/home66/gary/public_html/cloudy/c08_branch/source/optimize_subplx.cpp

Go to the documentation of this file.

/* This file is part of Cloudy and is copyright (C)1978-2008 by Gary J. Ferland and
 * others.  For conditions of distribution and use see copyright notice in license.txt */
#include "cddefines.h"
#include "optimize.h"

STATIC void calcc(long,realnum*,long,long,int,realnum[]);
STATIC double cdasum(long,realnum[],long);
STATIC void cdaxpy(long,double,realnum[],long,realnum[],long);
STATIC void cdcopy(long,realnum[],long,realnum[],long);
STATIC void csscal(long,double,realnum[],long);
STATIC double dist(long,realnum[],realnum[]);
STATIC void evalf(long,long[],realnum[],long,realnum[],realnum*,long*);
STATIC void fstats(double,long,int);
STATIC void newpt(long,double,realnum[],realnum[],int,realnum[],int*);
STATIC void order(long,realnum[],long*,long*,long*);
STATIC void partx(long,long[],realnum[],long*,long[]);
STATIC void setstp(long,long,realnum[],realnum[]);
STATIC void simplx(long,realnum[],long,long[],long,int,realnum[],
        realnum*,long*,realnum*,realnum[],long*);
STATIC void sortd(long,realnum[],long[]);
STATIC void start(long,realnum[],realnum[],long,long[],realnum*,int*);
STATIC void subopt(long);

/*lint -e801 goto is deprecated */
/*lint -e64 type mismatch */
/*********************************************************************
 *
 *     NB this is much the original coding and does not use strong typing
 *gjf mod: to avoid conflict with exemplar parallel version of lapack,
 *dasum changed to cdasum
 *daxpy changed to cdaxpy
 *dcopy changed to cdcopy
 *sscal changed to csscal
 *
 *********************************************************************** */
struct t_usubc {
        realnum alpha, 
          beta, 
          gamma, 
          delta, 
          psi, 
          omega;
        long int nsmin, 
          nsmax, 
          irepl, 
          ifxsw;
        realnum bonus, 
          fstop;
        long int nfstop, 
          nfxe;
        realnum fxstat[4], 
          ftest;
        int minf, 
          initx, 
          newx;
        }       usubc;
struct t_isubc {
        realnum fbonus, 
          sfstop, 
          sfbest;
        int IntNew;
        }       isubc;

void optimize_subplex(long int n, 
          double tol, 
          long int maxnfe, 
          long int mode, 
          realnum scale[], 
          realnum x[], 
          realnum *fx, 
          long int *nfe, 
          realnum work[], 
          long int iwork[], 
          long int *iflag)
{
        static int cmode;
        static long int i, 
          i_, 
          ifsptr, 
          ins, 
          insfnl, 
          insptr, 
          ipptr, 
          isptr,
          istep, 
          istptr, 
          nnn, 
          ns, 
          nsubs;
        static realnum dum,
          scl[1], 
          sfx, 
          xpscl, 
          xstop,
          xstop2;

        static const realnum bnsfac[2][3] = { {-1.0f,-2.0f,0.0f}, {1.0f,0.0f,2.0f} };

        DEBUG_ENTRY( "optimize_subplex()" );

        /*                                         Coded by Tom Rowan
         *                            Department of Computer Sciences
         *                              University of Texas at Austin
         *
         * Jason Ferguson:
         * Changed on 8/4/94, in order to incorparate into cloudy
         * changes made are: double precision to real,
         * a 2-D function f(n,x) into 1-D f(x),
         * and the termination check.
         *
         * optimize_subplex uses the subplex method to solve unconstrained
         * optimization problems.  The method is well suited for
         * optimizing objective functions that are noisy or are
         * discontinuous at the solution.
         *
         * optimize_subplex sets default optimization options by calling the
         * subroutine subopt.  The user can override these defaults
         * by calling subopt prior to calling optimize_subplex, changing the
         * appropriate common variables, and setting the value of
         * mode as indicated below.
         *
         * By default, optimize_subplex performs minimization.
         *
         * input
         *
         *   ffff   - user supplied function f(n,x) to be optimized,
         *            declared external in calling routine
         *   always uses optimize_func - change 97 dec 8
         *
         *   n      - problem dimension
         *
         *   tol    - relative error tolerance for x (tol .ge. 0.)
         *
         *   maxnfe - maximum number of function evaluations
         *
         *   mode   - integer mode switch with binary expansion
         *            (bit 1) (bit 0) :
         *            bit 0 = 0 : first call to optimize_subplex
         *                  = 1 : continuation of previous call
         *            bit 1 = 0 : use default options
         *                  = 1 : user set options
         *
         *   scale  - scale and initial stepsizes for corresponding
         *            components of x
         *            (If scale(1) .lt. 0.,
         *            ABS(scale(1)) is used for all components of x,
         *            and scale(2),...,scale(n) are not referenced.)
         *
         *   x      - starting guess for optimum
         *
         *   work   - real work array of dimension .ge.
         *            2*n + nsmax*(nsmax+4) + 1
         *            (nsmax is set in subroutine subopt.
         *            default: nsmax = MIN(5,n))
         *
         *   iwork  - integer work array of dimension .ge.
         *            n + INT(n/nsmin)
         *            (nsmin is set in subroutine subopt.
         *            default: nsmin = MIN(2,n))
         *
         * output
         *
         *   x      - computed optimum
         *
         *   fx     - value of f at x
         *
         *   nfe    - number of function evaluations
         *
         *   iflag  - error flag
         *            = -2 : invalid input
         *            = -1 : maxnfe exceeded
         *            =  0 : tol satisfied
         *            =  1 : limit of machine precision
         *            =  2 : fstop reached (fstop usage is determined
         *                   by values of options minf, nfstop, and
         *                   irepl. default: f(x) not tested against
         *                   fstop)
         *            iflag should not be reset between calls to
         *            optimize_subplex.
         *
         * common
         * */

