2][1];
	hv[i][0] = m[i][2] - m[i][0] * hv[i - 2][2] - hv[i][3] * hv[i - 1][1];
	if (hv[i][0] == 0) {
	    free(hv);
	    return FALSE;
	}
	hv[i][1] = (m[i][3] - hv[i][3] * hv[i - 1][2]) / hv[i][0];
	hv[i][2] = m[i][4] / hv[i][0];
    }

    hv[0][4] = 0;
    hv[1][4] = r[0] / hv[0][0];
    for (i = 1; i < n; i++) {
	hv[i + 1][4] = (r[i] - m[i][0] * hv[i - 1][4] - hv[i][3] * hv[i][4]) / hv[i][0];
    }

    x[n - 1] = hv[n][4];
    x[n - 2] = hv[n - 1][4] - hv[n - 2][1] * x[n - 1];
    for (i = n - 3; i >= 0; i--)
	x[i] = hv[i + 1][4] - hv[i][1] * x[i + 1] - hv[i][2] * x[i + 2];

    free(hv);
    return TRUE;
}


/*
 * Calculation of approximation cubic splines
 * Input:  x[i], y[i], weights z[i]
 *
 * Returns matrix of spline coefficients
 */
static spline_coeff *
cp_approx_spline(
    struct curve_points *plot,
    int first_point,		/* where to start in plot->points */
    int num_points)		/* to determine end in plot->points */
{
    spline_coeff *sc;
    five_diag *m;
    double *r, *x, *h, *xp, *yp;
    struct coordinate GPHUGE *this_points;
    int i;

    x_axis = plot->x_axis;
    y_axis = plot->y_axis;

    sc = gp_alloc((num_points) * sizeof(spline_coeff),
				   "spline matrix");

    if (num_points < 4)
	int_error(plot->token, "Can't calculate approximation splines, need at least 4 points");

    this_points = (plot->points) + first_point;

    for (i = 0; i < num_points; i++)
	if (this_points[i].z <= 0)
	    int_error(plot->token, "Can't calculate approximation splines, all weights have to be > 0");

    m = gp_alloc((num_points - 2) * sizeof(five_diag), "spline help matrix");

    r = gp_alloc((num_points - 2) * sizeof(double), "spline right side");
    x = gp_alloc((num_points - 2) * sizeof(double), "spline solution vector");
    h = gp_alloc((num_points - 1) * sizeof(double), "spline help vector");

    xp = gp_alloc((num_points) * sizeof(double), "x pos");
    yp = gp_alloc((num_points) * sizeof(double), "y pos");

    /* KB 981107: With logarithmic axis first convert back to linear scale */

    xp[0] = AXIS_DE_LOG_VALUE(x_axis, this_points[0].x);
    yp[0] = AXIS_DE_LOG_VALUE(y_axis, this_points[0].y);
    for (i = 1; i < num_points; i++) {
	xp[i] = AXIS_DE_LOG_VALUE(x_axis, this_points[i].x);
	yp[i] = AXIS_DE_LOG_VALUE(y_axis, this_points[i].y);
	h[i - 1] = xp[i] - xp[i - 1];
    }

    /* set up the matrix and the vector */

    for (i = 0; i <= num_points - 3; i++) {
	r[i] = 3 * ((yp[i + 2] - yp[i + 1]) / h[i + 1]
		    - (yp[i + 1] - yp[i]) / h[i]);

	if (i < 2)
	    m[i][0] = 0;
	else
	    m[i][0] = 6 / this_points[i].z / h[i - 1] / h[i];

	if (i < 1)
	    m[i][1] = 0;
	else
	    m[i][1] = h[i] - 6 / this_points[i].z / h[i] * (1 / h[i - 1] + 1 / h[i])
		- 6 / this_points[i + 1].z / h[i] * (1 / h[i] + 1 / h[i + 1]);

