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%%	options

copyright owner	=	Dirk Krause
copyright year	=	2018-xxxx
SPDX-License-Identifier:	BSD-3-Clause

%%	header

/**	@file	dk4iter.h	Iteration algorithms for root finding.

This module implements the following root finding iteration algorithms:
- Bisection
- Regula falsi (primitive form, Illinois, Pegasus, Anderson-Bjoerck)
- Newton
- fix point

The function to iterate must be implemented as dk4_iter_fct_t.
This function type returns an integer value (non-zero to indicate a
successful calculation, 0 to indicate an error).
The function expects the following arguments:
- Result address<br>
  Address of a variable or an array to store the calculation result.
  Functions for Newton iteration algorithm store 2 values: Function value
  and value of the first derivative.
- X position<br>
  The x value you want a function value for.
- Address of parameter set.
  Additional parameters probably required by the function. I.e. for a
  polynomial calculation function you may specify the coefficients here.
  This parameter is optional.

Details for an iteration may be specified in an iteration context.
The dk4iter_ctx_open() creates such a context and returns a pointer.
Use dk4iter_ctx_close() to release the context when done with it.
Alternatively use a automatic/static variable of the dk4_iter_ctx_t type
and initialize it using the dk4iter_ctx_init() function.

The dk4iter_ctx_set_algorithm() function chooses the iteration method.

For Newton and fixpoint the context may specify an x interval the
iteration must not leave. Leaving the interval results in abort.
The dk4iter_ctx_set_min(), dk4iter_ctx_set_exclusive_min(),
dk4iter_ctx_set_max() and dk4iter_ctx_set_exclusive_max() functions can
be used to set closed and open interval borders.

Tolerance values (epsilon) may be specified for
- y direction unless fixpoint is used and/or
- x direction.
Use dk4iter_ctx_set_eps_y() and dk4iter_ctx_set_eps_x() to set the
tolerances.
Use 0.0 or negative tolerances to skip one check (not both!).

Once the iteration reached allowed y difference you can decide
to stop when the x difference is in allowed tolerance range too or
to continue until full machine precision is reached (new x is exactly
the same as previous x).
I do not recommend attempts to iterate to full machine precision as this
(a) may fail for some functions resulting in an oscillation and
(b) uses a larger number of iteration passes and.
Use dk4iter_ctx_set_exact() to control this.

The dk4iter_ctx_set_maxpass() function sets the maximum number of
iteration passes (iteration steps). The iteration is aborted if there
is no success within this maximum number of passes.
Although you can use 0 to set an unlimited number of passes I do not recommend
to do so.

The dk4iter_interval() function runs an iteration for a specified interval.

The dk4iter_start_point() function runs an iteration if just one
starting x value is specified.

If no context is specified the following defaults are used:

Option | Default
:----: | :------
Algorithm | DK4_ITER_ALG_RF_ANDERSON_BJOERCK for dk4iter_interval(), DK4_ITER_ALG_NEWTON for dk4iter_start_point().
Max passes | 256.
Y tolerance | 1.0e-8
X tolerance | 1.0e-8
Full machine precision | No.
Restricted x interval | None.

*/



/**	Border values for maximum number of iteration passes.
*/
enum {
								/**	Number of passes for average cases.
								*/
	DK4_ITER_PASSES_REGULAR	=	256 ,

								/**	Do not set a limit on the number of passes.
									Warning: Might result in an endless loop!
								*/
	DK4_ITER_PASSES_UNLIMITED	=	0
};



/**	Iteration algorithms.
*/
enum {
									/**	Interval bisection.
									*/
	DK4_ITER_ALG_BISECTION			=	0 ,

									/**	Primitive from of regula falsi.
									*/
	DK4_ITER_ALG_RF_PRIMITIVE ,

									/**	Regula falsi, Illinois variant.
									*/
	DK4_ITER_ALG_RF_ILLINOIS ,

									/**	Regula falsi, Pegasus variant.
									*/
	DK4_ITER_ALG_RF_PEGASUS ,

									/**	Regula falsi, Anderson-Bjoerck variant.
									*/
	DK4_ITER_ALG_RF_ANDERSON_BJOERCK ,

									/**	Newton algorithm.
										The function must place 2 elements
										in the array specified by pointer:
										Function value and derivative value.
									*/
	DK4_ITER_ALG_NEWTON ,

									/**	Fix point algorithm.
										The function must be the phi(x)
										function from the
										phi(x) = x equation.
									*/
	DK4_ITER_ALG_FIX_POINT
};



/**	Iteration result.
*/
enum {
								/**	Iteration succeeded.
								*/
	DK4_ITER_RESULT_SUCCESS		=	1 ,

								/**	Error: Too many passes without success.
								*/
	DK4_ITER_RESULT_E_PASSES	=	0 ,

								/**	Error: Infinite value or NaN in calculation.
								*/
	DK4_ITER_RESULT_E_INFINITE	=	-1 ,

								/**	Error: X left initial interval.
									Can happen with Newton and fix point
									algorithm.
								*/
	DK4_ITER_RESULT_E_OOR		=	-2 ,

								/**	Error: Not converging.
									Fix point algorithm: Interval was enlarging.
								*/
	DK4_ITER_RESULT_E_CONV		=	-3 ,

								/**	Function calculation failed.
								*/
	DK4_ITER_RESULT_E_FCT		=	-4 ,

								/**	Invalid arguments passed to function call.
								*/
	DK4_ITER_RESULT_E_ARGS		=	-5 
};