        if( (mode%2) == 0 )
        {

                /*       first call, check input
                 * */
                if( n < 1 )
                        goto L_120;
                if( tol < 0.0f )
                        goto L_120;
                if( maxnfe < 1 )
                        goto L_120;
                if( scale[0] >= 0.0f )
                {
                        for( i=1; i <= n; i++ )
                        {
                                i_ = i - 1;
                                xpscl = x[i_] + scale[i_];
                                if( fp_equal( xpscl, x[i_] ) )
                                        goto L_120;
                        }
                }
                else
                {
                        scl[0] = (realnum)fabs(scale[0]);
                        for( i=1; i <= n; i++ )
                        {
                                i_ = i - 1;
                                xpscl = x[i_] + scl[0];
                                if( fp_equal( xpscl, x[i_] ) )
                                        goto L_120;
                        }
                }
                if( ((mode/2)%2) == 0 )
                {
                        subopt(n);
                }
                else if( usubc.alpha <= 0.0f )
                {
                        goto L_120;
                }
                else if( usubc.beta <= 0.0f || usubc.beta >= 1.0f )
                {
                        goto L_120;
                }
                else if( usubc.gamma <= 1.0f )
                {
                        goto L_120;
                }
                else if( usubc.delta <= 0.0f || usubc.delta >= 1.0f )
                {
                        goto L_120;
                }
                else if( usubc.psi <= 0.0f || usubc.psi >= 1.0f )
                {
                        goto L_120;
                }
                else if( usubc.omega <= 0.0f || usubc.omega >= 1.0f )
                {
                        goto L_120;
                }
                else if( (usubc.nsmin < 1 || usubc.nsmax < usubc.nsmin) || 
                  n < usubc.nsmax )
                {
                        goto L_120;
                }
                else if( n < ((n - 1)/usubc.nsmax + 1)*usubc.nsmin )
                {
                        goto L_120;
                }
                else if( usubc.irepl < 0 || usubc.irepl > 2 )
                {
                        goto L_120;
                }
                else if( usubc.ifxsw < 1 || usubc.ifxsw > 3 )
                {
                        goto L_120;
                }
                else if( usubc.bonus < 0.0f )
                {
                        goto L_120;
                }
                else if( usubc.nfstop < 0 )
                {
                        goto L_120;
                }

                /*       initialization
                 * */
                istptr = n + 1;
                isptr = istptr + n;
                ifsptr = isptr + usubc.nsmax*(usubc.nsmax + 3);
                insptr = n + 1;
                if( scale[0] > 0.0f )
                {
                        cdcopy(n,scale,1,work,1);
                        cdcopy(n,scale,1,&work[istptr-1],1);
                }
                else
                {
                        cdcopy(n,scl,0,work,1);
                        cdcopy(n,scl,0,&work[istptr-1],1);
                }
                for( i=1; i <= n; i++ )
                {
                        i_ = i - 1;
                        iwork[i_] = i;
                }
                *nfe = 0;
                usubc.nfxe = 1;
                if( usubc.irepl == 0 )
                {
                        isubc.fbonus = 0.0f;
                }
                else if( usubc.minf )
                {
                        isubc.fbonus = bnsfac[0][usubc.ifxsw-1]*usubc.bonus;
                }
                else
                {
                        isubc.fbonus = bnsfac[1][usubc.ifxsw-1]*usubc.bonus;
                }
                if( usubc.nfstop == 0 )
                {
                        isubc.sfstop = 0.0f;
                }
                else if( usubc.minf )
                {
                        isubc.sfstop = usubc.fstop;
                }
                else
                {
                        isubc.sfstop = -usubc.fstop;
                }
                usubc.ftest = 0.0f;
                cmode = false;
                isubc.IntNew = true;
                usubc.initx = true;
                nnn = 0;
                evalf(nnn,iwork,(realnum*)&dum,n,x,&sfx,nfe);
                usubc.initx = false;

                /*       continuation of previous call
                 * */
        }
        else if( *iflag == 2 )
        {
                if( usubc.minf )
                {
                        isubc.sfstop = usubc.fstop;
                }
                else
                {
                        isubc.sfstop = -usubc.fstop;
                }
                cmode = true;
                goto L_70;
        }
        else if( *iflag == -1 )
        {
                cmode = true;
                goto L_70;
        }
        else if( *iflag == 0 )
        {
                cmode = false;
                goto L_90;
        }
        else
        {
                return;
        }

        /*     subplex loop
         * */
L_40:

        for( i=0; i < n; i++ )
        {
                work[i] = (realnum)fabs(work[i]);
        }

        sortd(n,work,iwork);
        partx(n,iwork,work,&nsubs,&iwork[insptr-1]);
        cdcopy(n,x,1,work,1);
        ins = insptr;
        insfnl = insptr + nsubs - 1;
        ipptr = 1;

        /*       simplex loop
         * */

L_60:
        ns = iwork[ins-1];

L_70:
        simplx(n,&work[istptr-1],ns,&iwork[ipptr-1],maxnfe,cmode,x,&sfx,
          nfe,&work[isptr-1],&work[ifsptr-1],iflag);

        cmode = false;
        if( *iflag != 0 )
                goto L_121;

        if( ins < insfnl )
        {
                ins += 1;
                ipptr += ns;
                goto L_60;
        }

        /*       end simplex loop
         * */
        for( i=0; i < n; i++ )
        {
                work[i] = x[i] - work[i];
        }

        /*       check termination
         * */
L_90:
        /* new stop criterion: calculate distance between
         * previous and new best model, if distance is
         * smaller than tol return. */
        istep = istptr-1;
        xstop = 0.f;
        xstop2 = 0.f;
        for( i=0; i < n; i++ )
        {
                realnum ttemp;
                xstop += POW2(work[i]);
                ttemp = (realnum)fabs(work[istep]);
                /* chng to avoid side effects 
                 * xstop2 = MAX2(xstop2,(realnum)fabs(work[istep]));*/
                xstop2 = MAX2( xstop2 , ttemp );
                istep++;
        }

        if( sqrt(xstop) > tol || xstop2*usubc.psi > (realnum)tol )
        {
                setstp(nsubs,n,work,&work[istptr-1]);
                goto L_40;
        }

        /*     end subplex loop
         * */

        *iflag = 0;
L_121:
        if( usubc.minf )
        {
                *fx = sfx;
        }
        else
        {
                *fx = -sfx;
        }
        return;

        /*     invalid input
         * */
L_120:

        *iflag = -2;
        return;
}
/********************************************************************* */
STATIC void subopt(long int n)
{

        DEBUG_ENTRY( "subopt()" );


        /*                                         Coded by Tom Rowan
         *                            Department of Computer Sciences
         *                              University of Texas at Austin
         *
         * subopt sets options for optimize_subplex.
         *
         * input
         *
         *   n      - problem dimension
         *
         * common
         * */



        /* subroutines and functions
         *
         *   fortran
         *
         *-----------------------------------------------------------
         *
         ************************************************************
         * simplex method strategy parameters
         ************************************************************
         *
         * alpha  - reflection coefficient
         *          alpha .gt. 0
         * */
        usubc.alpha = 1.0f;

        /* beta   - contraction coefficient
         *          0 .lt. beta .lt. 1
         * */
        usubc.beta = .50f;

