optimization.cpp

//
// C++ Implementation: optimization
//
// Description: 
//
//
// Author: BUI Quang Minh, Steffen Klaere, Arndt von Haeseler <minh.bui@univie.ac.at>, (C) 2008
//
// Copyright: See COPYING file that comes with this distribution
//
//
#include "optimization.h"
#include <math.h>
#include <stdio.h>
#include <stdlib.h>
#include <iostream>
//#include "tools.h"


using namespace std;

const double ERROR_X = 1.0e-4;

double ran1(long *idum);
double *vector(long nl, long nh);
void free_vector(double *v, long nl, long nh);
double **matrix(long nrl, long nrh, long ncl, long nch);
void free_matrix(double **m, long nrl, long nrh, long ncl, long nch);
void fixBound(double x[], double lower[], double upper[], int n);

#define NR_END 1
#define FREE_ARG char*

#define GET_PSUM \
					for (n=1;n<=ndim;n++) {\
					for (sum=0.0,m=1;m<=mpts;m++) sum += p[m][n];\
					psum[n]=sum;}


#define IA 16807
#define IM 2147483647
#define AM (1.0/IM)
#define IQ 127773
#define IR 2836
#define NTAB 32
#define NDIV (1+(IM-1)/NTAB)
#define EPS 1.2e-7
#define RNMX (1.0-EPS)

double ran1(long *idum) {
	int j;
	long k;
	static long iy=0;
	static long iv[NTAB];
	double temp;

	if (*idum <= 0 || !iy) {
		if (-(*idum) < 1) *idum=1;
		else *idum = -(*idum);
		for (j=NTAB+7;j>=0;j--) {
			k=(*idum)/IQ;
			*idum=IA*(*idum-k*IQ)-IR*k;
			if (*idum < 0) *idum += IM;
			if (j < NTAB) iv[j] = *idum;
		}
		iy=iv[0];
	}
	k=(*idum)/IQ;
	*idum=IA*(*idum-k*IQ)-IR*k;
	if (*idum < 0) *idum += IM;
	j=iy/NDIV;
	iy=iv[j];
	iv[j] = *idum;
	if ((temp=AM*iy) > RNMX) return RNMX;
	else return temp;
}
#undef IA
#undef IM
#undef AM
#undef IQ
#undef IR
#undef NTAB
#undef NDIV
#undef EPS
#undef RNMX


long idum = 123456;
double tt;

void nrerror(const char *error_text)
/* Numerical Recipes standard error handler */
{
	cerr << "NUMERICAL ERROR: " << error_text << endl;
	//exit(1);
	throw error_text;
}

double *vector(long nl, long nh)
/* allocate a double vector with subscript range v[nl..nh] */
{
	double *v;

	v=(double *)malloc((size_t) ((nh-nl+1+NR_END)*sizeof(double)));
	if (!v) nrerror("allocation failure in vector()");
	return v-nl+NR_END;
}

double **matrix(long nrl, long nrh, long ncl, long nch)
/* allocate a double matrix with subscript range m[nrl..nrh][ncl..nch] */
{
	long i, nrow=nrh-nrl+1,ncol=nch-ncl+1;
	double **m;

	/* allocate pointers to rows */
	m=(double **) malloc((size_t)((nrow+NR_END)*sizeof(double*)));
	if (!m) nrerror("allocation failure 1 in matrix()");
	m += NR_END;
	m -= nrl;

	/* allocate rows and set pointers to them */
	m[nrl]=(double *) malloc((size_t)((nrow*ncol+NR_END)*sizeof(double)));
	if (!m[nrl]) nrerror("allocation failure 2 in matrix()");
	m[nrl] += NR_END;
	m[nrl] -= ncl;

	for(i=nrl+1;i<=nrh;i++) m[i]=m[i-1]+ncol;

	/* return pointer to array of pointers to rows */
	return m;
}


void free_vector(double *v, long nl, long nh)
/* free a double vector allocated with vector() */
{
	free((FREE_ARG) (v+nl-NR_END));
}

void free_matrix(double **m, long nrl, long nrh, long ncl, long nch)
/* free a double matrix allocated by dmatrix() */
{
	free((FREE_ARG) (m[nrl]+ncl-NR_END));
	free((FREE_ARG) (m+nrl-NR_END));
}

void fixBound(double x[], double lower[], double upper[], int n) {
	for (int i = 1; i <= n; i++) {
		if (x[i] < lower[i])
			x[i] = lower[i];
		else if (x[i] > upper[i])
			x[i] = upper[i];
	}
}