/**	Function to iterate.
	@param	d	Destination address. One element for all algorithms
	except DK4_ITER_ALG_NEWTON which saves function value and derivative
	value in a 2 elements array.
	@param	x	X position to calculate the function value for.
	@param	ps	Parameter set, may be NULL.
	@return	Non-zero value on success, 0 on error.
*/
typedef int dk4_iter_fct_t(double *d, double x, const void *ps);


/**	Iteration context.
*/
typedef struct {
	double			xmin;		/**< Minimum of x interval. */
	double			xmax;		/**< Maximum of x interval. */
	double			eps_x;		/**< Epsilon for x change. */
	double			eps_y;		/**< Epsilon for absolute y value. */
	unsigned long	maxpass;	/**< Maximum number of passes. */
	int				exact;		/**< Flag: End only if there is no x change. */
	int				algo;		/**< Iteration algorithm to use. */
	int				minmax;		/**< Flags: min(1), max(2) set. */
} dk4_iter_ctx_t;



#ifdef	__cplusplus
extern "C" {
#endif

/**	Open new iteration context.
	Allocate memory for new iteration context, set up default value.
	A context allocated by this function must be released after
	use by the dk4iter_ctx_close() function.
	@return	Valid pointer to new context on success, NULL on error.
*/

dk4_iter_ctx_t *
dk4iter_ctx_open(void);


/**	Initialize a context to default values.
	@param	ctx	Context to initialize.
*/

void
dk4iter_ctx_init(dk4_iter_ctx_t *ctx);


/**	Close iteration context.
	Release memory assigned to the context.
	@param	ctx	Context created by dk4iter_ctx_open().
*/

void
dk4iter_ctx_close(dk4_iter_ctx_t *ctx);


/**	Set epsilon value for x change.
	@param	ctx	Context to modify.
	@param	eps	Epsilon value for x change.
*/

void
dk4iter_ctx_set_eps_x(dk4_iter_ctx_t *ctx, double eps);


/**	Set maximum absolute y value.
	@param	ctx	Context to modify.
	@param	eps	Maximum absolute y value.
*/

void
dk4iter_ctx_set_eps_y(dk4_iter_ctx_t *ctx, double eps);


/**	Set maximum number of passes (iteration steps).
	@param	ctx	Context to modify.
	@param	passes	Maximum number of passes, 0 or negative values indicate
	an unlimited number of steps.
*/

void
dk4iter_ctx_set_maxpass(dk4_iter_ctx_t *ctx, unsigned long passes);


/**	Set flag for exact iteration.
	If the flag is activated, iteration is continued until there is
	no longer any change in the x value. For some functions you can
	retrieve a result in machine precision using this flag, for other
	functions the iteration may fail.
	@param	ctx	Context to modify.
	@param	flag	New flag value, 0=inactive, other=active.
	The recommended value is 0.
*/

void
dk4iter_ctx_set_exact(dk4_iter_ctx_t *ctx, int flag);


/**	Set up iteration algorithm.
	@param	ctx			Context to modify.
	@param	algorithm	Algorithm to use.
*/

void
dk4iter_ctx_set_algorithm(dk4_iter_ctx_t *ctx, int algorithm);


/**	Set interval minimum.
	The allowed interval is only used for Newton and fixed point
	algorithm.
	@param	ctx		Context to modify.
	@param	xmin	Minimum x value allowed.
*/

void
dk4iter_ctx_set_min(dk4_iter_ctx_t *ctx, double xmin);


/**	Set exlusive interval minimum (open interval border).
	The allowed interval is only used for Newton and fixed point
	algorithm.
	@param	ctx		Context to modify.
	@param	xmin	Minimum x value allowed.
*/

void
dk4iter_ctx_set_exclusive_min(dk4_iter_ctx_t *ctx, double xmin);


/**	Set interval maximum.
	The allowed interval is only used for Newton and fixed point
	algorithm.
	@param	ctx		Context to modify.
	@param	xmax	Maximum x value allowed.
*/

void
dk4iter_ctx_set_max(dk4_iter_ctx_t *ctx, double xmax);


/**	Set exclusive interval maximum (open interval border).
	The allowed interval is only used for Newton and fixed point
	algorithm.
	@param	ctx		Context to modify.
	@param	xmax	Maximum x value allowed.
*/

void
dk4iter_ctx_set_exclusive_max(dk4_iter_ctx_t *ctx, double xmax);


/**	Run iteration.
	The algorithm in the context must be one from:
	DK4_ITER_ALG_BISECTION, DK4_ITER_ALG_RF_PRIMITIVE, DK4_ITER_ALG_RF_ILLINOIS,
	DK4_ITER_ALG_RF_PEGASUS	DK4_ITER_ALG_RF_ANDERSON_BJOERCK.
	Without a context DK4_ITER_ALG_RF_ANDERSON_BJOERCK is used as default.
	@param	d	Address of result variable.
	@param	pp	Address of variable to store number of passes, may be NULL.
	@param	fct	Function to find root for.
	@param	ps	Parameter set for function, may be NULL.
	@param	a	One interval border.
	@param	b	Other interval border.
	@param	ctx	Iteration context.
	@return	DK4_ITER_RESULT_SUCCESS on success, one from
	DK4_ITER_RESULT_E_PASSES, DK4_ITER_RESULT_E_INFINITE,
	DK4_ITER_RESULT_E_OOR, DK4_ITER_RESULT_E_CONV or
	DK4_ITER_RESULT_E_FCT on error.
*/

int
dk4iter_interval(
	double					*d,
	unsigned long			*pp,
	dk4_iter_fct_t			*fct,
	void			const	*ps,
	double					 a,
	double					 b,
	dk4_iter_ctx_t	const	*ctx
);