        /* gamma  - expansion coefficient
         *          gamma .gt. 1
         * */
        usubc.gamma = 2.0f;

        /* delta  - shrinkage (massive contraction) coefficient
         *          0 .lt. delta .lt. 1
         * */
        usubc.delta = .50f;

        /************************************************************
         * subplex method strategy parameters
         ************************************************************
         *
         * psi    - simplex reduction coefficient
         *          0 .lt. psi .lt. 1
         * */
        usubc.psi = .250f;

        /* omega  - step reduction coefficient
         *          0 .lt. omega .lt. 1
         * */
        usubc.omega = .10f;

        /* nsmin and nsmax specify a range of subspace dimensions.
         * In addition to satisfying  1 .le. nsmin .le. nsmax .le. n,
         * nsmin and nsmax must be chosen so that n can be expressed
         * as a sum of positive integers where each of these integers
         * ns(i) satisfies   nsmin .le. ns(i) .ge. nsmax.
         * Specifically,
         *     nsmin*ceil(n/nsmax) .le. n   must be true.
         *
         * nsmin  - subspace dimension minimum
         * */
        usubc.nsmin = MIN2(2,n);

        /* nsmax  - subspace dimension maximum
         * */
        usubc.nsmax = MIN2(5,n);

        /************************************************************
         * subplex method special cases
         ************************************************************
         * nelder-mead simplex method with periodic restarts
         *   nsmin = nsmax = n
         ************************************************************
         * nelder-mead simplex method
         *   nsmin = nsmax = n, psi = small positive
         ************************************************************
         *
         * irepl, ifxsw, and bonus deal with measurement replication.
         * Objective functions subject to large amounts of noise can
         * cause an optimization method to halt at a false optimum.
         * An expensive solution to this problem is to evaluate f
         * several times at each point and return the average (or max
         * or min) of these trials as the function value.  optimize_subplex
         * performs measurement replication only at the current best
         * point. The longer a point is retained as best, the more
         * accurate its function value becomes.
         *
         * The common variable nfxe contains the number of function
         * evaluations at the current best point. fxstat contains the
         * mean, max, min, and standard deviation of these trials.
         *
         * irepl  - measurement replication switch
         * irepl  = 0, 1, or 2
         *        = 0 : no measurement replication
         *        = 1 : optimize_subplex performs measurement replication
         *        = 2 : user performs measurement replication
         *              (This is useful when optimizing on the mean,
         *              max, or min of trials is insufficient. Common
         *              variable initx is true for first function
         *              evaluation. newx is true for first trial at
         *              this point. The user uses subroutine fstats
         *              within his objective function to maintain
         *              fxstat. By monitoring newx, the user can tell
         *              whether to return the function evaluation
         *              (newx = .true.) or to use the new function
         *              evaluation to refine the function evaluation
         *              of the current best point (newx = .false.).
         *              The common variable ftest gives the function
         *              value that a new point must beat to be
         *              considered the new best point.)
         * */
        usubc.irepl = 0;

        /* ifxsw  - measurement replication optimization switch
         * ifxsw  = 1, 2, or 3
         *        = 1 : retain mean of trials as best function value
         *        = 2 : retain max
         *        = 3 : retain min
         * */
        usubc.ifxsw = 1;

        /* Since the current best point will also be the most
         * accurately evaluated point whenever irepl .gt. 0, a bonus
         * should be added to the function value of the best point
         * so that the best point is not replaced by a new point
         * that only appears better because of noise.
         * optimize_subplex uses bonus to determine how many multiples of
         * fxstat(4) should be added as a bonus to the function
         * evaluation. (The bonus is adjusted automatically by
         * optimize_subplex when ifxsw or minf is changed.)
         *
         * bonus  - measurement replication bonus coefficient
         *          bonus .ge. 0 (normally, bonus = 0 or 1)
         *        = 0 : bonus not used
         *        = 1 : bonus used
         * */
        usubc.bonus = 1.0f;

        /* nfstop = 0 : f(x) is not tested against fstop
         *        = 1 : if f(x) has reached fstop, optimize_subplex returns
         *              iflag = 2
         *        = 2 : (only valid when irepl .gt. 0)
         *              if f(x) has reached fstop and
         *              nfxe .gt. nfstop, optimize_subplex returns iflag = 2
         * */
        usubc.nfstop = 0;

        /* fstop  - f target value
         *          Its usage is determined by the value of nfstop.
         *
         * minf   - logical switch
         *        = .true.  : optimize_subplex performs minimization
         *        = .false. : optimize_subplex performs maximization
         * */
        usubc.minf = true;
        return;
}
/**********************************************************************/
/* >>chng 01 jan 03, cleaned up -1 and formatting in this routine */
STATIC void cdcopy(long int n, 
          realnum dx[], 
          long int incx, 
          realnum dy[], 
          long int incy)
{
        long int i, 
          ix, 
          iy, 
          m;

        DEBUG_ENTRY( "cdcopy()" );

        /* 
         * copies a vector, x, to a vector, y.
         * uses unrolled loops for increments equal to one.
         * Jack Dongarra, lapack, 3/11/78.
         */

        if( n > 0 )
        {
                if( incx == 1 && incy == 1 )
                {

                        /* code for both increments equal to 1 */

                        /* first the clean-up loop */
                        m = n%7;
                        if( m != 0 )
                        {
                                for( i=0; i < m; i++ )
                                {
                                        dy[i] = dx[i];
                                }
                                if( n < 7 )
                                { 
                                        return;
                                }
                        }

                        for( i=m; i < n; i += 7 )
                        {
                                dy[i] = dx[i];
                                dy[i+1] = dx[i+1];
                                dy[i+2] = dx[i+2];
                                dy[i+3] = dx[i+3];
                                dy[i+4] = dx[i+4];
                                dy[i+5] = dx[i+5];
                                dy[i+6] = dx[i+6];
                        }
                }
                else
                {

                        /* code for unequal increments or equal increments
                         * not equal to 1 */

                        ix = 1;
                        iy = 1;
                        if( incx < 0 )
                                ix = (-n + 1)*incx + 1;
                        if( incy < 0 )
                                iy = (-n + 1)*incy + 1;
                        for( i=0; i < n; i++ )
                        {
                                dy[iy-1] = dx[ix-1];
                                ix += incx;
                                iy += incy;
                        }
                }
        }
        return;
}
/********************************************************************* */
STATIC void evalf(long int ns, 
          long int ips[], 
          realnum xs[], 
          long int n, 
          realnum x[], 
          realnum *sfx, 
          long int *nfe)
{
        static int newbst;
        static long int i, 
          i_;
        static realnum fx;
        /* gary delete since in header */
        /*double optimize_func();*/