/**********************************************
	Optimization routines
**********************************************/
Optimization::Optimization()
{
}


Optimization::~Optimization()
{
}

/*****************************************************
	One dimensional optimization with Brent method
*****************************************************/
	
#define ITMAX 100
#define CGOLD 0.3819660
#define GOLD 1.618034
#define GLIMIT 100.0
#define TINY 1.0e-20
#define ZEPS 1.0e-10
#define SHFT(a,b,c,d) (a)=(b);(b)=(c);(c)=(d);
#define SIGN(a,b) ((b) >= 0.0 ? fabs(a) : -fabs(a))

/* Brents method in one dimension */
double Optimization::brent_opt (double ax, double bx, double cx, double tol,
                          double *foptx, double *f2optx, double fax, double fbx, double fcx) {
	int iter;
	double a,b,d=0,etemp,fu,fv,fw,fx,p,q,r,tol1,tol2,u,v,w,x,xm;
	double xw,wv,vx;
	double e=0.0;

	a=(ax < cx ? ax : cx);
	b=(ax > cx ? ax : cx);
	x=bx;
	fx=fbx;
	if (fax < fcx) {
		w=ax;
		fw=fax;
		v=cx;
		fv=fcx;
	} else {
		w=cx;
		fw=fcx;
		v=ax;
		fv=fax;
	}

	for (iter=1;iter<=ITMAX;iter++) {
		xm=0.5*(a+b);
		tol2=2.0*(tol1=tol*fabs(x)+ZEPS);
		if (fabs(x-xm) <= (tol2-0.5*(b-a))) {
			*foptx = fx;
			xw = x-w;
			wv = w-v;
			vx = v-x;
			*f2optx = 2.0*(fv*xw + fx*wv + fw*vx)/
			          (v*v*xw + x*x*wv + w*w*vx);
			return x;
		}

		if (fabs(e) > tol1) {
			r=(x-w)*(fx-fv);
			q=(x-v)*(fx-fw);
			p=(x-v)*q-(x-w)*r;
			q=2.0*(q-r);
			if (q > 0.0)
				p = -p;
			q=fabs(q);
			etemp=e;
			e=d;
			if (fabs(p) >= fabs(0.5*q*etemp) || p <= q*(a-x) || p >= q*(b-x))
				d=CGOLD*(e=(x >= xm ? a-x : b-x));
			else {
				d=p/q;
				u=x+d;
				if (u-a < tol2 || b-u < tol2)
					d=SIGN(tol1,xm-x);
			}
		} else {
			d=CGOLD*(e=(x >= xm ? a-x : b-x));
		}

		u=(fabs(d) >= tol1 ? x+d : x+SIGN(tol1,d));
		fu=computeFunction(u);
		if (fu <= fx) {
			if (u >= x)
				a=x;
			else
				b=x;

			SHFT(v,w,x,u)
			SHFT(fv,fw,fx,fu)
		} else {
			if (u < x)
				a=u;
			else
				b=u;
			if (fu <= fw || w == x) {
				v=w;
				w=u;
				fv=fw;
				fw=fu;
			} else
				if (fu <= fv || v == x || v == w) {
					v=u;
					fv=fu;
				}
		}
	}

	*foptx = fx;
	xw = x-w;
	wv = w-v;
	vx = v-x;
	*f2optx = 2.0*(fv*xw + fx*wv + fw*vx)/(v*v*xw + x*x*wv + w*w*vx);

	return x;
}

#undef ITMAX
#undef CGOLD
#undef ZEPS
#undef SHFT
#undef SIGN
#undef GOLD
#undef GLIMIT
#undef TINY


double Optimization::minimizeOneDimen(double xmin, double xguess, double xmax, double tolerance, double *fx, double *f2x) {
	double eps, optx, ax, bx, cx, fa, fb, fc;
	//int    converged;	/* not converged error flag */
		
	/* first attempt to bracketize minimum */
	if (xguess < xmin) xguess = xmin;
	if (xguess > xmax) xguess = xmax;
	eps = xguess*tolerance*50.0;
	ax = xguess - eps;
	if (ax < xmin) ax = xmin;
	bx = xguess;
	cx = xguess + eps;
	if (cx > xmax) cx = xmax;
	