/**	Run iteration.
	The algorithm in the context must be one from:
	DK4_ITER_ALG_NEWTON, DK4_ITER_ALG_FIX_POINT.
	Without a context, DK4_ITER_ALG_NEWTON is used as default.
	@param	d	Address of result variable.
	@param	pp	Address of variable to store number of passes, may be NULL.
	@param	fct	Function to find root for.
				For Newton iteration this function must set two values
				in the destination array: function value and derivative value.
				For fixpoint iteration the function calculates the
				phi(x) part of phi(x)=x.
	@param	ps	Parameter set for function, may be NULL.
	@param	x0	Start point.
	@param	ctx	Iteration context.
	@return	DK4_ITER_RESULT_SUCCESS on success, one from
	DK4_ITER_RESULT_E_PASSES, DK4_ITER_RESULT_E_INFINITE,
	DK4_ITER_RESULT_E_OOR, DK4_ITER_RESULT_E_CONV or
	DK4_ITER_RESULT_E_FCT on error.
*/

int
dk4iter_start_point(
	double					*d,
	unsigned long			*pp,
	dk4_iter_fct_t			*fct,
	void			const	*ps,
	double					 x0,
	dk4_iter_ctx_t	const	*ctx
);


#ifdef	__cplusplus
}
#endif

/* vim: set ai sw=4 ts=4 : */

%%	module


#include "dk4conf.h"

#include <stdio.h>

#if	DK4_HAVE_ASSERT_H
#ifndef	ASSERT_H_INCLUDED
#include <assert.h>
#define	ASSERT_H_INCLUDED 1
#endif
#endif

#if	DK4_HAVE_STDLIB_H
#ifndef	STDLIB_H_INCLUDED
#include <stdlib.h>
#define	STDLIB_H_INCLUDED 1
#endif
#endif

#if	DK4_HAVE_LIMITS_H
#ifndef	LIMITS_H_INCLUDED
#include <limits.h>
#define	LIMITS_H_INCLUDED 1
#endif
#endif

#if	DK4_HAVE_STDINT_H
#ifndef	STDINT_H_INCLUDED
#include <stdint.h>
#define	STDINT_H_INCLUDED 1
#endif
#endif


#include "dk4mem.h"
#include "dk4iter.h"
#include "dk4math.h"



$!trace-include



/**	Initial function value condition.
*/
enum {
		CONDITION_ILLEGAL		=	0 ,
								/**	fa < 0, fb >= 0.
								*/
		CONDITION_FA_LESS_ZERO ,

								/**	fa <= 0, fb > 0.
								*/
		CONDITION_FA_LEQ_ZERO ,

								/**	fa > 0, fb <= 0.
								*/
		CONDITION_FA_GREATER_ZERO ,

								/**	fa >= 0, fb < 0.
								*/
		CONDITION_FA_GEQ_ZERO
};



/**	Note which interval border was changed in previous step.
*/
enum {
					/**	No interval border change yet.
					*/
	CHANGED_NONE	=	 0,

					/**	Previous step changed interval border a.
					*/
	CHANGED_A		=	-1,

					/**	Previous step changed interval border b.
					*/
	CHANGED_B		=	 1
};



/**	Flags which restrictions apply to x values.
*/
enum {
						/**	Minimum specified.
						*/
	MINMAX_MINIMUM		=	0x0001 ,

						/**	Maximum specified.
						*/
	MINMAX_MAXIMUM		=	0x0002 ,

						/**	Specified minimum is exclusive.
						*/
	MINMAX_MIN_EXCL		=	0x0004 ,

						/**	Specified maximum is exclusive.
						*/
	MINMAX_MAX_EXCL		=	0x0008
};



void
dk4iter_ctx_init(dk4_iter_ctx_t *ctx)
{
#if	DK4_USE_ASSERT
  assert(NULL != ctx);
#endif
	if (NULL != ctx) {
		DK4_MEMRES(ctx,sizeof(dk4_iter_ctx_t));
		ctx->xmin		=	0.0;
		ctx->xmax		=	0.0;
		ctx->eps_x		=	1.0e-8;
		ctx->eps_y		=	1.0e-8;
		ctx->maxpass	=	(unsigned long)(DK4_ITER_PASSES_REGULAR);
		ctx->exact		=	0;
		ctx->algo		=	DK4_ITER_ALG_RF_ANDERSON_BJOERCK;
		ctx->minmax		=	0;
	}
}



dk4_iter_ctx_t *
dk4iter_ctx_open(void)
{
	dk4_iter_ctx_t	*back = NULL;
	back = dk4mem_new(dk4_iter_ctx_t,1,NULL);
	if (NULL != back) {
		dk4iter_ctx_init(back);
	}
	return back;
}



void
dk4iter_ctx_close(dk4_iter_ctx_t *ctx)
{
#if	DK4_USE_ASSERT
  assert(NULL != ctx);
#endif
	if (NULL != ctx) {
		dk4mem_free(ctx);
	}
}