        DEBUG_ENTRY( "evalf()" );
        /* gary add, not used, so trick compiler notes */
        i_ = n;

        /*>>chng 97 dec 07, implicit nonte, rid of first function argument
         *
         *                                         Coded by Tom Rowan
         *                            Department of Computer Sciences
         *                              University of Texas at Austin
         *
         * evalf evaluates the function f at a point defined by x
         * with ns of its components replaced by those in xs.
         *
         * input
         *
         *   f      - user supplied function f(n,x) to be optimized
         *   changed to optimize_func - cannot specify arbitrary function now
         *
         *   ns     - subspace dimension
         *
         *   ips    - permutation vector
         *
         *   xs     - real ns-vector to be mapped to x
         *
         *   n      - problem dimension
         *
         *   x      - real n-vector
         *
         *   nfe    - number of function evaluations
         *
         * output
         *
         *   sfx    - signed value of f evaluated at x
         *
         *   nfe    - incremented number of function evaluations
         *
         * common
         * */




        /* local variables
         * */


        /* subroutines and functions
         * */

        /*-----------------------------------------------------------
         * */
        for( i=1; i <= ns; i++ )
        {
                i_ = i - 1;
                x[ips[i_]-1] = xs[i_];
        }
        usubc.newx = isubc.IntNew || usubc.irepl != 2;
        fx = (realnum)optimize_func(x);
        /*      fx = f(n,x) */
        if( usubc.irepl == 0 )
        {
                if( usubc.minf )
                {
                        *sfx = fx;
                }
                else
                {
                        *sfx = -fx;
                }
        }
        else if( isubc.IntNew )
        {
                if( usubc.minf )
                {
                        *sfx = fx;
                        newbst = fx < usubc.ftest;
                }
                else
                {
                        *sfx = -fx;
                        newbst = fx > usubc.ftest;
                }
                if( usubc.initx || newbst )
                {
                        if( usubc.irepl == 1 )
                                fstats(fx,1,true);
                        usubc.ftest = fx;
                        isubc.sfbest = *sfx;
                }
        }
        else
        {
                if( usubc.irepl == 1 )
                {
                        fstats(fx,1,false);
                        fx = usubc.fxstat[usubc.ifxsw-1];
                }
                usubc.ftest = fx + isubc.fbonus*usubc.fxstat[3];
                if( usubc.minf )
                {
                        *sfx = usubc.ftest;
                        isubc.sfbest = fx;
                }
                else
                {
                        *sfx = -usubc.ftest;
                        isubc.sfbest = -fx;
                }
        }
        *nfe += 1;
        return;
}

/********************************************************************* */
STATIC void setstp(long int nsubs, 
          long int n, 
          realnum deltax[], 
          realnum step[])
{
        static long int i, 
          i_;
        static realnum stpfac;

        DEBUG_ENTRY( "setstp()" );


        /*                                         Coded by Tom Rowan
         *                            Department of Computer Sciences
         *                              University of Texas at Austin
         *
         * setstp sets the stepsizes for the corresponding components
         * of the solution vector.
         *
         * input
         *
         *   nsubs  - number of subspaces
         *
         *   n      - number of components (problem dimension)
         *
         *   deltax - vector of change in solution vector
         *
         *   step   - stepsizes for corresponding components of
         *            solution vector
         *
         * output
         *
         *   step   - new stepsizes
         *
         * common
         * */


        /* local variables
         * */


        /* subroutines and functions
         *
         *   blas */
        /*   fortran
         *
         *-----------------------------------------------------------
         *
         *     set new step
         * */
        if( nsubs > 1 )
        {       
                double a , b , c;
                a = cdasum(n,deltax,1);
                b = cdasum(n,step,1);
                c = MAX2(a/b ,usubc.omega);
                stpfac = (realnum)MIN2(c , 1.0f/usubc.omega);
        }
        else
        {
                stpfac = usubc.psi;
        }
        csscal(n,stpfac,step,1);

        /*     reorient simplex
         * */
        for( i=1; i <= n; i++ )
        {
                i_ = i - 1;
                if( deltax[i_] == 0.f )
                {
                        step[i_] = -step[i_];
                }
                else
                {
                        step[i_] = (realnum)sign(step[i_],deltax[i_]);
                }
        }
        return;
}

/********************************************************************* */
STATIC void sortd(long int n, 
          realnum xkey[], 
          long int ix[])
{
        long int i, 
          i_, 
          ifirst, 
          ilast, 
          iswap, 
          ixi, 
          ixip1;

        DEBUG_ENTRY( "sortd()" );


        /*                                         Coded by Tom Rowan
         *                            Department of Computer Sciences
         *                              University of Texas at Austin
         *
         * sortd uses the shakersort method to sort an array of keys
         * in decreasing order. The sort is performed implicitly by
         * modifying a vector of indices.
         *
         * For nearly sorted arrays, sortd requires O(n) comparisons.
         * for completely unsorted arrays, sortd requires O(n**2)
         * comparisons and will be inefficient unless n is small.
         *
         * input
         *
         *   n      - number of components
         *
         *   xkey   - real vector of keys
         *
         *   ix     - integer vector of indices
         *
         * output
         *
         *   ix     - indices satisfy xkey(ix(i)) .ge. xkey(ix(i+1))
         *            for i = 1,...,n-1
         *
         * local variables
         * */

        /*-----------------------------------------------------------
         * */
        ifirst = 1;
        iswap = 1;
        ilast = n - 1;
        while( ifirst <= ilast )
        {
                for( i=ifirst; i <= ilast; i++ )
                {
                        i_ = i - 1;
                        ixi = ix[i_];
                        ixip1 = ix[i_+1];
                        if( xkey[ixi-1] < xkey[ixip1-1] )
                        {
                                ix[i_] = ixip1;
                                ix[i_+1] = ixi;
                                iswap = i;
                        }
                }
                ilast = iswap - 1;
                for( i=ilast; i >= ifirst; i-- )
                {
                        i_ = i - 1;
                        ixi = ix[i_];
                        ixip1 = ix[i_+1];
                        if( xkey[ixi-1] < xkey[ixip1-1] )
                        {
                                ix[i_] = ixip1;
                                ix[i_+1] = ixi;
                                iswap = i;
                        }
                }
                ifirst = iswap + 1;
        }
        return;
}

/********************************************************************* */
STATIC void partx(long int n, 
          long int ip[], 
          realnum absdx[], 
          long int *nsubs, 
          long int nsvals[])
{
        static long int i, 
          limit, 
          nleft, 
          ns1, 
          ns1_, 
          ns2, 
          nused;
        static realnum as1, 
          as1max, 
          as2, 
          asleft, 
          gap, 
          gapmax;

