Logo Search packages:      
Sourcecode: openmx version File versions  Download package

lbfgs.c

/**********************************************************************
  lbfgs.c:

   lbfgs.c is a subroutine to optimize geometrical structure using the
   Limitied-memory Broyden-Fletcher-Goldfarb-Shanno (LBFGS) method.

   The original Fortran 77 code written by Jorge Nocedal was translated
   by f2c (version 20041007), and modified by Taisuke Ozakie so that it
   can run without including f2c.h

   See also the original webisite for the distribution
   http://www.ece.northwestern.edu/~nocedal/software.html

   The following is the message by Jorge Nocedal.

   ***********************************************************************
     Software for Large-scale Unconstrained Optimization

     L-BFGS is a limited-memory quasi-Newton code for unconstrained 
     optimization. The code has been developed at the Optimization
     Technology Center, a joint venture of Argonne National Laboratory
     and Northwestern University.

     Downloading and Installing L-BFGS

     You are welcome to grab the full Unix distribution, containing source
     code, makefile, and user guide. 

     Condition for Use: This software is freely available for educational
     or commercial purposes. We expect that all publications describing work
     using this software quote at least one of the references given below.

     References:

     C. Liu and J. Nocedal,
     On the limited memory BFGS method for large-scale optimization,
     Math. Programming 45 (1989), pp. 503--528. 

     J. Nocedal, Updating quasi-Newton matrices with limited storage,
     Math. Comp. 24 (1980), pp. 773--782. 
   ***********************************************************************

  Log of lbfgs.c:

     12/Nov/2005  Released by T.Ozaki

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

#include <stdio.h>
#include <stdlib.h>
#include <math.h>
#include "openmx_common.h"

 
#define min(a,b) ((a) <= (b) ? (a) : (b))
#define max(a,b) ((a) >= (b) ? (a) : (b))

#define TRUE_ (1)
#define FALSE_ (0)



/*external read, write*/
typedef struct
{     flag cierr;
      ftnint ciunit;
      flag ciend;
      char *cifmt;
      ftnint cirec;
} cilist;


 

/* Common Block Declarations */

struct lb3_1_ {
    integer mp, lp;
    doublereal gtol, stpmin, stpmax;
};

#define lb3_1 (*(struct lb3_1_ *) &lb3_)

/* Initialized data */

/*
struct {
    integer e_1[2];
    doublereal e_2[3];
    } lb3_ = { 6, 6, .9, 1.0e-20, 1.0e+20 };
*/


struct {
    integer e_1[2];
    doublereal e_2[3];
    } lb3_ = { 6, 6, .5, 0.0001, 0.6 };


/* Table of constant values */

static integer c__1 = 1;












/*    This file contains the LBFGS algorithm and supporting routines */

 int lbfgs(integer *n, integer *m, doublereal *x, doublereal 
         *f, doublereal *g, logical *diagco, doublereal *diag, integer *iprint,
         doublereal *eps, doublereal *xtol, doublereal *w, integer *iflag)
{
    /* Initialized data */

    static doublereal one = 1.;
    static doublereal zero = 0.;

    /* System generated locals */
    integer i__1;
    doublereal d__1;

    /* Local variables */
    static integer i__, cp;
    static doublereal sq, yr, ys, yy;
    extern /* Subroutine */ int lb1_(integer *, integer *, integer *, 
          doublereal *, integer *, integer *, doublereal *, doublereal *, 
          doublereal *, doublereal *, logical *);
    static integer npt;
    static doublereal stp, stp1, beta;
    static integer inmc;
    extern doublereal ddot_(integer *, doublereal *, integer *, doublereal *, 
          integer *);
    static integer info, iscn, nfev, iycn, iter;
    static doublereal ftol;
    static integer nfun, ispt, iypt, bound;
    static doublereal gnorm;
    extern /* Subroutine */ int daxpy_(integer *, doublereal *, doublereal *, 
          integer *, doublereal *, integer *);
    static integer point;
    static doublereal xnorm;
    extern /* Subroutine */ int mcsrch_(integer *, doublereal *, doublereal *,
           doublereal *, doublereal *, doublereal *, doublereal *, 
          doublereal *, integer *, integer *, integer *, doublereal *);
    static logical finish;
    static integer maxfev;

/*        LIMITED MEMORY BFGS METHOD FOR LARGE SCALE OPTIMIZATION */
/*                          JORGE NOCEDAL */
/*                        *** July 1990 *** */


/*     This subroutine solves the unconstrained minimization problem */

/*                      min F(x),    x= (x1,x2,...,xN), */

/*      using the limited memory BFGS method. The routine is especially */
/*      effective on problems involving a large number of variables. In */
/*      a typical iteration of this method an approximation Hk to the */
/*      inverse of the Hessian is obtained by applying M BFGS updates to */
/*      a diagonal matrix Hk0, using information from the previous M steps. */
/*      The user specifies the number M, which determines the amount of */
/*      storage required by the routine. The user may also provide the */
/*      diagonal matrices Hk0 if not satisfied with the default choice. */
/*      The algorithm is described in "On the limited memory BFGS method */
/*      for large scale optimization", by D. Liu and J. Nocedal, */
/*      Mathematical Programming B 45 (1989) 503-528. */

/*      The user is required to calculate the function value F and its */
/*      gradient G. In order to allow the user complete control over */
/*      these computations, reverse  communication is used. The routine */
/*      must be called repeatedly under the control of the parameter */
/*      IFLAG. */

/*      The steplength is determined at each iteration by means of the */
/*      line search routine MCVSRCH, which is a slight modification of */
/*      the routine CSRCH written by More' and Thuente. */

/*      The calling statement is */

/*          CALL LBFGS(N,M,X,F,G,DIAGCO,DIAG,IPRINT,EPS,XTOL,W,IFLAG) */

/*      where */

/*     N       is an INTEGER variable that must be set by the user to the */
/*             number of variables. It is not altered by the routine. */
/*             Restriction: N>0. */

/*     M       is an INTEGER variable that must be set by the user to */
/*             the number of corrections used in the BFGS update. It */
/*             is not altered by the routine. Values of M less than 3 are */
/*             not recommended; large values of M will result in excessive */
/*             computing time. 3<= M <=7 is recommended. Restriction: M>0. */

/*     X       is a DOUBLE PRECISION array of length N. On initial entry */
/*             it must be set by the user to the values of the initial */
/*             estimate of the solution vector. On exit with IFLAG=0, it */
/*             contains the values of the variables at the best point */
/*             found (usually a solution). */

/*     F       is a DOUBLE PRECISION variable. Before initial entry and on */
/*             a re-entry with IFLAG=1, it must be set by the user to */
/*             contain the value of the function F at the point X. */

/*     G       is a DOUBLE PRECISION array of length N. Before initial */
/*             entry and on a re-entry with IFLAG=1, it must be set by */
/*             the user to contain the components of the gradient G at */
/*             the point X. */