void
dk4iter_ctx_set_eps_x(dk4_iter_ctx_t *ctx, double eps)
{
#if	DK4_USE_ASSERT
  assert(NULL != ctx);
#endif
	if (NULL != ctx) {
		ctx->eps_x = eps;
	}
}



void
dk4iter_ctx_set_eps_y(dk4_iter_ctx_t *ctx, double eps)
{
#if	DK4_USE_ASSERT
  assert(NULL != ctx);
#endif
	if (NULL != ctx) {
		ctx->eps_y = eps;
	}
}



void
dk4iter_ctx_set_maxpass(dk4_iter_ctx_t *ctx, unsigned long passes)
{
#if	DK4_USE_ASSERT
  assert(NULL != ctx);
#endif
	if (NULL != ctx) {
		ctx->maxpass = passes;
	}
}



void
dk4iter_ctx_set_exact(dk4_iter_ctx_t *ctx, int flag)
{
#if	DK4_USE_ASSERT
  assert(NULL != ctx);
#endif
	if (NULL != ctx) {
		ctx->exact = flag;
	}
}



void
dk4iter_ctx_set_algorithm(dk4_iter_ctx_t *ctx, int algorithm)
{
#if	DK4_USE_ASSERT
  assert(NULL != ctx);
#endif
	if (NULL != ctx) {
		ctx->algo = algorithm;
	}
}



void
dk4iter_ctx_set_min(dk4_iter_ctx_t *ctx, double xmin)
{
#if	DK4_USE_ASSERT
  assert(NULL != ctx);
#endif
	if (NULL != ctx) {
		ctx->xmin = xmin;
		ctx->minmax |= MINMAX_MINIMUM;
	}
}



void
dk4iter_ctx_set_max(dk4_iter_ctx_t *ctx, double xmax)
{
#if	DK4_USE_ASSERT
  assert(NULL != ctx);
#endif
	if (NULL != ctx) {
		ctx->xmax = xmax;
		ctx->minmax |= MINMAX_MAXIMUM;
	}
}



void
dk4iter_ctx_set_exclusive_min(dk4_iter_ctx_t *ctx, double xmin)
{
#if	DK4_USE_ASSERT
  assert(NULL != ctx);
#endif
	if (NULL != ctx) {
		ctx->xmin = xmin;
		ctx->minmax |= (MINMAX_MINIMUM | MINMAX_MIN_EXCL);
	}
}



void
dk4iter_ctx_set_exclusive_max(dk4_iter_ctx_t *ctx, double xmax)
{
#if	DK4_USE_ASSERT
  assert(NULL != ctx);
#endif
	if (NULL != ctx) {
		ctx->xmax = xmax;
		ctx->minmax |= (MINMAX_MAXIMUM | MINMAX_MAX_EXCL);
	}
}



static
int
find_condition(double fa, double fb)
{
	int		back	=	CONDITION_ILLEGAL;

	if ((0.0 > fa) && (0.0 <= fb)) {
		back = CONDITION_FA_LESS_ZERO;
	}
	else {
		if ((0.0 < fa) && (0.0 >= fb)) {
			back = CONDITION_FA_GREATER_ZERO;
		}
		else {
			if ((0.0 >= fa) && (0.0 < fb)) {
				back = CONDITION_FA_LEQ_ZERO;
			}
			else {
				if ((0.0 <= fa) && (0.0 > fb)) {
					back = CONDITION_FA_GEQ_ZERO;
				}
			}
		}
	}
	return back;
}



/**	Calculate gamma value for PEGASUS variant.
	@param	fg	Function value at border last set.
	@param	fx	Function value at recent x position.
	@return	Gamma value.
*/
static
double
dk4iter_gamma_pegasus(double fg, double fx)
{
	double	back;
	back = fg / (fg + fx);
	if (0 == dk4ma_is_finite(back)) {
		back = 0.5;
	}
	else {
		if (0.0 >= back) {
			back = 0.5;
		}
#if	0
		/*	2018-07-06
			Can not happen, fg and fx have same sign.
		*/
		else {
			if (1.0 < back) {
				back = 1.0;
			}
		}
#endif
	}
	return back;
}



/**	Calculate gamma value for ANDERSON-BJOERCK variant.
	@param	fg	Function value at border last set.
	@param	fx	Function value at recent x position.
	@return	Gamma value.
*/
static
double
dk4iter_gamma_anderson_bjoerck(double fg, double fx)
{
	double back;
	back = 1.0 - fx / fg;
	if (0 == dk4ma_is_finite(back)) {
		back = 0.5;
	}
	else {
		if (0.0 >= back) {
			back = 0.5;
		}
#if 0
		/*	2018-07-06
			Can not happen as fx and fy have same sign.
		*/
		else {
			if (1.0 < back) {
				back = 1.0;
			}
		}
#endif
	}
	return back;
}