	m[i][2] = 2 * (h[i] + h[i + 1])
	    + 6 / this_points[i].z / h[i] / h[i]
	    + 6 / this_points[i + 1].z * (1 / h[i] + 1 / h[i + 1]) * (1 / h[i] + 1 / h[i + 1])
	    + 6 / this_points[i + 2].z / h[i + 1] / h[i + 1];

	if (i > num_points - 4)
	    m[i][3] = 0;
	else
	    m[i][3] = h[i + 1] - 6 / this_points[i + 1].z / h[i + 1] * (1 / h[i] + 1 / h[i + 1])
		- 6 / this_points[i + 2].z / h[i + 1] * (1 / h[i + 1] + 1 / h[i + 2]);

	if (i > num_points - 5)
	    m[i][4] = 0;
	else
	    m[i][4] = 6 / this_points[i + 2].z / h[i + 1] / h[i + 2];
    }

    /* solve the matrix */
    if (!solve_five_diag(m, r, x, num_points - 2)) {
	free(h);
	free(x);
	free(r);
	free(m);
	free(xp);
	free(yp);
	int_error(plot->token, "Can't calculate approximation splines");
    }
    sc[0][2] = 0;
    for (i = 1; i <= num_points - 2; i++)
	sc[i][2] = x[i - 1];
    sc[num_points - 1][2] = 0;

    sc[0][0] = yp[0] + 2 / this_points[0].z / h[0] * (sc[0][2] - sc[1][2]);
    for (i = 1; i <= num_points - 2; i++)
	sc[i][0] = yp[i] - 2 / this_points[i].z *
	    (sc[i - 1][2] / h[i - 1]
	     - sc[i][2] * (1 / h[i - 1] + 1 / h[i])
	     + sc[i + 1][2] / h[i]);
    sc[num_points - 1][0] = yp[num_points - 1]
	- 2 / this_points[num_points - 1].z / h[num_points - 2]
	* (sc[num_points - 2][2] - sc[num_points - 1][2]);

    for (i = 0; i <= num_points - 2; i++) {
	sc[i][1] = (sc[i + 1][0] - sc[i][0]) / h[i]
	    - h[i] / 3 * (sc[i + 1][2] + 2 * sc[i][2]);
	sc[i][3] = (sc[i + 1][2] - sc[i][2]) / 3 / h[i];
    }

    free(h);
    free(x);
    free(r);
    free(m);
    free(xp);
    free(yp);

    return (sc);
}


/*
 * Calculation of cubic splines
 *
 * This can be treated as a special case of approximation cubic splines, with
 * all weights -> infinity.
 *
 * Returns matrix of spline coefficients
 */
static spline_coeff *
cp_tridiag(struct curve_points *plot, int first_point, int num_points)
{
    spline_coeff *sc;
    tri_diag *m;
    double *r, *x, *h, *xp, *yp;
    struct coordinate GPHUGE *this_points;
    int i;

    x_axis = plot->x_axis;
    y_axis = plot->y_axis;
    if (num_points < 3)
	int_error(plot->token, "Can't calculate splines, need at least 3 points");

    this_points = (plot->points) + first_point;

    sc = gp_alloc((num_points) * sizeof(spline_coeff), "spline matrix");
    m = gp_alloc((num_points - 2) * sizeof(tri_diag), "spline help matrix");

    r = gp_alloc((num_points - 2) * sizeof(double), "spline right side");
    x = gp_alloc((num_points - 2) * sizeof(double), "spline solution vector");
    h = gp_alloc((num_points - 1) * sizeof(double), "spline help vector");

    xp = gp_alloc((num_points) * sizeof(double), "x pos");
    yp = gp_alloc((num_points) * sizeof(double), "y pos");