	/* check if this works */
	fa = computeFunction(ax);
	fb = computeFunction(bx);
	fc = computeFunction(cx);

	/* if it works use these borders else be conservative */
	if ((fa < fb) || (fc < fb)) {
		if (ax != xmin) fa = computeFunction(xmin);
		if (cx != xmax) fc = computeFunction(xmax);
		ax = xmin;
		cx = xmax;
	}
	/*
	const int MAX_ROUND = 10;
	for (i = 0; ((fa < fb-tolerance) || (fc < fb-tolerance)) && (i<MAX_ROUND); i++) {
		// brent method require that fb is smaller than both fa and fc
		// find some random values until fb achieve this
			bx = (((double)rand()) / RAND_MAX)*(cx-ax) + ax;
			fb = computeFunction(bx);
	}*/

/*
	if ((fa < fb) || (fc < fb)) {
		if (fa < fc) { bx = ax; fb = fa; } else { bx = cx; fb = fc; }
		//cout << "WARNING: Initial value for Brent method is set at bound " << bx << endl;
	}*/
	//	optx = brent_opt(xmin, xguess, xmax, tolerance, fx, f2x, fa, fb, fc);
	//} else
	optx = brent_opt(ax, bx, cx, tolerance, fx, f2x, fa, fb, fc);

	return optx; /* return optimal x */
}

double Optimization::minimizeOneDimenSafeMode(double xmin, double xguess, double xmax, double tolerance, double *f)
{
	double ferror;
	double optx = minimizeOneDimen(xmin, xguess, xmax, tolerance, f, &ferror);
	double fnew;
	// check value at the boundary
	if ((optx < xmax) && (fnew = computeFunction(xmax)) <= *f+tolerance) {
		//if (verbose_mode >= VB_MAX)
			//cout << "Note from Newton safe mode: " << optx << " (" << f << ") -> " << xmax << " ("<< fnew << ")" << endl;
		optx = xmax;
		*f = fnew;
	}
	if ((optx > xmin) && (fnew = computeFunction(xmin)) <= *f+tolerance) {
		//if (verbose_mode >= VB_MAX)
			//cout << "Note from Newton safe mode: " << optx << " -> " << xmin << endl;
		optx = xmin;
		*f = fnew;
	}
	return optx;
}

/*****************************************************
	One dimensional optimization with Newton Raphson 
	only applicable if 1st and 2nd derivatives are easy to compute
*****************************************************/


double Optimization::minimizeNewtonSafeMode(double xmin, double xguess, double xmax, double tolerance, double &f)
{
	double optx = minimizeNewton(xmin, xguess, xmax, tolerance, f);
	double fnew;
	// check value at the boundary
	if ((optx < xmax) && (fnew = computeFunction(xmax)) <= f+tolerance) {
		//if (verbose_mode >= VB_MAX)
			//cout << "Note from Newton safe mode: " << optx << " (" << f << ") -> " << xmax << " ("<< fnew << ")" << endl;
		optx = xmax;
		f = fnew;
	}
	if ((optx > xmin) && (fnew = computeFunction(xmin)) <= f+tolerance) {
		//if (verbose_mode >= VB_MAX)
			//cout << "Note from Newton safe mode: " << optx << " -> " << xmin << endl;
		optx = xmin;
		f = fnew;
	}
	return optx;
}

double Optimization::minimizeNewton(double x1, double xguess, double x2, double xacc, double &fm, double &d2l)
{
	const int MAXIT = 64;
	int j;
	double df,dx,dxold,f;
	double temp,xh,xl,rts, fold, finit;

	rts = xguess;
	if (rts < x1) rts = x1;
	if (rts > x2) rts = x2;
	finit = fold = fm = computeFuncDerv(rts,f,df);
	d2l = df;
	if (!isfinite(fm) || !isfinite(f) || !isfinite(df)) {
		nrerror("Wrong computeFuncDerv");
	}
	if (df >= 0.0 && fabs(f) < xacc) return rts;
	if (f < 0.0) {
		xl = rts;
		xh = x2;
	} else {
		xh = rts;
		xl = x1;	
	}