/*     DIAGCO  is a LOGICAL variable that must be set to .TRUE. if the */
/*             user  wishes to provide the diagonal matrix Hk0 at each */
/*             iteration. Otherwise it should be set to .FALSE., in which */
/*             case  LBFGS will use a default value described below. If */
/*             DIAGCO is set to .TRUE. the routine will return at each */
/*             iteration of the algorithm with IFLAG=2, and the diagonal */
/*              matrix Hk0  must be provided in the array DIAG. */


/*     DIAG    is a DOUBLE PRECISION array of length N. If DIAGCO=.TRUE., */
/*             then on initial entry or on re-entry with IFLAG=2, DIAG */
/*             it must be set by the user to contain the values of the */
/*             diagonal matrix Hk0.  Restriction: all elements of DIAG */
/*             must be positive. */

/*     IPRINT  is an INTEGER array of length two which must be set by the */
/*             user. */

/*             IPRINT(1) specifies the frequency of the output: */
/*                IPRINT(1) < 0 : no output is generated, */
/*                IPRINT(1) = 0 : output only at first and last iteration, */
/*                IPRINT(1) > 0 : output every IPRINT(1) iterations. */

/*             IPRINT(2) specifies the type of output generated: */
/*                IPRINT(2) = 0 : iteration count, number of function */
/*                                evaluations, function value, norm of the */
/*                                gradient, and steplength, */
/*                IPRINT(2) = 1 : same as IPRINT(2)=0, plus vector of */
/*                                variables and  gradient vector at the */
/*                                initial point, */
/*                IPRINT(2) = 2 : same as IPRINT(2)=1, plus vector of */
/*                                variables, */
/*                IPRINT(2) = 3 : same as IPRINT(2)=2, plus gradient vector. */


/*     EPS     is a positive DOUBLE PRECISION variable that must be set by */
/*             the user, and determines the accuracy with which the solution */
/*             is to be found. The subroutine terminates when */

/*                         ||G|| < EPS max(1,||X||), */

/*             where ||.|| denotes the Euclidean norm. */

/*     XTOL    is a  positive DOUBLE PRECISION variable that must be set by */
/*             the user to an estimate of the machine precision (e.g. */
/*             10**(-16) on a SUN station 3/60). The line search routine will */
/*             terminate if the relative width of the interval of uncertainty */
/*             is less than XTOL. */

/*     W       is a DOUBLE PRECISION array of length N(2M+1)+2M used as */
/*             workspace for LBFGS. This array must not be altered by the */
/*             user. */

/*     IFLAG   is an INTEGER variable that must be set to 0 on initial entry */
/*             to the subroutine. A return with IFLAG<0 indicates an error, */
/*             and IFLAG=0 indicates that the routine has terminated without */
/*             detecting errors. On a return with IFLAG=1, the user must */
/*             evaluate the function F and gradient G. On a return with */
/*             IFLAG=2, the user must provide the diagonal matrix Hk0. */

/*             The following negative values of IFLAG, detecting an error, */
/*             are possible: */

/*              IFLAG=-1  The line search routine MCSRCH failed. The */
/*                        parameter INFO provides more detailed information */
/*                        (see also the documentation of MCSRCH): */

/*                       INFO = 0  IMPROPER INPUT PARAMETERS. */

/*                       INFO = 2  RELATIVE WIDTH OF THE INTERVAL OF */
/*                                 UNCERTAINTY IS AT MOST XTOL. */

/*                       INFO = 3  MORE THAN 20 FUNCTION EVALUATIONS WERE */
/*                                 REQUIRED AT THE PRESENT ITERATION. */

/*                       INFO = 4  THE STEP IS TOO SMALL. */

/*                       INFO = 5  THE STEP IS TOO LARGE. */

/*                       INFO = 6  ROUNDING ERRORS PREVENT FURTHER PROGRESS. */
/*                                 THERE MAY NOT BE A STEP WHICH SATISFIES */
/*                                 THE SUFFICIENT DECREASE AND CURVATURE */
/*                                 CONDITIONS. TOLERANCES MAY BE TOO SMALL. */


/*              IFLAG=-2  The i-th diagonal element of the diagonal inverse */
/*                        Hessian approximation, given in DIAG, is not */
/*                        positive. */

/*              IFLAG=-3  Improper input parameters for LBFGS (N or M are */
/*                        not positive). */



/*    ON THE DRIVER: */

/*    The program that calls LBFGS must contain the declaration: */

/*                       EXTERNAL LB2 */

/*    LB2 is a BLOCK DATA that defines the default values of several */
/*    parameters described in the COMMON section. */



/*    COMMON: */

/*     The subroutine contains one common area, which the user may wish to */
/*    reference: */


/*    MP  is an INTEGER variable with default value 6. It is used as the */
/*        unit number for the printing of the monitoring information */
/*        controlled by IPRINT. */

/*    LP  is an INTEGER variable with default value 6. It is used as the */
/*        unit number for the printing of error messages. This printing */
/*        may be suppressed by setting LP to a non-positive value. */

/*    GTOL is a DOUBLE PRECISION variable with default value 0.9, which */
/*        controls the accuracy of the line search routine MCSRCH. If the */
/*        function and gradient evaluations are inexpensive with respect */
/*        to the cost of the iteration (which is sometimes the case when */
/*        solving very large problems) it may be advantageous to set GTOL */
/*        to a small value. A typical small value is 0.1.  Restriction: */
/*        GTOL should be greater than 1.D-04. */

/*    STPMIN and STPMAX are non-negative DOUBLE PRECISION variables which */
/*        specify lower and uper bounds for the step in the line search. */
/*        Their default values are 1.D-20 and 1.D+20, respectively. These */
/*        values need not be modified unless the exponents are too large */
/*        for the machine being used, or unless the problem is extremely */
/*        badly scaled (in which case the exponents should be increased). */


/*  MACHINE DEPENDENCIES */

/*        The only variables that are machine-dependent are XTOL, */
/*        STPMIN and STPMAX. */


/*  GENERAL INFORMATION */

/*    Other routines called directly:  DAXPY, DDOT, LB1, MCSRCH */

/*    Input/Output  :  No input; diagnostic messages on unit MP and */
/*                     error messages on unit LP. */



  /* Parameter adjustments */

  --diag;
  --g;
  --x;
  --w;
  --iprint;

  /* Function Body */
  /* INITIALIZE */


  if (*iflag == 0) {
    goto L10;
  }
  switch (*iflag) {
    case 1:  goto L172;
    case 2:  goto L100;
  }