    /* KB 981107: With logarithmic axis first convert back to linear scale */

    xp[0] = AXIS_DE_LOG_VALUE(x_axis,this_points[0].x);
    yp[0] = AXIS_DE_LOG_VALUE(y_axis,this_points[0].y);
    for (i = 1; i < num_points; i++) {
	xp[i] = AXIS_DE_LOG_VALUE(x_axis,this_points[i].x);
	yp[i] = AXIS_DE_LOG_VALUE(y_axis,this_points[i].y);
	h[i - 1] = xp[i] - xp[i - 1];
    }

    /* set up the matrix and the vector */

    for (i = 0; i <= num_points - 3; i++) {
	r[i] = 3 * ((yp[i + 2] - yp[i + 1]) / h[i + 1]
		    - (yp[i + 1] - yp[i]) / h[i]);

	if (i < 1)
	    m[i][0] = 0;
	else
	    m[i][0] = h[i];

	m[i][1] = 2 * (h[i] + h[i + 1]);

	if (i > num_points - 4)
	    m[i][2] = 0;
	else
	    m[i][2] = h[i + 1];
    }

    /* solve the matrix */
    if (!solve_tri_diag(m, r, x, num_points - 2)) {
	free(h);
	free(x);
	free(r);
	free(m);
	free(xp);
	free(yp);
	int_error(plot->token, "Can't calculate cubic splines");
    }
    sc[0][2] = 0;
    for (i = 1; i <= num_points - 2; i++)
	sc[i][2] = x[i - 1];
    sc[num_points - 1][2] = 0;

    for (i = 0; i <= num_points - 1; i++)
	sc[i][0] = yp[i];

    for (i = 0; i <= num_points - 2; i++) {
	sc[i][1] = (sc[i + 1][0] - sc[i][0]) / h[i]
	    - h[i] / 3 * (sc[i + 1][2] + 2 * sc[i][2]);
	sc[i][3] = (sc[i + 1][2] - sc[i][2]) / 3 / h[i];
    }

    free(h);
    free(x);
    free(r);
    free(m);
    free(xp);
    free(yp);

    return (sc);
}

static void
do_cubic(
    struct curve_points *plot,	/* still containes old plot->points */
    spline_coeff *sc,		/* generated by cp_tridiag */
    int first_point,		/* where to start in plot->points */
    int num_points,		/* to determine end in plot->points */
    struct coordinate *dest)	/* where to put the interpolated data */
{
    double xdiff, temp, x, y;
    double xstart, xend;	/* Endpoints of the sampled x range */
    int i, l;
    struct coordinate GPHUGE *this_points;

    /* min and max in internal (eg logged) co-ordinates. We update
     * these, then update the external extrema in user co-ordinates
     * at the end.
     */
    double ixmin, ixmax, iymin, iymax;
    double sxmin, sxmax, symin, symax;	/* starting values of above */

    x_axis = plot->x_axis;
    y_axis = plot->y_axis;

    ixmin = sxmin = AXIS_LOG_VALUE(x_axis, X_AXIS.min);
    ixmax = sxmax = AXIS_LOG_VALUE(x_axis, X_AXIS.max);
    iymin = symin = AXIS_LOG_VALUE(y_axis, Y_AXIS.min);
    iymax = symax = AXIS_LOG_VALUE(y_axis, Y_AXIS.max);

    this_points = (plot->points) + first_point;

    l = 0;

    /* HBB 20010727: Sample only across the actual x range, not the
     * full range of input data */
#if SAMPLE_CSPLINES_TO_FULL_RANGE
    xstart = this_points[0].x;
    xend = this_points[num_points - 1].x;
#else
    xstart = GPMAX(this_points[0].x, sxmin);
    xend = GPMIN(this_points[num_points - 1].x, sxmax);

    if (xstart >= xend)
	int_error(plot->token,
		  "Cannot smooth: no data within fixed xrange!");
#endif
    xdiff = (xend - xstart) / (samples_1 - 1);

    for (i = 0; i < samples_1; i++) {
	x = xstart + i * xdiff;

