/*
 *  R : A Computer Language for Statistical Data Analysis
 *  Copyright (C) 1999-2016  The R 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
 *  https://www.R-project.org/Licenses/
 */

/* from NETLIB c/brent.shar with max.iter, add'l info and convergence
   details hacked in by Peter Dalgaard */

/*************************************************************************
 *			    C math library
 * function ZEROIN - obtain a function zero within the given range
 *
 * Input
 *	double zeroin(ax,bx,f,info,Tol,Maxit)
 *	double ax;			Root will be seeked for within
 *	double bx;			a range [ax,bx]
 *	double (*f)(double x, void *info); Name of the function whose zero
 *					will be seeked for
 *	void *info;			Add'l info passed to f
 *	double *Tol;			Acceptable tolerance for the root
 *					value.
 *					May be specified as 0.0 to cause
 *					the program to find the root as
 *					accurate as possible
 *
 *	int *Maxit;			Max. iterations
 *
 *
 * Output
 *	Zeroin returns an estimate for the root with accuracy
 *	4*EPSILON*abs(x) + tol
 *	*Tol returns estimated precision
 *	*Maxit returns actual # of iterations, or -1 if maxit was
 *      reached without convergence.
 *
 * Algorithm
 *	G.Forsythe, M.Malcolm, C.Moler, Computer methods for mathematical
 *	computations. M., Mir, 1980, p.180 of the Russian edition
 *
 *	The function makes use of the bisection procedure combined with
 *	the linear or quadric inverse interpolation.
 *	At every step program operates on three abscissae - a, b, and c.
 *	b - the last and the best approximation to the root
 *	a - the last but one approximation
 *	c - the last but one or even earlier approximation than a that
 *		1) |f(b)| <= |f(c)|
 *		2) f(b) and f(c) have opposite signs, i.e. b and c confine
 *		   the root
 *	At every step Zeroin selects one of the two new approximations, the
 *	former being obtained by the bisection procedure and the latter
 *	resulting in the interpolation (if a,b, and c are all different
 *	the quadric interpolation is utilized, otherwise the linear one).
 *	If the latter (i.e. obtained by the interpolation) point is
 *	reasonable (i.e. lies within the current interval [b,c] not being
 *	too close to the boundaries) it is accepted. The bisection result
 *	is used in the other case. Therefore, the range of uncertainty is
 *	ensured to be reduced at least by the factor 1.6
 *
 ************************************************************************
 *
 * NOTE:  uniroot() --> do_zeroin2()  --- in  ../main/optimize.c
 *					      ~~~~~~~~~~~~~~~~~~
 */

#include <float.h>
#include <math.h>

#include <R_ext/Applic.h>

#define EPSILON DBL_EPSILON

/* R_zeroin2() is faster for "expensive" f(), in those typical cases where
 *             f(ax) and f(bx) are available anyway : */

double R_zeroin2(			/* An estimate of the root */
    double ax,				/* Left border | of the range	*/
    double bx,				/* Right border| the root is seeked*/
    double fa, double fb,		/* f(a), f(b) */
    double (*f)(double x, void *info),	/* Function under investigation	*/
    void *info,				/* Add'l info passed on to f	*/
    double *Tol,			/* Acceptable tolerance		*/
    int *Maxit)				/* Max # of iterations */
{
    double a,b,c, fc;			/* Abscissae, descr. see above,  f(c) */
    double tol;
    int maxit;

    a = ax;  b = bx;
    c = a;   fc = fa;
    maxit = *Maxit + 1; tol = * Tol;

    /* First test if we have found a root at an endpoint */
    if(fa == 0.0) {
	*Tol = 0.0;
	*Maxit = 0;
	return a;
    }
    if(fb ==  0.0) {
	*Tol = 0.0;
	*Maxit = 0;
	return b;
    }

    while(maxit--)		/* Main iteration loop	*/
    {
	double prev_step = b-a;		/* Distance from the last but one
					   to the last approximation	*/
	double tol_act;			/* Actual tolerance		*/
	double p;			/* Interpolation step is calcu- */
	double q;			/* lated in the form p/q; divi-
					 * sion operations is delayed
					 * until the last moment	*/
	double new_step;		/* Step at this iteration	*/

	if( fabs(fc) < fabs(fb) )
	{				/* Swap data for b to be the	*/
	    a = b;  b = c;  c = a;	/* best approximation		*/
	    fa=fb;  fb=fc;  fc=fa;
	}
	tol_act = 2*EPSILON*fabs(b) + tol/2;
	new_step = (c-b)/2;

	if( fabs(new_step) <= tol_act || fb == (double)0 )
	{
	    *Maxit -= maxit;
	    *Tol = fabs(c-b);
	    return b;			/* Acceptable approx. is found	*/
	}

	/* Decide if the interpolation can be tried	*/
	if( fabs(prev_step) >= tol_act	/* If prev_step was large enough*/
	    && fabs(fa) > fabs(fb) ) {	/* and was in true direction,
					 * Interpolation may be tried	*/
	    register double t1,cb,t2;
	    cb = c-b;
	    if( a==c ) {		/* If we have only two distinct	*/
					/* points linear interpolation	*/
		t1 = fb/fa;		/* can only be applied		*/
		p = cb*t1;
		q = 1.0 - t1;
	    }
	    else {			/* Quadric inverse interpolation*/

		q = fa/fc;  t1 = fb/fc;	 t2 = fb/fa;
		p = t2 * ( cb*q*(q-t1) - (b-a)*(t1-1.0) );
		q = (q-1.0) * (t1-1.0) * (t2-1.0);
	    }
	    if( p>(double)0 )		/* p was calculated with the */
		q = -q;			/* opposite sign; make p positive */
	    else			/* and assign possible minus to	*/
		p = -p;			/* q				*/

	    if( p < (0.75*cb*q-fabs(tol_act*q)/2) /* If b+p/q falls in [b,c]*/
		&& p < fabs(prev_step*q/2) )	/* and isn't too large	*/
		new_step = p/q;			/* it is accepted
						 * If p/q is too large then the
						 * bisection procedure can
						 * reduce [b,c] range to more
						 * extent */
	}

	if( fabs(new_step) < tol_act) {	/* Adjust the step to be not less*/
	    if( new_step > (double)0 )	/* than tolerance		*/
		new_step = tol_act;
	    else
		new_step = -tol_act;
	}
	a = b;	fa = fb;			/* Save the previous approx. */
	b += new_step;	fb = (*f)(b, info);	/* Do step to a new approxim. */
	if( (fb > 0 && fc > 0) || (fb < 0 && fc < 0) ) {
	    /* Adjust c for it to have a sign opposite to that of b */
	    c = a;  fc = fa;
	}

    }
    /* failed! */
    *Tol = fabs(c-b);
    *Maxit = -1;
    return b;
}