	dx=dxold=fabs(xh-xl);
	for (j=1;j<=MAXIT;j++) {
		if (
			(df <= 0.0) // function is concave
			|| (fm > fold + xacc) // increasing
			|| (((rts-xh)*df-f)*((rts-xl)*df-f) >= 0.0) // out of bound
			//|| (fabs(2.0*f) > fabs(dxold*df))  // converge too slow
			) {
			dxold=dx;
			dx=0.5*(xh-xl);
			rts=xl+dx;
            d2l = df;
			if (xl == rts) return rts;
		} else {
			dxold=dx;
			dx=f/df;
			temp=rts;
			rts -= dx;
			d2l = df;
			if (temp == rts) return rts;
		}
		if (fabs(dx) < xacc) { fm = computeFunction(rts); return rts; }
		fold = fm;
		fm = computeFuncDerv(rts,f,df);
		if (!isfinite(fm) || !isfinite(f) || !isfinite(df)) nrerror("Wrong computeFuncDerv");
		if (df > 0.0 && fabs(f) < xacc) {
			d2l = df;
			return rts;
		}
		if (f < 0.0)
			xl=rts;
		else
			xh=rts;
	}
	//return rts;
	nrerror("Maximum number of iterations exceeded in minimizeNewton");
	d2l = 0.0;
	return 0.0;
//return_ok:
	if (fm > finit) {
		//cout.precision(10);
		//cout << "revert xguess, fm=" << fm << " finit=" << finit << endl;
		fm = finit;
		return xguess;
	}
	return rts;
}

double Optimization::minimizeNewton(double x1, double xguess, double x2, double xacc, double &fm)
{
	double var;
	double optx = minimizeNewton(x1, xguess, x2, xacc, fm, var);
	return optx;
}

/*****************************************************
	Multi dimensional optimization with BFGS method
*****************************************************/

#define ALF 1.0e-4
#define TOLX 1.0e-7
static double maxarg1,maxarg2;
#define FMAX(a,b) (maxarg1=(a),maxarg2=(b),(maxarg1) > (maxarg2) ?\
        (maxarg1) : (maxarg2))

void Optimization::lnsrch(int n, double xold[], double fold, double g[], double p[], double x[],
                   double *f, double stpmax, int *check, double lower[], double upper[]) {
	int i;
	double a,alam,alam2=0,alamin,b,disc,f2=0,fold2=0,rhs1,rhs2,slope,sum,temp,
	test,tmplam;

	*check=0;
	for (sum=0.0,i=1;i<=n;i++) sum += p[i]*p[i];
	sum=sqrt(sum);
	if (sum > stpmax)
		for (i=1;i<=n;i++) p[i] *= stpmax/sum;
	for (slope=0.0,i=1;i<=n;i++)
		slope += g[i]*p[i];
	test=0.0;
	for (i=1;i<=n;i++) {
		temp=fabs(p[i])/FMAX(fabs(xold[i]),1.0);
		if (temp > test) test=temp;
	}
	alamin=TOLX/test;
	alam=1.0;
	/*
	int rep = 0;
	do {
		for (i=1;i<=n;i++) x[i]=xold[i]+alam*p[i];
		if (!checkRange(x))
			alam *= 0.5;
		else
			break;
		rep++;
	} while (rep < 10);
	*/
	bool first_time = true;
	for (;;) {
		for (i=1;i<=n;i++) x[i]=xold[i]+alam*p[i];
		fixBound(x, lower, upper, n);
		//checkRange(x);
		*f=targetFunk(x);
		if (alam < alamin) {
			for (i=1;i<=n;i++) x[i]=xold[i];
			*check=1;
			return;
		} else if (*f <= fold+ALF*alam*slope) return;
		else {
			if (first_time)
				tmplam = -slope/(2.0*(*f-fold-slope));
			else {
				rhs1 = *f-fold-alam*slope;
				rhs2=f2-fold2-alam2*slope;
				a=(rhs1/(alam*alam)-rhs2/(alam2*alam2))/(alam-alam2);
				b=(-alam2*rhs1/(alam*alam)+alam*rhs2/(alam2*alam2))/(alam-alam2);
				if (a == 0.0) tmplam = -slope/(2.0*b);
				else {
					disc=b*b-3.0*a*slope;
					if (disc<0.0) //nrerror("Roundoff problem in lnsrch.");
						tmplam = 0.5 * alam;
					else if (b <= 0.0) tmplam=(-b+sqrt(disc))/(3.0*a);
					else tmplam = -slope/(b+sqrt(disc));
				}
				if (tmplam>0.5*alam)
					tmplam=0.5*alam;
			}
		}
		alam2=alam;
		f2 = *f;
		fold2=fold;
		alam=FMAX(tmplam,0.1*alam);
		first_time = false;
	}
}
#undef ALF
#undef TOLX