	/* Move forward to the spline interval this point is in */
	while ((x >= this_points[l + 1].x) && (l < num_points - 2))
	    l++;

	/* KB 981107: With logarithmic x axis the values were
         * converted back to linear scale before calculating the
         * coefficients. Use exponential for log x values. */
	temp = AXIS_DE_LOG_VALUE(x_axis, x)
	    - AXIS_DE_LOG_VALUE(x_axis, this_points[l].x);

	/* Evaluate cubic spline polynomial */
	y = ((sc[l][3] * temp + sc[l][2]) * temp + sc[l][1]) * temp + sc[l][0];

	/* With logarithmic y axis, we need to convert from linear to
         * log scale now. */
	if (Y_AXIS.log) {
	    if (y > 0.)
		y = AXIS_DO_LOG(y_axis, y);
	    else
		y = symin - (symax - symin);
	}

	dest[i].type = INRANGE;
	STORE_AND_FIXUP_RANGE(dest[i].x, x, dest[i].type, ixmin, ixmax, X_AXIS.autoscale, NOOP, continue);
	STORE_AND_FIXUP_RANGE(dest[i].y, y, dest[i].type, iymin, iymax, Y_AXIS.autoscale, NOOP, NOOP);

	dest[i].xlow = dest[i].xhigh = dest[i].x;
	dest[i].ylow = dest[i].yhigh = dest[i].y;

	dest[i].z = -1;

    }

    UPDATE_RANGE(ixmax > sxmax, X_AXIS.max, ixmax, x_axis);
    UPDATE_RANGE(ixmin < sxmin, X_AXIS.min, ixmin, x_axis);
    UPDATE_RANGE(iymax > symax, Y_AXIS.max, iymax, y_axis);
    UPDATE_RANGE(iymin < symin, Y_AXIS.min, iymin, y_axis);

}


/*
 * do_freq() is like the other smoothers only in that it
 * needs to adjust the plot ranges. We don't have to copy
 * approximated curves or anything like that.
 */

static void
do_freq(
    struct curve_points *plot,	/* still contains old plot->points */
    int first_point,		/* where to start in plot->points */
    int num_points)		/* to determine end in plot->points */
{
    double x, y;
    int i;
    int x_axis = plot->x_axis;
    int y_axis = plot->y_axis;
    struct coordinate GPHUGE *this;

    /* min and max in internal (eg logged) co-ordinates. We update
     * these, then update the external extrema in user co-ordinates
     * at the end.
     */

    double ixmin, ixmax, iymin, iymax;
    double sxmin, sxmax, symin, symax;	/* starting values of above */

    ixmin = sxmin = AXIS_LOG_VALUE(x_axis, X_AXIS.min);
    ixmax = sxmax = AXIS_LOG_VALUE(x_axis, X_AXIS.max);
    iymin = symin = AXIS_LOG_VALUE(y_axis, Y_AXIS.min);
    iymax = symax = AXIS_LOG_VALUE(y_axis, Y_AXIS.max);

    this = (plot->points) + first_point;

    for (i=0; i<num_points; i++) {

	x = this[i].x;
	y = this[i].y;

	this[i].type = INRANGE;

	STORE_AND_FIXUP_RANGE(this[i].x, x, this[i].type, ixmin, ixmax, X_AXIS.autoscale, NOOP, continue);
	STORE_AND_FIXUP_RANGE(this[i].y, y, this[i].type, iymin, iymax, Y_AXIS.autoscale, NOOP, NOOP);

	this[i].xlow = this[i].xhigh = this[i].x;
	this[i].ylow = this[i].yhigh = this[i].y;
	this[i].z = -1;
    }

    UPDATE_RANGE(ixmax > sxmax, X_AXIS.max, ixmax, x_axis);
    UPDATE_RANGE(ixmin < sxmin, X_AXIS.min, ixmin, x_axis);
    UPDATE_RANGE(iymax > symax, Y_AXIS.max, iymax, y_axis);
    UPDATE_RANGE(iymin < symin, Y_AXIS.min, iymin, y_axis);
}