        DEBUG_ENTRY( "partx()" );


        /*                                         Coded by Tom Rowan
         *                            Department of Computer Sciences
         *                              University of Texas at Austin
         *
         * partx partitions the vector x by grouping components of
         * similar magnitude of change.
         *
         * input
         *
         *   n      - number of components (problem dimension)
         *
         *   ip     - permutation vector
         *
         *   absdx  - vector of magnitude of change in x
         *
         *   nsvals - integer array dimensioned .ge. INT(n/nsmin)
         *
         * output
         *
         *   nsubs  - number of subspaces
         *
         *   nsvals - integer array of subspace dimensions
         *
         * common
         * */


        /* local variables
         * */


        /* subroutines and functions
         *
         *
         *-----------------------------------------------------------
         * */
        *nsubs = 0;
        nused = 0;
        nleft = n;
        asleft = absdx[0];
        for( i=1; i < n; i++ )
        {
                asleft += absdx[i];
        }

        while( nused < n )
        {
                *nsubs += 1;
                as1 = 0.0f;
                for( i=0; i < (usubc.nsmin - 1); i++ )
                {
                        as1 += absdx[ip[nused+i]-1];
                }

                gapmax = -1.0f;
                limit = MIN2(usubc.nsmax,nleft);
                for( ns1=usubc.nsmin; ns1 <= limit; ns1++ )
                {
                        ns1_ = ns1 - 1;
                        as1 += absdx[ip[nused+ns1_]-1];
                        ns2 = nleft - ns1;
                        if( ns2 > 0 )
                        {
                                if( ns2 >= ((ns2 - 1)/usubc.nsmax + 1)*usubc.nsmin )
                                {
                                        as2 = asleft - as1;
                                        gap = as1/ns1 - as2/ns2;
                                        if( gap > gapmax )
                                        {
                                                gapmax = gap;
                                                nsvals[*nsubs-1] = ns1;
                                                as1max = as1;
                                        }
                                }
                        }
                        else if( as1/ns1 > gapmax )
                        {
                                goto L_21;
                        }
                }
                nused += nsvals[*nsubs-1];
                nleft = n - nused;
                asleft -= as1max;
        }
        return;
L_21:
        nsvals[*nsubs-1] = ns1;
        return;
}

/********************************************************************* */

#undef S
#define S(I_,J_)        (*(s+(I_)*(ns)+(J_)))

STATIC void simplx(long int n, 
          realnum step[], 
          long int ns, 
          long int ips[], 
          long int maxnfe, 
          int cmode, 
          realnum x[], 
          realnum *fx, 
          long int *nfe, 
          realnum *s, 
          realnum fs[], 
          long int *iflag)
{
        static int small, 
          updatc;

        static long int i, 
          icent, 
          ih, 
          il, 
          inew = 0, 
          is, 
          itemp, 
          j, 
          j_, 
          npts;

        static realnum dum, 
          fc, 
          fe, 
          fr, 
          tol;

        DEBUG_ENTRY( "simplx()" );


        /*                                         Coded by Tom Rowan
         *                            Department of Computer Sciences
         *                              University of Texas at Austin
         *
         * simplx uses the Nelder-Mead simplex method to minimize the
         * function f on a subspace.
         *
         * input
         *
         *   ffff   - function to be minimized, declared external in
         *            calling routine
         *   removed - always calls optimize_func
         *
         *   n      - problem dimension
         *
         *   step   - stepsizes for corresponding components of x
         *
         *   ns     - subspace dimension
         *
         *   ips    - permutation vector
         *
         *   maxnfe - maximum number of function evaluations
         *
         *   cmode  - logical switch
         *            = .true.  : continuation of previous call
         *            = .false. : first call
         *
         *   x      - starting guess for minimum
         *
         *   fx     - value of f at x
         *
         *   nfe    - number of function evaluations
         *
         *   s      - real work array of dimension .ge.
         *            ns*(ns+3) used to store simplex
         *
         *   fs     - real work array of dimension .ge.
         *            ns+1 used to store function values of simplex
         *            vertices
         *
         * output
         *
         *   x      - computed minimum
         *
         *   fx     - value of f at x
         *
         *   nfe    - incremented number of function evaluations
         *
         *   iflag  - error flag
         *            = -1 : maxnfe exceeded
         *            =  0 : simplex reduced by factor of psi
         *            =  1 : limit of machine precision
         *            =  2 : reached fstop
         *
         * common
         * */




        /* local variables
         * */


        /* subroutines and functions
         *
         *     external optimize_func,calcc,dist,evalf,newpt,order,start */
        /*   blas */

        /*-----------------------------------------------------------
         * */
        if( cmode )
                goto L_50;
        npts = ns + 1;
        icent = ns + 2;
        itemp = ns + 3;
        updatc = false;
        start(n,x,step,ns,ips,s,&small);
        if( small )
        {
                *iflag = 1;
                return;
        }
        else
        {
                if( usubc.irepl > 0 )
                {
                        isubc.IntNew = false;
                        evalf(ns,ips,&S(0,0),n,x,&fs[0],nfe);
                }
                else
                {
                        fs[0] = *fx;
                }
                isubc.IntNew = true;
                for( j=2; j <= npts; j++ )
                {
                        j_ = j - 1;
                        evalf(ns,ips,&S(j_,0),n,x,&fs[j_],nfe);
                }
                il = 1;
                order(npts,fs,&il,&is,&ih);
                tol = (realnum)(usubc.psi*dist(ns,&S(ih-1,0),&S(il-1,0)));
        }

        /*     main loop
         * */
L_20:
        calcc(ns,s,ih,inew,updatc,&S(icent-1,0));
        updatc = true;
        inew = ih;

        /*       reflect
         * */
        newpt(ns,usubc.alpha,&S(icent-1,0),&S(ih-1,0),true,&S(itemp-1,0),
          &small);
        if( !small )
        {
                evalf(ns,ips,&S(itemp-1,0),n,x,&fr,nfe);
                if( fr < fs[il-1] )
                {

                        /*         expand
                         * */
                        newpt(ns,-usubc.gamma,&S(icent-1,0),&S(itemp-1,0),true,
                          &S(ih-1,0),&small);
                        if( small )
                                goto L_40;
                        evalf(ns,ips,&S(ih-1,0),n,x,&fe,nfe);
                        if( fe < fr )
                        {
                                fs[ih-1] = fe;
                        }
                        else
                        {
                                cdcopy(ns,&S(itemp-1,0),1,&S(ih-1,0),1);
                                fs[ih-1] = fr;
                        }
                }
                else if( fr < fs[is-1] )
                {