 L10:
  iter = 0;
  if (*n <= 0 || *m <= 0) {
    goto L196;
  }
  if (lb3_1.gtol <= 1e-4) {
    if (lb3_1.lp > 0) {
      printf("  GTOL IS LESS THAN OR EQUAL TO 1.D-04\n"); 
      printf("  IT HAS BEEN RESET TO 9.D-01\n"); 
    }
    lb3_1.gtol = .9;
  }

  nfun = 1;
  point = 0;
  finish = FALSE_;
  if (*diagco) {
    i__1 = *n;
    for (i__ = 1; i__ <= i__1; ++i__) {
      /* L30: */
      if (diag[i__] <= zero) {
      goto L195;
      }
    }
  }

  else {
    i__1 = *n;
    for (i__ = 1; i__ <= i__1; ++i__) {
      diag[i__] = 1.;
    }
  }

  /*     THE WORK VECTOR W IS DIVIDED AS FOLLOWS: */
  /*     --------------------------------------- */
  /*     THE FIRST N LOCATIONS ARE USED TO STORE THE GRADIENT AND */
  /*         OTHER TEMPORARY INFORMATION. */
  /*     LOCATIONS (N+1)...(N+M) STORE THE SCALARS RHO. */
  /*     LOCATIONS (N+M+1)...(N+2M) STORE THE NUMBERS ALPHA USED */
  /*         IN THE FORMULA THAT COMPUTES H*G. */
  /*     LOCATIONS (N+2M+1)...(N+2M+NM) STORE THE LAST M SEARCH */
  /*         STEPS. */
  /*     LOCATIONS (N+2M+NM+1)...(N+2M+2NM) STORE THE LAST M */
  /*         GRADIENT DIFFERENCES. */
  /*     THE SEARCH STEPS AND GRADIENT DIFFERENCES ARE STORED IN A */
  /*     CIRCULAR ORDER CONTROLLED BY THE PARAMETER POINT. */

  ispt = *n + (*m << 1);
  iypt = ispt + *n * *m;
  i__1 = *n;
  for (i__ = 1; i__ <= i__1; ++i__) {
    w[ispt + i__] = -g[i__] * diag[i__];
  }
  gnorm = sqrt(ddot_(n, &g[1], &c__1, &g[1], &c__1));
  stp1 = one / gnorm;

  /* PARAMETERS FOR LINE SEARCH ROUTINE */

  ftol = 1e-4;
  maxfev = 20;

  if (iprint[1] >= 0) {
    lb1_(&iprint[1], &iter, &nfun, &gnorm, n, m, &x[1], f, &g[1], &stp, &finish);
  }

  /* MAIN ITERATION LOOP */

L80:

  ++iter;
  info = 0;
  bound = iter - 1;
  if (iter == 1) {
    goto L165;
  }

  if (iter > *m) {
    bound = *m;
  }

  ys = ddot_(n, &w[iypt + npt + 1], &c__1, &w[ispt + npt + 1], &c__1);
  if (! (*diagco)) {
    yy = ddot_(n, &w[iypt + npt + 1], &c__1, &w[iypt + npt + 1], &c__1);
    i__1 = *n;
    for (i__ = 1; i__ <= i__1; ++i__) {
      diag[i__] = ys / yy;
    }
  }

  else {
    *iflag = 2;
    return 0;
  }






L100:

  if (*diagco) {
    i__1 = *n;
    for (i__ = 1; i__ <= i__1; ++i__) {
      if (diag[i__] <= zero) {
      goto L195;
      }
    }
  }

  /*  COMPUTE -H*G USING THE FORMULA GIVEN IN: Nocedal, J. 1980, */
  /*  "Updating quasi-Newton matrices with limited storage",     */
  /*  Mathematics of Computation, Vol.24, No.151, pp. 773-782.   */

  cp = point;
  if (point == 0) {
    cp = *m;
  }
  w[*n + cp] = one / ys;
  i__1 = *n;
  for (i__ = 1; i__ <= i__1; ++i__) {
    w[i__] = -g[i__];
  }

  cp = point;
  i__1 = bound;
  for (i__ = 1; i__ <= i__1; ++i__) {
    --cp;
    if (cp == -1) {
      cp = *m - 1;
    }
    sq = ddot_(n, &w[ispt + cp * *n + 1], &c__1, &w[1], &c__1);
    inmc = *n + *m + cp + 1;
    iycn = iypt + cp * *n;
    w[inmc] = w[*n + cp + 1] * sq;
    d__1 = -w[inmc];
    daxpy_(n, &d__1, &w[iycn + 1], &c__1, &w[1], &c__1);
  }

  i__1 = *n;
  for (i__ = 1; i__ <= i__1; ++i__) {
    w[i__] = diag[i__] * w[i__];
  }

  i__1 = bound;
  for (i__ = 1; i__ <= i__1; ++i__) {
    yr = ddot_(n, &w[iypt + cp * *n + 1], &c__1, &w[1], &c__1);
    beta = w[*n + cp + 1] * yr;
    inmc = *n + *m + cp + 1;
    beta = w[inmc] - beta;
    iscn = ispt + cp * *n;
    daxpy_(n, &beta, &w[iscn + 1], &c__1, &w[1], &c__1);
    ++cp;
    if (cp == *m) {
      cp = 0;
    }
  }

  /* STORE THE NEW SEARCH DIRECTION */

  i__1 = *n;
  for (i__ = 1; i__ <= i__1; ++i__) {
    w[ispt + point * *n + i__] = w[i__];
  }

  /*     OBTAIN THE ONE-DIMENSIONAL MINIMIZER OF THE FUNCTION */
  /*     BY USING THE LINE SEARCH ROUTINE MCSRCH */

L165:

  nfev = 0;
  stp = one;
  if (iter == 1) {
    stp = stp1;
  }
  i__1 = *n;
  for (i__ = 1; i__ <= i__1; ++i__) {
    w[i__] = g[i__];
  }





L172:

  mcsrch_(n, &x[1], f, &g[1], &w[ispt + point * *n + 1], &stp, &ftol, xtol, 
           &maxfev, &info, &nfev, &diag[1]);

  if (info == -1) {
    *iflag = 1;
    return 0;
  }
  if (info != 1) {
    goto L190;
  }

  nfun += nfev;

  /* COMPUTE THE NEW STEP AND GRADIENT CHANGE */

  npt = point * *n;
  i__1 = *n;
  for (i__ = 1; i__ <= i__1; ++i__) {
    w[ispt + npt + i__] = stp * w[ispt + npt + i__];
    w[iypt + npt + i__] = g[i__] - w[i__];
  }

  ++point;
  if (point == *m) {
    point = 0;
  }

  /* TERMINATION TEST */

  gnorm = sqrt(ddot_(n, &g[1], &c__1, &g[1], &c__1));
  xnorm = sqrt(ddot_(n, &x[1], &c__1, &x[1], &c__1));
  xnorm = max(1.,xnorm);

  if (gnorm / xnorm <= *eps) {
    finish = TRUE_;
  }

  if (iprint[1] >= 0) {
    lb1_(&iprint[1], &iter, &nfun, &gnorm, n, m, &x[1], f, &g[1], &stp, &finish);
  }

  if (finish) {
   *iflag = 0;
    return 0;
  }
  goto L80;