const int MAX_ITER = 3;
extern double random_double();

double Optimization::minimizeMultiDimen(double guess[], int ndim, double lower[], double upper[], bool bound_check[], double gtol) {
	int i, iter;
	double fret, minf = 10000000.0;
	double *minx = new double [ndim+1];
	int count = 0;
	bool restart;
	do {
		dfpmin(guess, ndim, lower, upper, gtol, &iter, &fret);
		if (fret < minf) {
 			minf = fret;
			for (i = 1; i <= ndim; i++)
				minx[i] = guess[i];
		}
		count++;
		// restart the search if at the boundary
		// it's likely to end at a local optimum at the boundary
		restart = false;
		
		
		for (i = 1; i <= ndim; i++)
			if (bound_check[i])
			if (fabs(guess[i]-lower[i]) < 1e-4 || fabs(guess[i]-upper[i]) < 1e-4) {
				restart = true;
				break;
			}
		
		if (!restart)
			break;

		if (count == MAX_ITER)
			break;
			
		do {
			for (i = 1; i <= ndim; i++) {
				guess[i] = random_double() * (upper[i] - lower[i])/3 + lower[i];
			}
		} while (false);
		cout << "Restart estimation at the boundary... " << std::endl;
	} while (count < MAX_ITER);
	if (count > 1) {
		for (i = 1; i <= ndim; i++)
			guess[i] = minx[i];
		fret = minf;
	}
	delete [] minx;
	
	return fret;
}


#define ITMAX 500
static double sqrarg;
#define SQR(a) ((sqrarg=(a)) == 0.0 ? 0.0 : sqrarg*sqrarg)
#define EPS 3.0e-8
#define TOLX (4*EPS)
#define STPMX 100.0

#define FREEALL free_vector(xi,1,n);free_vector(pnew,1,n); \
free_matrix(hessin,1,n,1,n);free_vector(hdg,1,n);free_vector(g,1,n); \
free_vector(dg,1,n);



void Optimization::dfpmin(double p[], int n, double lower[], double upper[], double gtol, int *iter, double *fret) {
	int check,i,its,j;
	double den,fac,fad,fae,fp,stpmax,sum=0.0,sumdg,sumxi,temp,test;
	double *dg,*g,*hdg,**hessin,*pnew,*xi;

	dg=vector(1,n);
	g=vector(1,n);
	hdg=vector(1,n);
	hessin=matrix(1,n,1,n);
	pnew=vector(1,n);
	xi=vector(1,n);
	fp = derivativeFunk(p,g);
	for (i=1;i<=n;i++) {
		for (j=1;j<=n;j++) hessin[i][j]=0.0;
		hessin[i][i]=1.0;
		xi[i] = -g[i];
		sum += p[i]*p[i];
	}
	//checkBound(p, xi, lower, upper, n);
	//checkDirection(p, xi);