int
dk4iter_interval(
	double					*d,
	unsigned long			*pp,
	dk4_iter_fct_t			*fct,
	void			const	*ps,
	double					 a,
	double					 b,
	dk4_iter_ctx_t	const	*ctx
)
{
	dk4_iter_ctx_t		mctx;		/* Copy of context */
	double			fa		=	0.0;	/* Function value for border a */
	double			fb		=	0.0;	/* Function value for border b */
	double			x		=	0.0;	/* Text x value */
	double			fx		=	0.0;	/* Function value for x */
	double			afx		=	0.0;	/* Absolute function value for x */
	double			xo		=	0.0;	/* Previous step x value */
	double			gamma	=	0.5;	/* Correction factor */
	unsigned long	passno	=	0UL;	/* Number of current pass */
	int				res		=	0;	/* Operation result */
	int				cond	=	0;	/* Condition of interval borders */
	int				cc		=	0;	/* 1=continue, 0=finished, -1=abort */
	int				pc		=	CHANGED_NONE;	/* Previous border change */
	int				nc		=	CHANGED_NONE;	/* Next change */
	int		 		back	=	DK4_ITER_RESULT_E_ARGS;
	$? "+ dk4iter_interval"
	/*	Check function call arguments
	*/
#if	DK4_USE_ASSERT
  assert(NULL != d);
  assert(NULL != fct);
#endif
	if ((NULL == d) || (NULL == fct)) {		$? "! d or fct"
		goto finished;
	}
	if (a == b) {							$? "! zero length interval"
		goto finished;
	}

	/*	Copy or initialize context
	*/
	if (NULL != ctx) {						$? ". use context"
		DK4_MEMCPY(&mctx,ctx,sizeof(dk4_iter_ctx_t));
	}
	else {									$? ". no context, use defaults"
		dk4iter_ctx_init(&mctx);
	}

	/*	Check algorithm specification
	*/
	if (DK4_ITER_ALG_BISECTION != mctx.algo) {
		if (DK4_ITER_ALG_RF_PRIMITIVE != mctx.algo) {
			if (DK4_ITER_ALG_RF_ILLINOIS != mctx.algo) {
				if (DK4_ITER_ALG_RF_PEGASUS != mctx.algo) {
					if (DK4_ITER_ALG_RF_ANDERSON_BJOERCK != mctx.algo) {
						goto finished;		$? "! illegal algorithm"
					}
				}
			}
		}
	}

	/*	Calculate borders at beginning, check
	*/
	res = (*fct)(&fa, a, ps);
	if ((0 == res) || (!(dk4ma_is_finite(fa)))) {
		back = DK4_ITER_RESULT_E_FCT;		$? "! function calculation for a"
		goto finished;
	}
	res = (*fct)(&fb, b, ps);
	if ((0 == res) || (!(dk4ma_is_finite(fb)))) {
		back = DK4_ITER_RESULT_E_FCT;		$? "! function calculation for b"
		goto finished;
	}
	cond = find_condition(fa, fb);
	if (CONDITION_ILLEGAL == cond) {		$? "! invalid interval"
		goto finished;
	}

	/*	Prepare iteration loop
	*/
	passno = 0UL;
	x = xo = 0.0;
	cc = 1;
	pc = CHANGED_NONE;