  /* END OF MAIN ITERATION LOOP. ERROR EXITS. */


L190:
  *iflag = -1;
  if (lb3_1.lp > 0) {
    printf(" IFLAG= -1, LINE SEARCH FAILED. SEE\n");
    printf(" DOCUMENTATION OF ROUTINE MCSRCH, ERROR RETURN\n");
    printf(" OF LINE SEARCH: INFO= %2d\n",info);
    printf(" POSSIBLE CAUSES: FUNCTION OR GRADIENT ARE INCORRECT\n");
    printf(" OR INCORRECT TOLERANCES\n");
  }
  return 0;

L195:

  *iflag = -2;
  if (lb3_1.lp > 0) {
    printf(" IFLAG= -2, THE, %2d, -TH DIAGONAL ELEMENT OF THE\n",i__);
    printf(" INVERSE HESSIAN APPROXIMATION IS NOT POSITIVE\n");
  }
  return 0;

L196:

  *iflag = -3;
  if (lb3_1.lp > 0) {
    printf(" IFLAG= -3, IMPROPER INPUT PARAMETERS (N OR M\n");
    printf(" ARE NOT POSITIVE)\n");
  }

  return 0;

} /* lbfgs_ */
















int lb1_(integer *iprint, integer *iter, integer *nfun, 
       doublereal *gnorm, integer *n, integer *m, doublereal *x, doublereal *
       f, doublereal *g, doublereal *stp, logical *finish)
{

  /* System generated locals */
  integer i__1;

  /* Local variables */
  static integer i__;


  /*     ------------------------------------------------------------- */
  /*      THIS ROUTINE PRINTS MONITORING INFORMATION. THE FREQUENCY AND */
  /*     AMOUNT OF OUTPUT ARE CONTROLLED BY IPRINT. */
  /*     ------------------------------------------------------------- */


  /* Parameter adjustments */
  --iprint;
  --g;
  --x;

  /* Function Body */
  if (*iter == 0) {

    printf("*************************************************\n");
    printf("  N= %3d   NUMBER OF CORRECTIONS= %3d\n",*n,*m);
    printf("            INITIAL VALUES\n");
    printf("  F= %15.12f    GNORM= %15.12f\n",*f,*gnorm);

    if (iprint[2] >= 1) {

      printf(" VECTOR X=\n");
      i__1 = *n;
      for (i__ = 1; i__ <= i__1; ++i__) {
      printf("       i=%4d  x=%15.12f\n",i__,x[i__]);
      }

      printf(" GRADIENT VECTOR G=\n");
      i__1 = *n;
      for (i__ = 1; i__ <= i__1; ++i__) {
      printf("       i=%4d  g=%15.12f\n",i__,g[i__]);
      }
    }

    printf("*************************************************\n");
    printf("   I   NFN   FUNC  GNORM   STEPLENGTH\n");


  }

 else {

   if (iprint[1] == 0 && (*iter != 1 && ! (*finish))) {
     return 0;
   }

   if (iprint[1] != 0) {
     if ((*iter - 1) % iprint[1] == 0 || *finish) {
       if (iprint[2] > 1 && *iter > 1) {
       printf("   I   NFN   FUNC  GNORM   STEPLENGTH\n");
       }
       printf("A %3d  %3d  %18.12f %18.12f %18.12f\n",*iter,*nfun,*f,*gnorm,*stp);
     } else {
       return 0;
     }
   } else {

     if (iprint[2] > 1 && *finish) {
       printf("   I   NFN   FUNC  GNORM   STEPLENGTH\n");
       printf("B %3d  %3d  %18.12f %18.12f %18.12f\n",*iter,*nfun,*f,*gnorm,*stp);
     }
   }
   if (iprint[2] == 2 || iprint[2] == 3) {
     if (*finish) {
       printf(" FINAL POINT X= \n");
     }
     else {
       printf(" VECTOR X=\n");
     }

     i__1 = *n;
     for (i__ = 1; i__ <=i__1; ++i__) {
       printf("       i=%4d  x=%15.12f\n",i__,x[i__]);
     }

     if (iprint[2] == 3) {

       printf(" GRADIENT VECTOR G=\n");
       i__1 = *n;
       for (i__ = 1; i__ <= i__1; ++i__) {
       printf("       i=%4d  g=%15.12f\n",i__,g[i__]);
       }
     }
   }
   if (*finish) {
     printf(" THE MINIMIZATION TERMINATED WITHOUT DETECTING ERRORS\n");
     printf(" IFLAG = 0\n");
   }
 }

  return 0;
} /* lb1_ */









int lb2_(void)
{
  return 0;
} /* lb2_ */
















int daxpy_(integer *n, doublereal *da, doublereal *dx, 
           integer *incx, doublereal *dy, integer *incy)
{
  /* System generated locals */
  integer i__1;

  /* Local variables */
  static integer i__, m, ix, iy, mp1;


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


    /* Parameter adjustments */
    --dy;
    --dx;

    /* Function Body */
    if (*n <= 0) {
      return 0;
    }
    if ( fabs(*da) <1.0e-50 ) {
      return 0;
    }
    if (*incx == 1 && *incy == 1) {
      goto L20;
    }

/*        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;
    }
    i__1 = *n;
    for (i__ = 1; i__ <= i__1; ++i__) {
      dy[iy] += *da * dx[ix];
      ix += *incx;
      iy += *incy;
/* L10: */
    }
    return 0;

/*        code for both increments equal to 1 */


/*        clean-up loop */

L20:
    m = *n % 4;
    if (m == 0) {
      goto L40;
    }
    i__1 = m;
    for (i__ = 1; i__ <= i__1; ++i__) {
      dy[i__] += *da * dx[i__];
/* L30: */
    }
    if (*n < 4) {
      return 0;
    }
L40:
    mp1 = m + 1;
    i__1 = *n;
    for (i__ = mp1; i__ <= i__1; i__ += 4) {
      dy[i__] += *da * dx[i__];
      dy[i__ + 1] += *da * dx[i__ + 1];
      dy[i__ + 2] += *da * dx[i__ + 2];
      dy[i__ + 3] += *da * dx[i__ + 3];
/* L50: */
    }
    return 0;
} /* daxpy_ */



/*   ---------------------------------------------------------- */

doublereal ddot_(integer *n, doublereal *dx, integer *incx, doublereal *dy, 
      integer *incy)
{
    /* System generated locals */
    integer i__1;
    doublereal ret_val;

    /* Local variables */
    static integer i__, m, ix, iy, mp1;
    static doublereal dtemp;


/*     forms the dot product of two vectors. */
/*     uses unrolled loops for increments equal to one. */
/*     jack dongarra, linpack, 3/11/78. */


    /* Parameter adjustments */
    --dy;
    --dx;