/*
 * Frequency plots have don't need new points allocated; we just need
 * to adjust the plot ranges. Wedging this into gen_interp() would
 * make that code even harder to read.
 */

void
gen_interp_frequency(struct curve_points *plot)
{
    int i, j, curves;
    int first_point, num_points;
    double y, y_total;

    curves = num_curves(plot);

    if (plot->plot_smooth == SMOOTH_CUMULATIVE_NORMALISED) {
	y_total = 0.0;
	first_point = 0;

	for (i = 0; i < curves; i++) {
	    num_points = next_curve(plot, &first_point);

	    for (j = first_point; j < first_point + num_points; j++) {
		if (plot->points[j].type == UNDEFINED) 
		    continue;
	    
		y_total += plot->points[j].y;
	    }
	    first_point += num_points + 1;
	}
    }


    first_point = 0;
    for (i = 0; i < curves; i++) {
        num_points = next_curve(plot, &first_point);

        /* If cumulative, replace the current y-value with the
           sum of all previous y-values. This assumes that the
           data has already been sorted by x-values. */
        if( plot->plot_smooth == SMOOTH_CUMULATIVE ) {
            y = 0;
            for (j = first_point; j < first_point + num_points; j++) {
                if (plot->points[j].type == UNDEFINED) 
                    continue;

                y += plot->points[j].y;
                plot->points[j].y = y;
            }
        }

	/* Alternatively, cumulative normalised means replace the
	   current y-value with the sum of all previous y-values
	   divided by the total sum of all values.  This assumes the
	   data is sorted as before.  Normalising in this way allows
	   comparison of the CDF of data sets with differing total
	   numbers of samples.  */

	if (plot->plot_smooth == SMOOTH_CUMULATIVE_NORMALISED) {
	    y = 0;

	    for (j = first_point; j < first_point + num_points; j++) {
		if (plot->points[j].type == UNDEFINED) 
		    continue;

		y += plot->points[j].y;
		plot->points[j].y = y / y_total;
	    }
	}


        do_freq(plot, first_point, num_points);
        first_point += num_points + 1;
    }
    return;
}

/*
 * This is the main entry point used for everything except frequencies.
 * As stated in the header, it is fine, but I'm not too happy with it.
 */

void
gen_interp(struct curve_points *plot)
{

    spline_coeff *sc;
    double *bc;
    struct coordinate *new_points;
    int i, curves;
    int first_point, num_points;

    curves = num_curves(plot);
    new_points = gp_alloc((samples_1 + 1) * curves * sizeof(struct coordinate),
			  "interpolation table");

    first_point = 0;
    for (i = 0; i < curves; i++) {
	num_points = next_curve(plot, &first_point);
	switch (plot->plot_smooth) {
	case SMOOTH_CSPLINES:
	    sc = cp_tridiag(plot, first_point, num_points);
	    do_cubic(plot, sc, first_point, num_points,
		     new_points + i * (samples_1 + 1));
	    free(sc);
	    break;
	case SMOOTH_ACSPLINES:
	    sc = cp_approx_spline(plot, first_point, num_points);
	    do_cubic(plot, sc, first_point, num_points,
		     new_points + i * (samples_1 + 1));
	    free(sc);
	    break;

	case SMOOTH_BEZIER:
	case SMOOTH_SBEZIER:
	    bc = cp_binomial(num_points);
	    do_bezier(plot, bc, first_point, num_points,
		      new_points + i * (samples_1 + 1));
	    free((char *) bc);
	    break;
	case SMOOTH_KDENSITY:
	  do_kdensity( plot, first_point, num_points, 
		       new_points + i * (samples_1 + 1));
	  break;
	default:		/* keep gcc -Wall quiet */
	    ;
	}
	new_points[(i + 1) * (samples_1 + 1) - 1].type = UNDEFINED;
	first_point += num_points;
    }