                        /*         accept reflected point
                         * */
                        cdcopy(ns,&S(itemp-1,0),1,&S(ih-1,0),1);
                        fs[ih-1] = fr;
                }
                else
                {

                        /*         contract
                         * */
                        if( fr > fs[ih-1] )
                        {
                                newpt(ns,-usubc.beta,&S(icent-1,0),&S(ih-1,0),true,
                                  &S(itemp-1,0),&small);
                        }
                        else
                        {
                                newpt(ns,-usubc.beta,&S(icent-1,0),&S(itemp-1,0),false,
                                  (realnum*)&dum,&small);
                        }
                        if( small )
                                goto L_40;
                        evalf(ns,ips,&S(itemp-1,0),n,x,&fc,nfe);
                        if( fc < (realnum)MIN2(fr,fs[ih-1]) )
                        {
                                cdcopy(ns,&S(itemp-1,0),1,&S(ih-1,0),1);
                                fs[ih-1] = fc;
                        }
                        else
                        {

                                /*           shrink simplex
                                 * */
                                for( j=1; j <= npts; j++ )
                                {
                                        j_ = j - 1;
                                        if( j != il )
                                        {
                                                newpt(ns,-usubc.delta,&S(il-1,0),&S(j_,0),
                                                  false,(realnum*)&dum,&small);
                                                if( small )
                                                        goto L_40;
                                                evalf(ns,ips,&S(j_,0),n,x,&fs[j_],nfe);
                                        }
                                }
                        }
                        updatc = false;
                }
                order(npts,fs,&il,&is,&ih);
        }

        /*       check termination
         * */

L_40:
        if( usubc.irepl == 0 )
        {
                *fx = fs[il-1];
        }
        else
        {
                *fx = isubc.sfbest;
        }

L_50:
        if( (usubc.nfstop > 0 && *fx <= isubc.sfstop) && usubc.nfxe >= 
          usubc.nfstop )
                goto L_51;
        if( *nfe >= maxnfe )
                goto L_52;
        if( !(dist(ns,&S(ih-1,0),&S(il-1,0)) <= tol || small) )
                goto L_20;
        *iflag = 0;
        goto L_53;

L_52:
        *iflag = -1;
        goto L_53;

L_51:
        *iflag = 2;

        /*     end main loop, return best point
         * */

L_53:
        for( i=0; i < ns; i++ )
        {
                x[ips[i]-1] = S(il-1,i);
        }
        return;
}

/********************************************************************* */
STATIC void fstats(double fx, 
          long int ifxwt, 
          int reset)
{
        static long int nsv;
        static realnum f1sv, 
          fscale;

        DEBUG_ENTRY( "fstats()" );


        /*                                         Coded by Tom Rowan
         *                            Department of Computer Sciences
         *                              University of Texas at Austin
         *
         * fstats modifies the common /usubc/ variables nfxe,fxstat.
         *
         * input
         *
         *   fx     - most recent evaluation of f at best x
         *
         *   ifxwt  - integer weight for fx
         *
         *   reset  - logical switch
         *            = .true.  : initialize nfxe,fxstat
         *            = .false. : update nfxe,fxstat
         *
         * common
         * */


        /* local variables
         * */


        /* subroutines and functions
         *
         *
         *-----------------------------------------------------------
         * */
        if( reset )
        {
                usubc.nfxe = ifxwt;
                usubc.fxstat[0] = (realnum)fx;
                usubc.fxstat[1] = (realnum)fx;
                usubc.fxstat[2] = (realnum)fx;
                usubc.fxstat[3] = 0.0f;
        }
        else
        {
                nsv = usubc.nfxe;
                f1sv = usubc.fxstat[0];
                usubc.nfxe += ifxwt;
                usubc.fxstat[0] += (realnum)(ifxwt*(fx - usubc.fxstat[0])/usubc.nfxe);
                usubc.fxstat[1] = MAX2(usubc.fxstat[1],(realnum)fx);
                usubc.fxstat[2] = MIN2(usubc.fxstat[2],(realnum)fx);
                fscale = (realnum)MAX3(fabs(usubc.fxstat[1]),fabs(usubc.fxstat[2]),1.);
                usubc.fxstat[3] = (realnum)(fscale*sqrt(((nsv-1)*POW2(usubc.fxstat[3]/
                  fscale)+nsv*POW2((usubc.fxstat[0]-f1sv)/fscale)+ifxwt*
                  POW2((fx-usubc.fxstat[0])/fscale))/(usubc.nfxe-1)));
        }
        return;
}

STATIC double cdasum(long int n, 
          realnum dx[], 
          long int incx)
{
        /*
         *
         *     takes the sum of the absolute values.
         *     uses unrolled loops for increment equal to one.
         *     jack dongarra, lapack, 3/11/78.
         *     modified to correct problem with negative increment, 8/21/90.
         *
         */
        long int i, 
          ix, 
          m;
        realnum cdasum_v, 
          dtemp;

        DEBUG_ENTRY( "cdasum()" );

        cdasum_v = 0.00f;
        dtemp = 0.00f;
        if( n > 0 )
        {
                if( incx == 1 )
                {

                        /*        code for increment equal to 1
                         *
                         *
                         *        clean-up loop
                         * */
                        m = n%6;
                        if( m != 0 )
                        {
                                for( i=0; i < m; i++ )
                                {
                                        dtemp += (realnum)fabs(dx[i]);
                                }
                                if( n < 6 )
                                        goto L_60;
                        }

                        for( i=m; i < n; i += 6 )
                        {
                                dtemp += (realnum)(fabs(dx[i]) + fabs(dx[i+1]) + fabs(dx[i+2]) + 
                                  fabs(dx[i+3]) + fabs(dx[i+4]) + fabs(dx[i+5]));
                        }
L_60:
                        cdasum_v = dtemp;
                }
                else
                {

                        /*        code for increment not equal to 1
                         * */
                        ix = 1;

                        if( incx < 0 )
                                ix = (-n + 1)*incx + 1;

                        for( i=0; i < n; i++ )
                        {
                                dtemp += (realnum)fabs(dx[ix-1]);
                                ix += incx;
                        }
                        cdasum_v = dtemp;
                }
        }
        return( cdasum_v );
}

STATIC void csscal(long int n, 
          double da, 
          realnum dx[], 
          long int incx)
{
        long int 
          i, 
          i_, 
          m, 
          mp1, 
          nincx;

        DEBUG_ENTRY( "csscal()" );