    /* Function Body */
    ret_val = 0.;
    dtemp = 0.;
    if (*n <= 0) {
      return ret_val;
    }
    if (*incx == 1 && *incy == 1) {
      goto L20;
    }

/*        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;
    }
    i__1 = *n;
    for (i__ = 1; i__ <= i__1; ++i__) {
      dtemp += dx[ix] * dy[iy];
      ix += *incx;
      iy += *incy;
/* L10: */
    }
    ret_val = dtemp;
    return ret_val;

/*        code for both increments equal to 1 */


/*        clean-up loop */

L20:
    m = *n % 5;
    if (m == 0) {
      goto L40;
    }
    i__1 = m;
    for (i__ = 1; i__ <= i__1; ++i__) {
      dtemp += dx[i__] * dy[i__];
/* L30: */
    }
    if (*n < 5) {
      goto L60;
    }
L40:
    mp1 = m + 1;
    i__1 = *n;
    for (i__ = mp1; i__ <= i__1; i__ += 5) {
      dtemp = dtemp + dx[i__] * dy[i__] + dx[i__ + 1] * dy[i__ + 1] + dx[
            i__ + 2] * dy[i__ + 2] + dx[i__ + 3] * dy[i__ + 3] + dx[i__ + 
            4] * dy[i__ + 4];
/* L50: */
    }
L60:
    ret_val = dtemp;
    return ret_val;
} /* ddot_ */

/*    ------------------------------------------------------------------ */

/*     ************************** */
/*     LINE SEARCH ROUTINE MCSRCH */
/*     ************************** */

/* Subroutine */

int mcsrch_(integer *n, doublereal *x, doublereal *f, 
          doublereal *g, doublereal *s, doublereal *stp, doublereal *ftol, 
          doublereal *xtol, integer *maxfev, integer *info, integer *nfev, 
          doublereal *wa)
{
    /* Initialized data */

    static doublereal p5 = .5;
    static doublereal p66 = .66;
    static doublereal xtrapf = 4.;
    static doublereal zero = 0.;

    /* System generated locals */
    integer i__1;
    doublereal d__1;

    /* Local variables */
    static integer j;
    static doublereal dg, fm, fx, fy, dgm, dgx, dgy, fxm, fym, stx, sty, dgxm,
           dgym;
    static integer infoc;
    static doublereal finit, width, stmin, stmax;
    static logical stage1;
    static doublereal width1, ftest1;
    static logical brackt;
    static doublereal dginit, dgtest;
    extern /* Subroutine */ int mcstep_(doublereal *, doublereal *, 
          doublereal *, doublereal *, doublereal *, doublereal *, 
          doublereal *, doublereal *, doublereal *, logical *, doublereal *,
           doublereal *, integer *);


/*                     SUBROUTINE MCSRCH */

/*     A slight modification of the subroutine CSRCH of More' and Thuente. */
/*     The changes are to allow reverse communication, and do not affect */
/*     the performance of the routine. */

/*     THE PURPOSE OF MCSRCH IS TO FIND A STEP WHICH SATISFIES */
/*     A SUFFICIENT DECREASE CONDITION AND A CURVATURE CONDITION. */

/*     AT EACH STAGE THE SUBROUTINE UPDATES AN INTERVAL OF */
/*     UNCERTAINTY WITH ENDPOINTS STX AND STY. THE INTERVAL OF */
/*     UNCERTAINTY IS INITIALLY CHOSEN SO THAT IT CONTAINS A */
/*     MINIMIZER OF THE MODIFIED FUNCTION */

/*          F(X+STP*S) - F(X) - FTOL*STP*(GRADF(X)'S). */

/*     IF A STEP IS OBTAINED FOR WHICH THE MODIFIED FUNCTION */
/*     HAS A NONPOSITIVE FUNCTION VALUE AND NONNEGATIVE DERIVATIVE, */
/*     THEN THE INTERVAL OF UNCERTAINTY IS CHOSEN SO THAT IT */
/*     CONTAINS A MINIMIZER OF F(X+STP*S). */

/*     THE ALGORITHM IS DESIGNED TO FIND A STEP WHICH SATISFIES */
/*     THE SUFFICIENT DECREASE CONDITION */

/*           F(X+STP*S) .LE. F(X) + FTOL*STP*(GRADF(X)'S), */

/*     AND THE CURVATURE CONDITION */

/*           ABS(GRADF(X+STP*S)'S)) .LE. GTOL*ABS(GRADF(X)'S). */

/*     IF FTOL IS LESS THAN GTOL AND IF, FOR EXAMPLE, THE FUNCTION */
/*     IS BOUNDED BELOW, THEN THERE IS ALWAYS A STEP WHICH SATISFIES */
/*     BOTH CONDITIONS. IF NO STEP CAN BE FOUND WHICH SATISFIES BOTH */
/*     CONDITIONS, THEN THE ALGORITHM USUALLY STOPS WHEN ROUNDING */
/*     ERRORS PREVENT FURTHER PROGRESS. IN THIS CASE STP ONLY */
/*     SATISFIES THE SUFFICIENT DECREASE CONDITION. */

/*     THE SUBROUTINE STATEMENT IS */

/*        SUBROUTINE MCSRCH(N,X,F,G,S,STP,FTOL,XTOL, MAXFEV,INFO,NFEV,WA) */
/*     WHERE */

/*       N IS A POSITIVE INTEGER INPUT VARIABLE SET TO THE NUMBER */
/*         OF VARIABLES. */

/*       X IS AN ARRAY OF LENGTH N. ON INPUT IT MUST CONTAIN THE */
/*         BASE POINT FOR THE LINE SEARCH. ON OUTPUT IT CONTAINS */
/*         X + STP*S. */

/*       F IS A VARIABLE. ON INPUT IT MUST CONTAIN THE VALUE OF F */
/*         AT X. ON OUTPUT IT CONTAINS THE VALUE OF F AT X + STP*S. */

/*       G IS AN ARRAY OF LENGTH N. ON INPUT IT MUST CONTAIN THE */
/*         GRADIENT OF F AT X. ON OUTPUT IT CONTAINS THE GRADIENT */
/*         OF F AT X + STP*S. */

/*       S IS AN INPUT ARRAY OF LENGTH N WHICH SPECIFIES THE */
/*         SEARCH DIRECTION. */

/*       STP IS A NONNEGATIVE VARIABLE. ON INPUT STP CONTAINS AN */
/*         INITIAL ESTIMATE OF A SATISFACTORY STEP. ON OUTPUT */
/*         STP CONTAINS THE FINAL ESTIMATE. */

/*       FTOL AND GTOL ARE NONNEGATIVE INPUT VARIABLES. (In this reverse */
/*         communication implementation GTOL is defined in a COMMON */
/*         statement.) TERMINATION OCCURS WHEN THE SUFFICIENT DECREASE */
/*         CONDITION AND THE DIRECTIONAL DERIVATIVE CONDITION ARE */
/*         SATISFIED. */