	stpmax=STPMX*FMAX(sqrt(sum),(double)n);
	for (its=1;its<=ITMAX;its++) {
		*iter=its;
		lnsrch(n,p,fp,g,xi,pnew,fret,stpmax,&check, lower, upper);
		fp = *fret;
		for (i=1;i<=n;i++) {
			xi[i]=pnew[i]-p[i];
			p[i]=pnew[i];
		}
		test=0.0;
		for (i=1;i<=n;i++) {
			temp=fabs(xi[i])/FMAX(fabs(p[i]),1.0);
			if (temp > test) test=temp;
		}
		if (test < TOLX) {
			FREEALL
			return;
		}
		for (i=1;i<=n;i++) dg[i]=g[i];
		derivativeFunk(p,g);
		test=0.0;
		den=FMAX(*fret,1.0);
		for (i=1;i<=n;i++) {
			temp=fabs(g[i])*FMAX(fabs(p[i]),1.0)/den;
			if (temp > test) test=temp;
		}
		if (test < gtol) {
			FREEALL
			return;
		}
		for (i=1;i<=n;i++) dg[i]=g[i]-dg[i];
		for (i=1;i<=n;i++) {
			hdg[i]=0.0;
			for (j=1;j<=n;j++) hdg[i] += hessin[i][j]*dg[j];
		}
		fac=fae=sumdg=sumxi=0.0;
		for (i=1;i<=n;i++) {
			fac += dg[i]*xi[i];
			fae += dg[i]*hdg[i];
			sumdg += SQR(dg[i]);
			sumxi += SQR(xi[i]);
		}
		if (fac*fac > EPS*sumdg*sumxi)
		{
			fac=1.0/fac;
			fad=1.0/fae;
			for (i=1;i<=n;i++) dg[i]=fac*xi[i]-fad*hdg[i];
			for (i=1;i<=n;i++) {
				for (j=1;j<=n;j++) {
					hessin[i][j] += fac*xi[i]*xi[j]
					                -fad*hdg[i]*hdg[j]+fae*dg[i]*dg[j];
				}
			}
		}
		for (i=1;i<=n;i++) {
			xi[i]=0.0;
			for (j=1;j<=n;j++) xi[i] -= hessin[i][j]*g[j];
		}
		//checkBound(p, xi, lower, upper, n);
		//checkDirection(p, xi);
		//if (*iter > 200) cout << "iteration=" << *iter << endl;
	}
	nrerror("too many iterations in dfpmin");
	FREEALL
}
#undef ITMAX
#undef SQR
#undef EPS
#undef TOLX
#undef STPMX
#undef FREEALL
#undef FMAX


/**
	the approximated derivative function
	@param x the input vector x
	@param dfx the derivative at x
	@return the function value at x
*/
double Optimization::derivativeFunk(double x[], double dfx[]) {
	/*
	if (!checkRange(x))
		return INFINITIVE;
	*/
	double fx = targetFunk(x);
	int ndim = getNDim();
	double h, temp;
	for (int dim = 1; dim <= ndim; dim++ ){
		temp = x[dim];
		h = ERROR_X * fabs(temp);
		if (h == 0.0) h = ERROR_X;
		x[dim] = temp + h;
		h = x[dim] - temp;
		dfx[dim] = (targetFunk(x) - fx) / h;
		x[dim] = temp;
	}
	return fx;
}


/*#define NRANSI
#define ITMAX 100
#define CGOLD 0.3819660
#define ZEPS 1.0e-10
#define SHFT(a,b,c,d) (a)=(b);(b)=(c);(c)=(d);
#define SIGN(a,b) ((b) >= 0.0 ? fabs(a) : -fabs(a))

double Optimization::brent(double ax, double bx, double cx, double tol,
	double *xmin)
{
	int iter;
	double a,b,d=0.0,etemp,fu,fv,fw,fx,p,q,r,tol1,tol2,u,v,w,x,xm;
	double e=0.0;

	a=(ax < cx ? ax : cx);
	b=(ax > cx ? ax : cx);
	x=w=v=bx;
	fw=fv=fx=computeFunction(x);
	for (iter=1;iter<=ITMAX;iter++) {
		xm=0.5*(a+b);
		tol2=2.0*(tol1=tol*fabs(x)+ZEPS);
		if (fabs(x-xm) <= (tol2-0.5*(b-a))) {
			*xmin=x;
			return fx;
		}
		if (fabs(e) > tol1) {
			r=(x-w)*(fx-fv);
			q=(x-v)*(fx-fw);
			p=(x-v)*q-(x-w)*r;
			q=2.0*(q-r);
			if (q > 0.0) p = -p;
			q=fabs(q);
			etemp=e;
			e=d;
			if (fabs(p) >= fabs(0.5*q*etemp) || p <= q*(a-x) || p >= q*(b-x))
				d=CGOLD*(e=(x >= xm ? a-x : b-x));
			else {
				d=p/q;
				u=x+d;
				if (u-a < tol2 || b-u < tol2)
					d=SIGN(tol1,xm-x);
			}
		} else {
			d=CGOLD*(e=(x >= xm ? a-x : b-x));
		}
		u=(fabs(d) >= tol1 ? x+d : x+SIGN(tol1,d));
		fu=computeFunction(u);
		if (fu <= fx) {
			if (u >= x) a=x; else b=x;
			SHFT(v,w,x,u)
			SHFT(fv,fw,fx,fu)
		} else {
			if (u < x) a=u; else b=u;
			if (fu <= fw || w == x) {
				v=w;
				w=u;
				fv=fw;
				fw=fu;
			} else if (fu <= fv || v == x || v == w) {
				v=u;
				fv=fu;
			}
		}
	}
	nrerror("Too many iterations in brent");
	*xmin=x;
	return fx;
}