        /*     scales a vector by a constant.
         *     uses unrolled loops for increment equal to one.
         *     jack dongarra, lapack, 3/11/78.
         *     modified 3/93 to return if incx .le. 0.
         *     modified 12/3/93, array(1) declarations changed to array(*)
         *     changed to single precisions
         * */

        if( !(n <= 0 || incx <= 0) )
        {
                if( incx == 1 )
                {

                        /*        code for increment equal to 1
                         *
                         *
                         *        clean-up loop
                         * */
                        m = n%5;
                        if( m != 0 )
                        {
                                for( i=1; i <= m; i++ )
                                {
                                        i_ = i - 1;
                                        dx[i_] *= (realnum)da;
                                }
                                if( n < 5 )
                                { 
                                        return;
                                }
                        }
                        mp1 = m + 1;
                        for( i=mp1; i <= n; i += 5 )
                        {
                                i_ = i - 1;
                                dx[i_] *= (realnum)da;
                                dx[i_+1] *= (realnum)da;
                                dx[i_+2] *= (realnum)da;
                                dx[i_+3] *= (realnum)da;
                                dx[i_+4] *= (realnum)da;
                        }
                }
                else
                {

                        /*        code for increment not equal to 1
                         * */
                        nincx = n*incx;
                        /*for( i=1, _do0=DOCNT(i,nincx,_do1 = incx); _do0 > 0; i += _do1, _do0-- )*/
                        /* gary change forc */
                        for( i=0; i<nincx; i=i+incx)
                        /*for( i=1, _do0=DOCNT(i,nincx,_do1 = incx); _do0 > 0; i += _do1, _do0-- )*/
                        {
                                dx[i] *= (realnum)da;
                        }
                }
        }
        return;
}

/********************************************************************* */

#undef S
#define S(I_,J_)        (*(s+(I_)*(ns)+(J_)))

STATIC void start(long int n, 
          realnum x[], 
          realnum step[], 
          long int ns, 
          long int ips[], 
          realnum *s, 
          int *small)
{
        long int i, 
          i_, 
          j, 
          j_;

        DEBUG_ENTRY( "start()" );

        /* gary add, not used so set to i_ to mute compiler note */
        i_ = n;


        /*                                         Coded by Tom Rowan
         *                            Department of Computer Sciences
         *                              University of Texas at Austin
         *
         * start creates the initial simplex for simplx minimization.
         *
         * input
         *
         *   n      - problem dimension
         *
         *   x      - current best point
         *
         *   step   - stepsizes for corresponding components of x
         *
         *   ns     - subspace dimension
         *
         *   ips    - permutation vector
         *
         *
         * output
         *
         *   s      - first ns+1 columns contain initial simplex
         *
         *   small  - logical flag
         *            = .true.  : coincident points
         *            = .false. : otherwise
         *
         * local variables
         * */

        /* subroutines and functions
         *
         *   blas */
        /*   fortran */

        /*-----------------------------------------------------------
         * */
        for( i=1; i <= ns; i++ )
        {
                i_ = i - 1;
                S(0,i_) = x[ips[i_]-1];
        }

        for( j=2; j <= (ns + 1); j++ )
        {
                j_ = j - 1;
                cdcopy(ns,&S(0,0),1,&S(j_,0),1);
                S(j_,j_-1) = S(0,j_-1) + step[ips[j_-1]-1];
        }

        /* check for coincident points
         * */
        for( j=2; j <= (ns + 1); j++ )
        {
                j_ = j - 1;
                if( (double)(S(j_,j_-1)) == (double)(S(0,j_-1)) )
                        goto L_40;
        }
        *small = false;
        return;

        /* coincident points
         * */
L_40:
        *small = true;
        return;
}

/********************************************************************* */
STATIC void order(long int npts, 
          realnum fs[], 
          long int *il, 
          long int *is, 
          long int *ih)
{
        long int i, 
          il0, 
          j;

        DEBUG_ENTRY( "order()" );


        /*                                         Coded by Tom Rowan
         *                            Department of Computer Sciences
         *                              University of Texas at Austin
         *
         * order determines the indices of the vertices with the
         * lowest, second highest, and highest function values.
         *
         * input
         *
         *   npts   - number of points in simplex
         *
         *   fs     - real vector of function values of
         *            simplex
         *
         *   il     - index to vertex with lowest function value
         *
         * output
         *
         *   il     - new index to vertex with lowest function value
         *
         *   is     - new index to vertex with second highest
         *            function value
         *
         *   ih     - new index to vertex with highest function value
         *
         * local variables
         * */

        /* subroutines and functions
         *
         *
         *-----------------------------------------------------------
         * */
        il0 = *il;
        j = (il0%npts) + 1;

        if( fs[j-1] >= fs[*il-1] )
        {
                *ih = j;
                *is = il0;
        }
        else
        {
                *ih = il0;
                *is = j;
                *il = j;
        }

        for( i=il0 + 1; i <= (il0 + npts - 2); i++ )
        {
                j = (i%npts) + 1;
                if( fs[j-1] >= fs[*ih-1] )
                {
                        *is = *ih;
                        *ih = j;
                }
                else if( fs[j-1] > fs[*is-1] )
                {
                        *is = j;
                }
                else if( fs[j-1] < fs[*il-1] )
                {
                        *il = j;
                }
        }
        return;
}

STATIC double dist(long int n, 
          realnum x[], 
          realnum y[])
{
        /*
         *
         */
        long int i, 
          i_;
        realnum absxmy, 
          dist_v, 
          scale, 
          sum;

        DEBUG_ENTRY( "dist()" );

        /*                                         Coded by Tom Rowan
         *                            Department of Computer Sciences
         *                              University of Texas at Austin
         *
         * dist calculates the distance between the points x,y.
         *
         * input
         *
         *   n      - number of components
         *
         *   x      - point in n-space
         *
         *   y      - point in n-space
         *
         * local variables
         * */

        /* subroutines and functions
         *
         *   fortran
         *
         *-----------------------------------------------------------
         * */
        absxmy = (realnum)fabs(x[0]-y[0]);
        if( absxmy <= 1.0f )
        {
                sum = absxmy*absxmy;
                scale = 1.0f;
        }
        else
        {
                sum = 1.0f;
                scale = absxmy;
        }

        for( i=2; i <= n; i++ )
        {
                i_ = i - 1;
                absxmy = (realnum)fabs(x[i_]-y[i_]);
                if( absxmy <= scale )
                {
                        sum += POW2(absxmy/scale);
                }
                else
                {
                        sum = 1.0f + sum*POW2(scale/absxmy);
                        scale = absxmy;
                }
        }
        dist_v = (realnum)(scale*sqrt(sum));
        return( dist_v );
}