    free(plot->points);
    plot->points = new_points;
    plot->p_max = curves * (samples_1 + 1);
    plot->p_count = plot->p_max - 1;

    return;
}

/*
 * sort_points
 *
 * sort data succession for further evaluation by plot_splines, etc.
 * This routine is mainly introduced for compilers *NOT* supporting the
 * UNIX qsort() routine. You can then easily replace it by more convenient
 * stuff for your compiler.
 * (MGR 1992)
 */

/* HBB 20010720: To avoid undefined behaviour that would be caused by
 * casting functions pointers around, changed arguments to what
 * qsort() *really* wants */
/* HBB 20010720: removed 'static' to avoid HP-sUX gcc bug */
int
compare_points(SORTFUNC_ARGS arg1, SORTFUNC_ARGS arg2)
{
    struct coordinate const *p1 = arg1;
    struct coordinate const *p2 = arg2;

    if (p1->x > p2->x)
	return (1);
    if (p1->x < p2->x)
	return (-1);
    return (0);
}

void
sort_points(struct curve_points *plot)
{
    int first_point, num_points;

    first_point = 0;
    while ((num_points = next_curve(plot, &first_point)) > 0) {
	/* Sort this set of points, does qsort handle 1 point correctly? */
	/* HBB 20010720: removed casts -- they don't help a thing, but
	 * may hide problems */
	qsort(plot->points + first_point, num_points,
	      sizeof(struct coordinate), compare_points);
	first_point += num_points;
    }
    return;
}


/*
 * cp_implode() if averaging is selected this function computes the new
 *              entries and shortens the whole thing to the necessary
 *              size
 * MGR Addendum
 */

void
cp_implode(struct curve_points *cp)
{
    int first_point, num_points;
    int i, j, k;
    double x = 0., y = 0., sux = 0., slx = 0., suy = 0., sly = 0.;
    double weight; /* used for acsplines */
    TBOOLEAN all_inrange = FALSE;

    x_axis = cp->x_axis;
    y_axis = cp->y_axis;
    j = 0;
    first_point = 0;
    while ((num_points = next_curve(cp, &first_point)) > 0) {
	k = 0;
	for (i = first_point; i < first_point + num_points; i++) {
	    /* HBB 20020801: don't try to use undefined datapoints */
	    if (cp->points[i].type == UNDEFINED)
	        continue;
	    if (!k) {
		x = cp->points[i].x;
		y = cp->points[i].y;
		sux = cp->points[i].xhigh;
		slx = cp->points[i].xlow;
		suy = cp->points[i].yhigh;
		sly = cp->points[i].ylow;
		weight = cp->points[i].z;
		all_inrange = (cp->points[i].type == INRANGE);
		k = 1;
	    } else if (cp->points[i].x == x) {
		y += cp->points[i].y;
		sux += cp->points[i].xhigh;
		slx += cp->points[i].xlow;
		suy += cp->points[i].yhigh;
		sly += cp->points[i].ylow;
		weight += cp->points[i].z;
		if (cp->points[i].type != INRANGE)
		    all_inrange = FALSE;
		k++;
	    } else {
		cp->points[j].x = x;
 		if ( cp->plot_smooth == SMOOTH_FREQUENCY ||
 		     cp->plot_smooth == SMOOTH_CUMULATIVE ||
		     cp->plot_smooth == SMOOTH_CUMULATIVE_NORMALISED )
		    k = 1;
		cp->points[j].y = y /= (double) k;
		cp->points[j].xhigh = sux / (double) k;
		cp->points[j].xlow = slx / (double) k;
		cp->points[j].yhigh = suy / (double) k;
		cp->points[j].ylow = sly / (double) k;
		cp->points[j].z = weight / (double) k;
		/* HBB 20000405: I wanted to use STORE_AND_FIXUP_RANGE
		 * here, but won't: it assumes we want to modify the
		 * range, and that the range is given in 'input'
		 * coordinates.  For logarithmic axes, the overhead
		 * would be larger than the possible gain, so write it
		 * out explicitly, instead:
		 * */
		cp->points[j].type = INRANGE;
		if (! all_inrange) {
		    if (X_AXIS.log) {
			if (x <= -VERYLARGE) {
			    cp->points[j].type = OUTRANGE;
			    goto is_outrange;
			}
			x = AXIS_UNDO_LOG(x_axis, x);
		    }
		    if (((x < X_AXIS.min) && !(X_AXIS.autoscale & AUTOSCALE_MIN))
			|| ((x > X_AXIS.max) && !(X_AXIS.autoscale & AUTOSCALE_MAX))) {
			cp->points[j].type = OUTRANGE;
			goto is_outrange;
		    }
		    if (Y_AXIS.log) {
			if (y <= -VERYLARGE) {
			    cp->points[j].type = OUTRANGE;
			    goto is_outrange;
			}
			y = AXIS_UNDO_LOG(y_axis, y);
		    }
		    if (((y < Y_AXIS.min) && !(Y_AXIS.autoscale & AUTOSCALE_MIN))
			|| ((y > Y_AXIS.max) && !(Y_AXIS.autoscale & AUTOSCALE_MAX)))
			cp->points[j].type = OUTRANGE;
		is_outrange:
		    ;
		} /* if(! all inrange) */