/*       XTOL IS A NONNEGATIVE INPUT VARIABLE. TERMINATION OCCURS */
/*         WHEN THE RELATIVE WIDTH OF THE INTERVAL OF UNCERTAINTY */
/*         IS AT MOST XTOL. */

/*       STPMIN AND STPMAX ARE NONNEGATIVE INPUT VARIABLES WHICH */
/*         SPECIFY LOWER AND UPPER BOUNDS FOR THE STEP. (In this reverse */
/*         communication implementatin they are defined in a COMMON */
/*         statement). */

/*       MAXFEV IS A POSITIVE INTEGER INPUT VARIABLE. TERMINATION */
/*         OCCURS WHEN THE NUMBER OF CALLS TO FCN IS AT LEAST */
/*         MAXFEV BY THE END OF AN ITERATION. */

/*       INFO IS AN INTEGER OUTPUT VARIABLE SET AS FOLLOWS: */

/*         INFO = 0  IMPROPER INPUT PARAMETERS. */

/*         INFO =-1  A RETURN IS MADE TO COMPUTE THE FUNCTION AND GRADIENT. */

/*         INFO = 1  THE SUFFICIENT DECREASE CONDITION AND THE */
/*                   DIRECTIONAL DERIVATIVE CONDITION HOLD. */

/*         INFO = 2  RELATIVE WIDTH OF THE INTERVAL OF UNCERTAINTY */
/*                   IS AT MOST XTOL. */

/*         INFO = 3  NUMBER OF CALLS TO FCN HAS REACHED MAXFEV. */

/*         INFO = 4  THE STEP IS AT THE LOWER BOUND STPMIN. */

/*         INFO = 5  THE STEP IS AT THE UPPER BOUND STPMAX. */

/*         INFO = 6  ROUNDING ERRORS PREVENT FURTHER PROGRESS. */
/*                   THERE MAY NOT BE A STEP WHICH SATISFIES THE */
/*                   SUFFICIENT DECREASE AND CURVATURE CONDITIONS. */
/*                   TOLERANCES MAY BE TOO SMALL. */

/*       NFEV IS AN INTEGER OUTPUT VARIABLE SET TO THE NUMBER OF */
/*         CALLS TO FCN. */

/*       WA IS A WORK ARRAY OF LENGTH N. */

/*     SUBPROGRAMS CALLED */

/*       MCSTEP */

/*       FORTRAN-SUPPLIED...ABS,MAX,MIN */

/*     ARGONNE NATIONAL LABORATORY. MINPACK PROJECT. JUNE 1983 */
/*     JORGE J. MORE', DAVID J. THUENTE */

/*     ********** */
    /* Parameter adjustments */
    --wa;
    --s;
    --g;
    --x;

    /* Function Body */
    if (*info == -1) {
      goto L45;
    }
    infoc = 1;

/*     CHECK THE INPUT PARAMETERS FOR ERRORS. */

    if (*n <= 0 || *stp <= zero || *ftol < zero || lb3_1.gtol < zero || *xtol 
          < zero || lb3_1.stpmin < zero || lb3_1.stpmax < lb3_1.stpmin || *
          maxfev <= 0) {
      return 0;
    }

/*     COMPUTE THE INITIAL GRADIENT IN THE SEARCH DIRECTION */
/*     AND CHECK THAT S IS A DESCENT DIRECTION. */

    dginit = zero;
    i__1 = *n;
    for (j = 1; j <= i__1; ++j) {
      dginit += g[j] * s[j];
/* L10: */
    }
    if (dginit >= zero) {
      printf("  THE SEARCH DIRECTION IS NOT A DESCENT DIRECTION\n");
      return 0;
    }

/*     INITIALIZE LOCAL VARIABLES. */

    brackt = FALSE_;
    stage1 = TRUE_;
    *nfev = 0;
    finit = *f;
    dgtest = *ftol * dginit;
    width = lb3_1.stpmax - lb3_1.stpmin;
    width1 = width / p5;
    i__1 = *n;
    for (j = 1; j <= i__1; ++j) {
      wa[j] = x[j];
    }

/*     THE VARIABLES STX, FX, DGX CONTAIN THE VALUES OF THE STEP, */
/*     FUNCTION, AND DIRECTIONAL DERIVATIVE AT THE BEST STEP. */
/*     THE VARIABLES STY, FY, DGY CONTAIN THE VALUE OF THE STEP, */
/*     FUNCTION, AND DERIVATIVE AT THE OTHER ENDPOINT OF */
/*     THE INTERVAL OF UNCERTAINTY. */
/*     THE VARIABLES STP, F, DG CONTAIN THE VALUES OF THE STEP, */
/*     FUNCTION, AND DERIVATIVE AT THE CURRENT STEP. */

    stx = zero;
    fx = finit;
    dgx = dginit;
    sty = zero;
    fy = finit;
    dgy = dginit;

/*     START OF ITERATION. */

L30:

/*        SET THE MINIMUM AND MAXIMUM STEPS TO CORRESPOND */
/*        TO THE PRESENT INTERVAL OF UNCERTAINTY. */

    if (brackt) {
      stmin = min(stx,sty);
      stmax = max(stx,sty);
    } else {
      stmin = stx;
      stmax = *stp + xtrapf * (*stp - stx);
    }

    /*  FORCE THE STEP TO BE WITHIN THE BOUNDS STPMAX AND STPMIN. */

    *stp = max(*stp,lb3_1.stpmin);
    *stp = min(*stp,lb3_1.stpmax);

/*        IF AN UNUSUAL TERMINATION IS TO OCCUR THEN LET */
/*        STP BE THE LOWEST POINT OBTAINED SO FAR. */

    if (brackt && (*stp <= stmin || *stp >= stmax) || *nfev >= *maxfev - 1 || 
          infoc == 0 || brackt && stmax - stmin <= *xtol * stmax) {
      *stp = stx;
    }

/*        EVALUATE THE FUNCTION AND GRADIENT AT STP */
/*        AND COMPUTE THE DIRECTIONAL DERIVATIVE. */
/*        We return to main program to obtain F and G. */

    i__1 = *n;
    for (j = 1; j <= i__1; ++j) {
      x[j] = wa[j] + *stp * s[j];
/* L40: */
    }
    *info = -1;
    return 0;

L45:

    *info = 0;
    ++(*nfev);
    dg = zero;
    i__1 = *n;
    for (j = 1; j <= i__1; ++j) {
      dg += g[j] * s[j];
    }
    ftest1 = finit + *stp * dgtest;