/********************************************************************* */

#undef S
#define S(I_,J_)        (*(s+(I_)*(ns)+(J_)))

STATIC void calcc(long int ns, 
          realnum *s, 
          long int ih, 
          long int inew, 
          int updatc, 
          realnum c[])
{
        long int i, 
          j, 
          j_;
        /* >>chng 99 apr 21, following was not initialized, caught by Peter van Hoof */
        realnum xNothing[1] = { 0.0f };

        DEBUG_ENTRY( "calcc()" );


        /*                                         Coded by Tom Rowan
         *                            Department of Computer Sciences
         *                              University of Texas at Austin
         *
         * calcc calculates the centroid of the simplex without the
         * vertex with highest function value.
         *
         * input
         *
         *   ns     - subspace dimension
         *
         *   s      - real work space of dimension .ge.
         *            ns*(ns+3) used to store simplex
         *
         *   ih     - index to vertex with highest function value
         *
         *   inew   - index to new point
         *
         *   updatc - logical switch
         *            = .true.  : update centroid
         *            = .false. : calculate centroid from scratch
         *
         *   c      - centroid of the simplex without vertex with
         *            highest function value
         *
         * output
         *
         *   c      - new centroid
         *
         * local variables
         * */
        /*     added to get prototypes to work */

        /* subroutines and functions
         *
         *   blas */

        /*-----------------------------------------------------------
         * */
        if( !updatc )
        {
                /*       call cdcopy (ns,0.0,0,c,1)
                 *       xNothing will not be used since 0 is increment */
                cdcopy(ns,xNothing,0,c,1);
                for( j=1; j <= (ns + 1); j++ )
                {
                        j_ = j - 1;
                        if( j != ih )
                                cdaxpy(ns,1.0f,&S(j_,0),1,c,1);
                }
                csscal(ns,1.0f/ns,c,1);
        }
        else if( ih != inew )
        {
                for( i=0; i < ns; i++ )
                {
                        c[i] += (S(inew-1,i) - S(ih-1,i))/ns;
                }
        }
        return;
}

/********************************************************************* */
STATIC void newpt(long int ns, 
          double coef, 
          realnum xbase[], 
          realnum xold[], 
          int IntNew, 
          realnum xnew[], 
          int *small)
{
        int eqbase, 
          eqold;
        long int i;
        realnum xoldi;

        DEBUG_ENTRY( "newpt()" );


        /*                                         Coded by Tom Rowan
         *                            Department of Computer Sciences
         *                              University of Texas at Austin
         *
         * newpt performs reflections, expansions, contractions, and
         * shrinkages (massive contractions) by computing:
         *
         * xbase + coef * (xbase - xold)
         *
         * The result is stored in xnew if IntNew .eq. .true.,
         * in xold otherwise.
         *
         * use :  coef .gt. 0 to reflect
         *        coef .lt. 0 to expand, contract, or shrink
         *
         * input
         *
         *   ns     - number of components (subspace dimension)
         *
         *   coef   - one of four simplex method coefficients
         *
         *   xbase  - real ns-vector representing base
         *            point
         *
         *   xold   - real ns-vector representing old
         *            point
         *
         *   IntNew    - logical switch
         *            = .true.  : store result in xnew
         *            = .false. : store result in xold, xnew is not
         *                        referenced
         *
         * output
         *
         *   xold   - unchanged if IntNew .eq. .true., contains new
         *            point otherwise
         *
         *   xnew   - real ns-vector representing new
         *            point if  new .eq. .true., not referenced
         *            otherwise
         *
         *   small  - logical flag
         *            = .true.  : coincident points
         *            = .false. : otherwise
         *
         * local variables
         * */

        /* subroutines and functions
         *
         *   fortran */

        /*-----------------------------------------------------------
         * */
        eqbase = true;
        eqold = true;
        if( IntNew )
        {
                for( i=0; i < ns; i++ )
                {
                        xnew[i] = (realnum)(xbase[i] + coef*(xbase[i] - xold[i]));
                        eqbase = eqbase && ((double)(xnew[i]) == (double)(xbase[i]));
                        eqold = eqold && ((double)(xnew[i]) == (double)(xold[i]));
                }
        }
        else
        {
                for( i=0; i < ns; i++ )
                {
                        xoldi = xold[i];
                        xold[i] = (realnum)(xbase[i] + coef*(xbase[i] - xold[i]));
                        eqbase = eqbase && ((double)(xold[i]) == (double)(xbase[i]));
                        eqold = eqold && ((double)(xold[i]) == (double)(xoldi));
                }
        }
        *small = eqbase || eqold;
        return;
}
/********************************************************************* */
STATIC void cdaxpy(long int n, 
          double da, 
          realnum dx[], 
          long int incx, 
          realnum dy[], 
          long int incy)
{
        long int i, 
          i_, 
          ix, 
          iy, 
          m;

        DEBUG_ENTRY( "cdaxpy()" );

        /*     constant times a vector plus a vector.
         *     uses unrolled loops for increments equal to one.
         *     jack dongarra, lapack, 3/11/78.
         * */

        if( n > 0 )
        {
                if( da != 0.00f )
                {
                        if( incx == 1 && incy == 1 )
                        {

                                /*        code for both increments equal to 1
                                 *
                                 *
                                 *        clean-up loop
                                 * */
                                m = n%4;
                                if( m != 0 )
                                {
                                        for( i=1; i <= m; i++ )
                                        {
                                                i_ = i - 1;
                                                dy[i_] += (realnum)(da*dx[i_]);
                                        }
                                        if( n < 4 )
                                        {
                                                return;
                                        }
                                }

                                for( i=m; i < n; i += 4 )
                                {
                                        dy[i] += (realnum)(da*dx[i]);
                                        dy[i+1] += (realnum)(da*dx[i+1]);
                                        dy[i+2] += (realnum)(da*dx[i+2]);
                                        dy[i+3] += (realnum)(da*dx[i+3]);
                                }
                        }
                        else
                        {

                                /*        code for unequal increments or equal increments
                                 *          not equal to 1
                                 * */
                                ix = 1;
                                iy = 1;
                                if( incx < 0 )
                                        ix = (-n + 1)*incx + 1;
                                if( incy < 0 )
                                        iy = (-n + 1)*incy + 1;
                                for( i=0; i < n; i++ )
                                {
                                        dy[iy-1] += (realnum)(da*dx[ix-1]);
                                        ix += incx;
                                        iy += incy;
                                }
                        }
                }
        }
        return;
}
/*lint +e801 goto is deprecated */
/*lint +e64 type mismatch */

---