	/*	Run iteration loop
	*/
	do {
		/*	Keep previous x position
		*/
		xo = x;
		/*	Increase pass number, but avoid wrapping
		*/
		if (ULONG_MAX > passno) { passno++; }	$? ". pass %lu", passno
		/*	Calculate x position
		*/
		$? ". a = %lg   fa = %lg      b = %lg   fb = %lg", a, fa, b, fb
		switch (mctx.algo) {
			case DK4_ITER_ALG_RF_PRIMITIVE :
			case DK4_ITER_ALG_RF_ILLINOIS :
			case DK4_ITER_ALG_RF_PEGASUS :
			case DK4_ITER_ALG_RF_ANDERSON_BJOERCK : {	/* Regula falsi */
				x = (a * fb - b * fa) / (fb - fa);
			} break;
			default : {									/* Bisection */
				x = (a + b) / 2.0;
			} break;
		}
		if (0 == dk4ma_is_finite(x)) {
			back = DK4_ITER_RESULT_E_INFINITE;
			cc = -1;
		}
		else {								$? ". x = %lg", x
			/*	Calculate function for x, if x is usable
			*/
			res = (*fct)(&fx, x, ps);
			if (0 == res) {					$? "! function calculation for x"
				back = DK4_ITER_RESULT_E_FCT;
				cc = -1;
			}
			if (0 == dk4ma_is_finite(fx)) {	$? "! function calculation for x"
				back = DK4_ITER_RESULT_E_INFINITE;
				cc = -1;
			}
			afx = fabs(fx);
			if (0 == dk4ma_is_finite(afx)) {	$? "! absolute value for fx"
				back = DK4_ITER_RESULT_E_INFINITE;
				cc = -1;
			}
			/*	Check whether we are done, if we could continue
			*/
			if (1 == cc) {					$? ". y = %lg", fx
				$!trace-code if (1UL < passno) {
				$? ". x-delta = %lg", fabs(x - xo)
				$!trace-code }
				if (isgreater(mctx.eps_y,0.0)) {
					/* Must check y */
					if (afx < mctx.eps_y) {	$? ". y small enough"
						if ((x == xo) && (1UL < passno)) {
							cc = 0;			$? ". OK exact match"
						}
						else {
							if (0 == mctx.exact) {
								if(isgreater(mctx.eps_x,0.0)) {
									if (isless(fabs(x - xo),mctx.eps_x)) {
										if (1UL < passno) {
											cc = 0;	$? ". OK small step"
										}
									}
								}
								else {
									cc = 0;	$? ". OK no x step restriction"
								}
							}
						}
					}
				}
				else {						$? ". no y check required"
					/* No y check */
					if ((x == xo) && (1UL < passno)) {
						cc = 0;				$? ". OK exact match"
					}
					else {
						if (0 == mctx.exact) {
							if(isgreater(mctx.eps_x,0.0)) {
								if (isless(fabs(x - xo),mctx.eps_x)) {
									if (1UL < passno) {
										cc = 0;	$? ". OK x step small enough"
									}
								}
							}
							else {
								/*	Neither eps_x nor eps_y,
									so we wait for x=xo.
								*/
							}
						}
					}
				}
			}
			/*	Continue for usable y values only
			*/
			if (1 == cc) {
				/*	Find direction for next change
				*/
				nc = CHANGED_NONE;
				switch (cond) {
					case CONDITION_FA_LESS_ZERO : {
						if (0.0 > fx) {
							nc = CHANGED_A;
						}
						else {
							nc = CHANGED_B;
						}
					} break;
					case CONDITION_FA_LEQ_ZERO : {
						if (0.0 >= fx) {
							nc = CHANGED_A;
						}
						else {
							nc = CHANGED_B;
						}
					} break;
					case CONDITION_FA_GREATER_ZERO : {
						if (0.0 < fx) {
							nc = CHANGED_A;
						}
						else {
							nc = CHANGED_B;
						}
					} break;
					case CONDITION_FA_GEQ_ZERO : {
						if (0.0 <= fx) {
							nc = CHANGED_A;
						}
						else {
							nc = CHANGED_B;
						}
					} break;
				}
				/*	Apply interval border change
				*/
				switch (nc) {
					case CHANGED_A: {		$? ". use x as a"
						if (CHANGED_A == pc) {
							switch (mctx.algo) {
								case DK4_ITER_ALG_RF_ILLINOIS : {
									fb = 0.5 * fb;
									if (0 == dk4ma_is_finite(fb)) {
										back = DK4_ITER_RESULT_E_INFINITE;
										cc = -1;		$? "! fb"
									}
								} break;
								case DK4_ITER_ALG_RF_PEGASUS : {
									gamma = dk4iter_gamma_pegasus(fa, fx);
									fb *= gamma;
									if (0 == dk4ma_is_finite(fb)) {
										back = DK4_ITER_RESULT_E_INFINITE;
										cc = -1;		$? "! fb"
									}
								} break;
								case DK4_ITER_ALG_RF_ANDERSON_BJOERCK : {
									gamma = dk4iter_gamma_anderson_bjoerck(
										fa, fx
									);
									fb *= gamma;
									if (0 == dk4ma_is_finite(fb)) {
										back = DK4_ITER_RESULT_E_INFINITE;
										cc = -1;		$? "! fb"
									}
								} break;
							}
						}
						a = x;
						fa = fx;
						pc = CHANGED_A;
					} break;
					case CHANGED_B: {		$? ". use x as b"
						if (CHANGED_B == pc) {
							switch (mctx.algo) {
								case DK4_ITER_ALG_RF_ILLINOIS : {
									fa = 0.5 * fa;
									if (0 == dk4ma_is_finite(fa)) {
										back = DK4_ITER_RESULT_E_INFINITE;
										cc = -1;		$? "! fa"
									}
								} break;
								case DK4_ITER_ALG_RF_PEGASUS : {
									gamma = dk4iter_gamma_pegasus(
										fb, fx
									);
									fa *= gamma;
									if (0 == dk4ma_is_finite(fa)) {
										back = DK4_ITER_RESULT_E_INFINITE;
										cc = -1;		$? "! fa"
									}
								} break;
								case DK4_ITER_ALG_RF_ANDERSON_BJOERCK : {
									gamma = dk4iter_gamma_anderson_bjoerck(
										fb, fx
									);
									fa *= gamma;
									if (0 == dk4ma_is_finite(fa)) {
										back = DK4_ITER_RESULT_E_INFINITE;
										cc = -1;		$? "! fa"
									}
								} break;
							}
						}
						b = x;
						fb = fx;
						pc = CHANGED_B;
					} break;
					default: {
						/* ERROR: Must not happen */
						back = DK4_ITER_RESULT_E_CONV;	$? "! bug"
						cc = -1;
					} break;
				}
			}
		}
		/*	Check number of passes
		*/
		if ((1 == cc) && (0UL < mctx.maxpass) && (passno >= mctx.maxpass)) {
			back = DK4_ITER_RESULT_E_PASSES;	$? "! too many passes"
			cc = -1;
		}
	} while (1 == cc);

	/*	Success
	*/
	if (0 == cc) {
		*d = x;
		if (NULL != pp)  { *pp = passno; }
		back = DK4_ITER_RESULT_SUCCESS;
	}