    /*  TEST FOR CONVERGENCE. */

    if (brackt && (*stp <= stmin || *stp >= stmax) || infoc == 0) {
      *info = 6;
    }
    if ( fabs(*stp-lb3_1.stpmax)<1.0e-50 && *f <= ftest1 && dg <= dgtest) {
      *info = 5;
    }
    if ( fabs(*stp-lb3_1.stpmin)<1.0e-50 && (*f > ftest1 || dg >= dgtest)) {
      *info = 4;
    }
    if (*nfev >= *maxfev) {
      *info = 3;
    }
    if (brackt && stmax - stmin <= *xtol * stmax) {
      *info = 2;
    }
    if (*f <= ftest1 && abs(dg) <= lb3_1.gtol * (-dginit)) {
      *info = 1;
    }

    /* CHECK FOR TERMINATION. */

    if (*info != 0) {
      return 0;
    }

/*        IN THE FIRST STAGE WE SEEK A STEP FOR WHICH THE MODIFIED */
/*        FUNCTION HAS A NONPOSITIVE VALUE AND NONNEGATIVE DERIVATIVE. */

    if (stage1 && *f <= ftest1 && dg >= min(*ftol,lb3_1.gtol) * dginit) {
      stage1 = FALSE_;
    }

/*        A MODIFIED FUNCTION IS USED TO PREDICT THE STEP ONLY IF */
/*        WE HAVE NOT OBTAINED A STEP FOR WHICH THE MODIFIED */
/*        FUNCTION HAS A NONPOSITIVE FUNCTION VALUE AND NONNEGATIVE */
/*        DERIVATIVE, AND IF A LOWER FUNCTION VALUE HAS BEEN */
/*        OBTAINED BUT THE DECREASE IS NOT SUFFICIENT. */

    if (stage1 && *f <= fx && *f > ftest1) {

/*           DEFINE THE MODIFIED FUNCTION AND DERIVATIVE VALUES. */

      fm = *f - *stp * dgtest;
      fxm = fx - stx * dgtest;
      fym = fy - sty * dgtest;
      dgm = dg - dgtest;
      dgxm = dgx - dgtest;
      dgym = dgy - dgtest;

/*           CALL CSTEP TO UPDATE THE INTERVAL OF UNCERTAINTY */
/*           AND TO COMPUTE THE NEW STEP. */

      mcstep_(&stx, &fxm, &dgxm, &sty, &fym, &dgym, stp, &fm, &dgm, &brackt,
             &stmin, &stmax, &infoc);

/*           RESET THE FUNCTION AND GRADIENT VALUES FOR F. */

      fx = fxm + stx * dgtest;
      fy = fym + sty * dgtest;
      dgx = dgxm + dgtest;
      dgy = dgym + dgtest;
    }

    else {

/*           CALL MCSTEP TO UPDATE THE INTERVAL OF UNCERTAINTY */
/*           AND TO COMPUTE THE NEW STEP. */

      mcstep_(&stx, &fx, &dgx, &sty, &fy, &dgy, stp, f, &dg, &brackt, &
            stmin, &stmax, &infoc);
    }

/*        FORCE A SUFFICIENT DECREASE IN THE SIZE OF THE */
/*        INTERVAL OF UNCERTAINTY. */

    if (brackt) {
      if ((d__1 = sty - stx, abs(d__1)) >= p66 * width1) {
          *stp = stx + p5 * (sty - stx);
      }
      width1 = width;
      width = (d__1 = sty - stx, abs(d__1));
    }

/*        END OF ITERATION. */

    goto L30;

/*     LAST LINE OF SUBROUTINE MCSRCH. */

} /* mcsrch_ */








int mcstep_(doublereal *stx, doublereal *fx, doublereal *dx, 
            doublereal *sty, doublereal *fy, doublereal *dy, doublereal *stp,  
          doublereal *fp, doublereal *dp, logical *brackt, doublereal *stpmin, 
          doublereal *stpmax, integer *info)
{
    /* System generated locals */
    doublereal d__1, d__2, d__3;

    /* Local variables */
    static doublereal p, q, r__, s, sgnd, stpc, stpf, stpq, gamma, theta;
    static logical bound;

/*     SUBROUTINE MCSTEP */

/*     THE PURPOSE OF MCSTEP IS TO COMPUTE A SAFEGUARDED STEP FOR */
/*     A LINESEARCH AND TO UPDATE AN INTERVAL OF UNCERTAINTY FOR */
/*     A MINIMIZER OF THE FUNCTION. */

/*     THE PARAMETER STX CONTAINS THE STEP WITH THE LEAST FUNCTION */
/*     VALUE. THE PARAMETER STP CONTAINS THE CURRENT STEP. IT IS */
/*     ASSUMED THAT THE DERIVATIVE AT STX IS NEGATIVE IN THE */
/*     DIRECTION OF THE STEP. IF BRACKT IS SET TRUE THEN A */
/*     MINIMIZER HAS BEEN BRACKETED IN AN INTERVAL OF UNCERTAINTY */
/*     WITH ENDPOINTS STX AND STY. */

/*     THE SUBROUTINE STATEMENT IS */

/*       SUBROUTINE MCSTEP(STX,FX,DX,STY,FY,DY,STP,FP,DP,BRACKT, */
/*                        STPMIN,STPMAX,INFO) */

/*     WHERE */

/*       STX, FX, AND DX ARE VARIABLES WHICH SPECIFY THE STEP, */
/*         THE FUNCTION, AND THE DERIVATIVE AT THE BEST STEP OBTAINED */
/*         SO FAR. THE DERIVATIVE MUST BE NEGATIVE IN THE DIRECTION */
/*         OF THE STEP, THAT IS, DX AND STP-STX MUST HAVE OPPOSITE */
/*         SIGNS. ON OUTPUT THESE PARAMETERS ARE UPDATED APPROPRIATELY. */

/*       STY, FY, AND DY ARE VARIABLES WHICH SPECIFY THE STEP, */
/*         THE FUNCTION, AND THE DERIVATIVE AT THE OTHER ENDPOINT OF */
/*         THE INTERVAL OF UNCERTAINTY. ON OUTPUT THESE PARAMETERS ARE */
/*         UPDATED APPROPRIATELY. */

/*       STP, FP, AND DP ARE VARIABLES WHICH SPECIFY THE STEP, */
/*         THE FUNCTION, AND THE DERIVATIVE AT THE CURRENT STEP. */
/*         IF BRACKT IS SET TRUE THEN ON INPUT STP MUST BE */
/*         BETWEEN STX AND STY. ON OUTPUT STP IS SET TO THE NEW STEP. */

/*       BRACKT IS A LOGICAL VARIABLE WHICH SPECIFIES IF A MINIMIZER */
/*         HAS BEEN BRACKETED. IF THE MINIMIZER HAS NOT BEEN BRACKETED */
/*         THEN ON INPUT BRACKT MUST BE SET FALSE. IF THE MINIMIZER */
/*         IS BRACKETED THEN ON OUTPUT BRACKT IS SET TRUE. */