#undef SIGN
#undef ITMAX
#undef CGOLD
#undef ZEPS
#undef SHFT
#undef NRANSI*/

/*#define JMAX 20

double Optimization::minimizeNewton(double xmin, double xguess, double xmax, double tolerance, double &f)
{
	return rtsafe(xmin, xguess, xmax, tolerance, f);
	//double fe;
	//return minimizeOneDimen(xmin, rtn, xmax, tolerance, &f, &fe);

	int j;
	double df,ddf,dx,rtn,rtnold, fstart=0, fnew;

	rtn=xguess;
	if (rtn < xmin) rtn = xmin;
	if (rtn > xmax) rtn = xmax;
	

	for (j=1;j<=JMAX;j++) {
		f = computeFuncDerv(rtn,df,ddf);
		if (!isfinite(f)) 
			return 0;
		if (j == 1) fstart = f;
		if (ddf == 0.0) break;
		dx=(df/fabs(ddf));
		if (fabs(dx) <= tolerance) break;

		rtnold = rtn; rtn = rtn-dx;
		if (rtn < xmin) rtn = xmin;
		if (rtn > xmax) rtn = xmax;
		dx = rtnold-rtn;

		while (fabs(dx) > tolerance && (fnew = computeFunction(rtn)) > f + tolerance) {
			dx /= 2;
			rtn = rtnold - dx;
		}
		if (fabs(dx) <= tolerance) { rtn = rtnold; break; }
	}
	//if (j > JMAX)
		//nrerror("Maximum number of iterations exceeded in Newton-Raphson");
	if (f <= fstart && j <= JMAX && (j > 1 || xguess > xmin+tolerance)) 
		return rtn;
	// Newton does not work, turn to other method
	double fe;
	return minimizeOneDimen(xmin, xguess, xmax, tolerance, &f, &fe);
}*/

/*
double Optimization::minimizeNewton(double xmin, double xguess, double xmax, double tolerance, double &f)
{
	int j;
	double df,ddf,dx,dxold,rtn,temp, fold, fstart;
	double xmin_orig = xmin, xmax_orig = xmax;

	rtn=xguess;
	dx=dxold=(xmax-xmin);
	fstart = fold = f = computeFuncDerv(rtn,df,ddf);

	for (j=1;j<=JMAX;j++) {
		if (ddf <= 0.0) break;
		if ((((rtn-xmax)*ddf-df) * ((rtn-xmin)*ddf-df) > 0) || // run out of range
			(fabs(2.0*df) > fabs(dxold*ddf)) // dx not decreasing fast enough
			) // f even increase
		{
			dxold = dx;
			dx = 0.5*(xmax-xmin);
			rtn = xmin+dx;
			if (xmin == rtn) break;
		} else {
			dxold=dx;
			if (ddf == 0.0)
				nrerror("2nd derivative is zero");
			dx=df/ddf;
			temp=rtn;
			//if (f > fold) dx /= 2.0;
			//if (ddf < 0) dx = -dx;
			rtn -= dx;
			//if (rtn < xmin) rtn = xmin;
			//if (rtn > xmax) rtn = xmax;
			dx = temp - rtn;
			if (temp == rtn) break;
		}
		if (fabs(dx) < tolerance) break;
		fold = f;
		f = computeFuncDerv(rtn,df,ddf);
		//if (f > fold) break; // Does not decrease function, escape
		if (df < 0.0) 
			xmin = rtn;
		else
			xmax = rtn;
	}
	if (j > JMAX)
	nrerror("Maximum number of iterations exceeded in Newton-Raphson");
	if (f <= fstart) return rtn;
	// Newton does not work (find a max instead of min), turn to other method
	double fe;
	return minimizeOneDimen(xmin_orig, xguess, xmax_orig, tolerance, &f, &fe);
	//return 0.0;
}
#undef JMAX*/