  finished:
  	$? "- dk4iter_interval %d", back
	return back;
}



/**	Run Newton iteration.
	@param	d	Destination (address of result variable).
	@param	pp	Address of variable to store number of passes on success.
	@param	fct	Iteration function, returns two values into the array
				at address d: the function value and the first derivative
				value.
	@param	ps	Parameter set, may be NULL if fct does not use it.
	@param	x0	Start point.
	@param	ctx	Iteration context, may be NULL.
	@return	DK4_ITER_RESULT_SUCCESS on success, one from
	DK4_ITER_RESULT_E_PASSES, DK4_ITER_RESULT_E_INFINITE,
	DK4_ITER_RESULT_E_OOR, DK4_ITER_RESULT_E_CONV, DK4_ITER_RESULT_E_FCT,
	or DK4_ITER_RESULT_E_ARGS on error.
*/
static
int
dk4iter_newton(
	double					*d,
	unsigned long			*pp,
	dk4_iter_fct_t			*fct,
	void			const	*ps,
	double					 x0,
	dk4_iter_ctx_t	const	*ctx
)
{
	double				v[2];
	double				xn		=	0.0;
	unsigned long		passno	=	0UL;
	int		 			back	=	DK4_ITER_RESULT_E_ARGS;
	int					cc		=	1;
	int					res		=	0;
	$? "+ dk4iter_newton"
#if	DK4_USE_ASSERT
  assert(NULL != d);
  assert(NULL != fct);
#endif
	res = (*fct)(v, x0, ps);
	if (0 == res) {
		cc = -1;								$? "! function calculation"
		back = DK4_ITER_RESULT_E_FCT;
	}
	if (!((dk4ma_is_finite(v[0])) && (dk4ma_is_finite(v[1])))) {
		cc = -1;								$? "! infinite values"
		back = DK4_ITER_RESULT_E_INFINITE;
	}
	while (1 == cc) {
		if (ULONG_MAX > passno) { passno++; }	$? ". begin pass %lu", passno
		xn = x0 - v[0] / v[1];
		if (dk4ma_is_finite(xn)) {				$? ". x     = %lg", xn
			if (0 != (MINMAX_MINIMUM & (ctx->minmax))) {
				if (0 != (MINMAX_MIN_EXCL & (ctx->minmax))) {
					if (xn <= ctx->xmin) {
						cc = -1;			$? "! out of range, too small"
						back = DK4_ITER_RESULT_E_OOR;
					}
				}
				else {
					if (xn < ctx->xmin) {
						cc = -1;			$? "! out of range, too small"
						back = DK4_ITER_RESULT_E_OOR;
					}
				}
			}
			if (0 != (MINMAX_MAXIMUM & (ctx->minmax))) {
				if (0 != (MINMAX_MAX_EXCL & (ctx->minmax))) {
					if (xn >= ctx->xmax) {
						cc = -1;				$? "! out of range, too large"
						back = DK4_ITER_RESULT_E_OOR;
					}
				}
				else {
					if (xn > ctx->xmax) {
						cc = -1;				$? "! out of range, too large"
						back = DK4_ITER_RESULT_E_OOR;
					}
				}
			}
			if (1 == cc) {
				res = (*fct)(v, xn, ps);
				if (0 == res) {
					cc = -1;					$? "! function calculation"
					back = DK4_ITER_RESULT_E_FCT;
				}
				if (!((dk4ma_is_finite(v[0])) && (dk4ma_is_finite(v[1])))) {
					cc = -1;					$? "! infinite values"
					back = DK4_ITER_RESULT_E_INFINITE;
				}
				if (1 == cc) {
					$? ". y     = %lg", v[0]
					$? ". dy/dx = %lg", v[1]
					if (isgreater(ctx->eps_y,0.0)) {	$? ". check y"
						if (fabs(v[0]) < ctx->eps_y) {	$? ". y in range"
							if (xn == x0) {
								cc = 0;					$? ". OK exact value"
							}
							else {
								if (0 == ctx->exact) {	$? ". check step"
									if(isgreater(ctx->eps_x,0.0)) {
										if (isless(fabs(xn - x0),ctx->eps_x)) {
											cc = 0;	$? ". OK small step"
										}
									}
									else {
										cc = 0;	$? ". OK no x step constraint"
									}
								}
							}
						}
					}
					else {								$? ". no y check"
						if (xn == x0) {
							cc = 0;	$? ". OK exact match"
						}
						else {
							if (0 == ctx->exact) {
								if (isgreater(ctx->eps_x,0.0)) {
									if (isless(fabs(xn - x0),ctx->eps_x)) {
										cc = 0; $? ". OK x step small enough"
									}
								}
								else {
									/*	Neither eps_x nor eps_y,
										so we wait for x=x0.
									*/
								}
							}
						}
					}
				}
				x0 = xn;
			}
		}
		else {
			cc = -1;						$? "! infinite value"
			back = DK4_ITER_RESULT_E_INFINITE;
		}
		if ((1 == cc) && (0UL < ctx->maxpass) && (passno >= ctx->maxpass)) {
			cc = -1;
			back = DK4_ITER_RESULT_E_PASSES;
		}
	}
	if (0 == cc) {
		*d = xn;
		if (NULL != pp) { *pp = passno; }
		back = DK4_ITER_RESULT_SUCCESS;
	}
	$? "- dk4iter_newton %d", back
	return back;
}