		j++;		/* next valid entry */
		k = 0;		/* to read */
		i--;		/* from this (-> last after for(;;)) entry */
	    } /* else (same x position) */
	} /* for(points in curve) */

	if (k) {
	    cp->points[j].x = x;
	    if ( cp->plot_smooth == SMOOTH_FREQUENCY ||
		 cp->plot_smooth == SMOOTH_CUMULATIVE ||
		 cp->plot_smooth == SMOOTH_CUMULATIVE)
		k = 1;
	    cp->points[j].y = y /= (double) k;
	    cp->points[j].xhigh = sux / (double) k;
	    cp->points[j].xlow = slx / (double) k;
	    cp->points[j].yhigh = suy / (double) k;
	    cp->points[j].ylow = sly / (double) k;
	    cp->points[j].z = weight / (double) k;
	    cp->points[j].type = INRANGE;
	    if (! all_inrange) {
		    if (X_AXIS.log) {
			if (x <= -VERYLARGE) {
			    cp->points[j].type = OUTRANGE;
			    goto is_outrange2;
			}
			x = AXIS_UNDO_LOG(x_axis, x);
		    }
		    if (((x < X_AXIS.min) && !(X_AXIS.autoscale & AUTOSCALE_MIN))
			|| ((x > X_AXIS.max) && !(X_AXIS.autoscale & AUTOSCALE_MAX))) {
			cp->points[j].type = OUTRANGE;
			goto is_outrange2;
		    }
		    if (Y_AXIS.log) {
			if (y <= -VERYLARGE) {
			    cp->points[j].type = OUTRANGE;
			    goto is_outrange2;
			}
			y = AXIS_UNDO_LOG(y_axis, y);
		    }
		    if (((y < Y_AXIS.min) && !(Y_AXIS.autoscale & AUTOSCALE_MIN))
			|| ((y > Y_AXIS.max) && !(Y_AXIS.autoscale & AUTOSCALE_MAX)))
			cp->points[j].type = OUTRANGE;
		is_outrange2:
		    ;
	    }
	    j++;		/* next valid entry */
	}
	/* insert invalid point to separate curves */
	if (j < cp->p_count) {
	    cp->points[j].type = UNDEFINED;
	    j++;
	}
	first_point += num_points;
    }				/* end while */
    cp->p_count = j;
    cp_extend(cp, j);
}