/**	Run fix point iteration.
	@param	d	Destination (address of result variable).
	@param	pp	Address of variable to store number of passes on success.
	@param	fct	Iteration function, the phi part of phi(x)=x.
	@param	ps	Parameter set, may be NULL if fct does not use it.
	@param	x0	Start point.
	@param	ctx	Iteration context, may be NULL.
	@return	DK4_ITER_RESULT_SUCCESS on success, one from
	DK4_ITER_RESULT_E_PASSES, DK4_ITER_RESULT_E_INFINITE,
	DK4_ITER_RESULT_E_OOR, DK4_ITER_RESULT_E_CONV, DK4_ITER_RESULT_E_FCT,
	or DK4_ITER_RESULT_E_ARGS on error.
*/
static
int
dk4iter_fix_point(
	double					*d,
	unsigned long			*pp,
	dk4_iter_fct_t			*fct,
	void			const	*ps,
	double					 x0,
	dk4_iter_ctx_t	const	*ctx
)
{
#if 0
	double				a;					/* Border from previous steps */
	double				b;					/* Border from previous steps */
#endif
	double				xn		=	0.0;	/* New x value */
	unsigned long		passno	=	0UL;	/* Iteration step number */
	int					res		=	0;		/* Function evaluation result */
	int					cc		=	1;		/* Flag: Can continue */
	int		 			back	=	DK4_ITER_RESULT_E_ARGS;
	$? "+ dk4iter_fix_point"
#if	DK4_USE_ASSERT
  assert(NULL != d);
  assert(NULL != fct);
#endif
	while (1 == cc) {
		if (ULONG_MAX > passno) { passno++; }	$? ". begin pass %lu", passno
		/*
			Calculate new x, check result
		*/
		res = (*fct)(&xn, x0, ps);
		if (0 == res) {
			cc = -1;							$? "! function evaluation"
			back = DK4_ITER_RESULT_E_FCT;
		}
		if (!(dk4ma_is_finite(xn))) {
			cc = -1;							$? "! infinite result"
			back = DK4_ITER_RESULT_E_INFINITE;
		}
		if (1 == cc) {							$? ". x = %lg", xn
			/*
				Check whether specified interval is exceeded
			*/
			if (0 != (MINMAX_MINIMUM & (ctx->minmax))) {
				if (0 != (MINMAX_MIN_EXCL & (ctx->minmax))) {
					if (xn <= ctx->xmin) {
						cc = -1;			$? "! out of range, too small"
						back = DK4_ITER_RESULT_E_OOR;
					}
				}
				else {
					if (xn < ctx->xmin) {
						cc = -1;			$? "! out of range, too small"
						back = DK4_ITER_RESULT_E_OOR;
					}
				}
			}
			if (0 != (MINMAX_MAXIMUM & (ctx->minmax))) {
				if (0 != (MINMAX_MAX_EXCL & (ctx->minmax))) {
					if (xn >= ctx->xmax) {
						cc = -1;				$? "! out of range, too large"
						back = DK4_ITER_RESULT_E_OOR;
					}
				}
				else {
					if (xn > ctx->xmax) {
						cc = -1;				$? "! out of range, too large"
						back = DK4_ITER_RESULT_E_OOR;
					}
				}
			}
			if (1 == cc) {
				if (1 == cc) {
					/*
						Check whether we are finished
					*/
					if (xn == x0) {
						cc = 0;				$? ". OK exact match"
					}
					else {
						if ((0 == ctx->exact) && (0.0 < ctx->eps_x)) {
							if (isless(fabs(xn-x0),ctx->eps_x)) {
								cc = 0;		$? ". OK change small enough"
							}
#if	TRACE_DEBUG
							else {			$? ". change %lg", fabs(xn-x0)
							}
#endif
						}
						else {	 $? "! No condition to check %lg", (xn-x0)
						}
					}
				}
			}
			x0 = xn;
		}
		/*	Stop if too many passes
		*/
		if (1 == cc) {
			if ((0UL < ctx->maxpass) && (passno >= ctx->maxpass)) {
				cc = -1;
				back = DK4_ITER_RESULT_E_PASSES;
			}
		}
	}
	if (0 == cc) {
		*d = xn;
		if (NULL != pp) { *pp = passno; }
		back = DK4_ITER_RESULT_SUCCESS;
	}
	$? "- dk4iter_fix_point %d", back
	return back;
}



int
dk4iter_start_point(
	double					*d,
	unsigned long			*pp,
	dk4_iter_fct_t			*fct,
	void			const	*ps,
	double					 x0,
	dk4_iter_ctx_t	const	*ctx
)
{
	dk4_iter_ctx_t		mctx;
	int		 			back	=	DK4_ITER_RESULT_E_ARGS;

	/*	Check function call arguments
	*/
#if	DK4_USE_ASSERT
  assert(NULL != d);
  assert(NULL != fct);
#endif
	if ((NULL == d) || (NULL == fct)) {
		goto finished;
	}
	/*	Copy or initialize context
	*/
	if (NULL != ctx) {
		DK4_MEMCPY(&mctx,ctx,sizeof(dk4_iter_ctx_t));
	}
	else {
		dk4iter_ctx_init(&mctx);
		mctx.algo = DK4_ITER_ALG_NEWTON;
	}
	/*	Check x0 in interval
	*/
	if (0 != (MINMAX_MINIMUM & (mctx.minmax))) {
		if (0 != (MINMAX_MIN_EXCL & (mctx.minmax))) {
			if (x0 <= mctx.xmin) {
				goto finished;
			}
		}
		else {
			if (x0 < mctx.xmin) {
				goto finished;
			}
		}
	}
	if (0 != (MINMAX_MAXIMUM & (mctx.minmax))) {
		if (0 != (MINMAX_MAX_EXCL & (mctx.minmax))) {
			if (x0 >= mctx.xmax) {
				goto finished;
			}
		}
		else {
			if (x0 > mctx.xmax) {
				goto finished;
			}
		}
	}
	/*	Call function for specified algorithm.
	*/
	switch (mctx.algo) {
		case DK4_ITER_ALG_NEWTON : {
			back = dk4iter_newton(d, pp, fct, ps, x0, &mctx);
		} break;
		case DK4_ITER_ALG_FIX_POINT : {
			back = dk4iter_fix_point(d, pp, fct, ps, x0, &mctx);
		} break;
	}

  finished:
	return back;
}




/* vim: set ai sw=4 ts=4 : */