/*       STPMIN AND STPMAX ARE INPUT VARIABLES WHICH SPECIFY LOWER */
/*         AND UPPER BOUNDS FOR THE STEP. */

/*       INFO IS AN INTEGER OUTPUT VARIABLE SET AS FOLLOWS: */
/*         IF INFO = 1,2,3,4,5, THEN THE STEP HAS BEEN COMPUTED */
/*         ACCORDING TO ONE OF THE FIVE CASES BELOW. OTHERWISE */
/*         INFO = 0, AND THIS INDICATES IMPROPER INPUT PARAMETERS. */

/*     SUBPROGRAMS CALLED */

/*       FORTRAN-SUPPLIED ... ABS,MAX,MIN,SQRT */

/*     ARGONNE NATIONAL LABORATORY. MINPACK PROJECT. JUNE 1983 */
/*     JORGE J. MORE', DAVID J. THUENTE */

    *info = 0;

/*     CHECK THE INPUT PARAMETERS FOR ERRORS. */

    if (*brackt && (*stp <= min(*stx,*sty) || *stp >= max(*stx,*sty)) || *dx *
           (*stp - *stx) >= 0.f || *stpmax < *stpmin) {
      return 0;
    }

/*     DETERMINE IF THE DERIVATIVES HAVE OPPOSITE SIGN. */

    sgnd = *dp * (*dx / abs(*dx));

/*     FIRST CASE. A HIGHER FUNCTION VALUE. */
/*     THE MINIMUM IS BRACKETED. IF THE CUBIC STEP IS CLOSER */
/*     TO STX THAN THE QUADRATIC STEP, THE CUBIC STEP IS TAKEN, */
/*     ELSE THE AVERAGE OF THE CUBIC AND QUADRATIC STEPS IS TAKEN. */

    if (*fp > *fx) {

      *info = 1;
      bound = TRUE_;
      theta = (*fx - *fp) * 3 / (*stp - *stx) + *dx + *dp;

        /* Computing MAX */

      d__1 = abs(theta), d__2 = abs(*dx), d__1 = max(d__1,d__2), d__2 = abs(*dp);
      s = max(d__1,d__2);

        /* Computing 2nd power */

      d__1 = theta / s;
      gamma = s * sqrt(d__1 * d__1 - *dx / s * (*dp / s));
      if (*stp < *stx) {
          gamma = -gamma;
      }

      p = gamma - *dx + theta;
      q = gamma - *dx + gamma + *dp;
      r__ = p / q;
      stpc = *stx + r__ * (*stp - *stx);
      stpq = *stx + *dx / ((*fx - *fp) / (*stp - *stx) + *dx) / 2 * (*stp - *stx);

      if ((d__1 = stpc - *stx, abs(d__1)) < (d__2 = stpq - *stx, abs(d__2))){

        stpf = stpc;
      }
 
        else {
        stpf = stpc + (stpq - stpc) / 2;
      }

      *brackt = TRUE_;

/*     SECOND CASE. A LOWER FUNCTION VALUE AND DERIVATIVES OF */
/*     OPPOSITE SIGN. THE MINIMUM IS BRACKETED. IF THE CUBIC */
/*     STEP IS CLOSER TO STX THAN THE QUADRATIC (SECANT) STEP, */
/*     THE CUBIC STEP IS TAKEN, ELSE THE QUADRATIC STEP IS TAKEN. */

    } 

    else if (sgnd < 0.f) {

      *info = 2;
      bound = FALSE_;
      theta = (*fx - *fp) * 3 / (*stp - *stx) + *dx + *dp;
/* Computing MAX */
      d__1 = abs(theta), d__2 = abs(*dx), d__1 = max(d__1,d__2), d__2 = abs(
            *dp);
      s = max(d__1,d__2);
/* Computing 2nd power */
      d__1 = theta / s;
      gamma = s * sqrt(d__1 * d__1 - *dx / s * (*dp / s));
      if (*stp > *stx) {
          gamma = -gamma;
      }
      p = gamma - *dp + theta;
      q = gamma - *dp + gamma + *dx;
      r__ = p / q;
      stpc = *stp + r__ * (*stx - *stp);
      stpq = *stp + *dp / (*dp - *dx) * (*stx - *stp);
      if ((d__1 = stpc - *stp, abs(d__1)) > (d__2 = stpq - *stp, abs(d__2)))
             {
          stpf = stpc;
      } else {
          stpf = stpq;
      }
      *brackt = TRUE_;

/*     THIRD CASE. A LOWER FUNCTION VALUE, DERIVATIVES OF THE */
/*     SAME SIGN, AND THE MAGNITUDE OF THE DERIVATIVE DECREASES. */
/*     THE CUBIC STEP IS ONLY USED IF THE CUBIC TENDS TO INFINITY */
/*     IN THE DIRECTION OF THE STEP OR IF THE MINIMUM OF THE CUBIC */
/*     IS BEYOND STP. OTHERWISE THE CUBIC STEP IS DEFINED TO BE */
/*     EITHER STPMIN OR STPMAX. THE QUADRATIC (SECANT) STEP IS ALSO */
/*     COMPUTED AND IF THE MINIMUM IS BRACKETED THEN THE THE STEP */
/*     CLOSEST TO STX IS TAKEN, ELSE THE STEP FARTHEST AWAY IS TAKEN. */

    }

    else if (abs(*dp) < abs(*dx)) {

      *info = 3;
      bound = TRUE_;
      theta = (*fx - *fp) * 3 / (*stp - *stx) + *dx + *dp;
/* Computing MAX */
      d__1 = abs(theta), d__2 = abs(*dx), d__1 = max(d__1,d__2), d__2 = abs(
            *dp);
      s = max(d__1,d__2);

/*        THE CASE GAMMA = 0 ONLY ARISES IF THE CUBIC DOES NOT TEND */
/*        TO INFINITY IN THE DIRECTION OF THE STEP. */

/* Computing MAX */
/* Computing 2nd power */
      d__3 = theta / s;
      d__1 = 0., d__2 = d__3 * d__3 - *dx / s * (*dp / s);
      gamma = s * sqrt((max(d__1,d__2)));
      if (*stp > *stx) {
          gamma = -gamma;
      }
      p = gamma - *dp + theta;
      q = gamma + (*dx - *dp) + gamma;
      r__ = p / q;
      if (r__ < 0.f && 1.0e-50<fabs(gamma)) {
          stpc = *stp + r__ * (*stx - *stp);
      } else if (*stp > *stx) {
          stpc = *stpmax;
      } else {
          stpc = *stpmin;
      }
      stpq = *stp + *dp / (*dp - *dx) * (*stx - *stp);
      if (*brackt) {
          if ((d__1 = *stp - stpc, abs(d__1)) < (d__2 = *stp - stpq, abs(
                d__2))) {
            stpf = stpc;
          } else {
            stpf = stpq;
          }
      } else {
          if ((d__1 = *stp - stpc, abs(d__1)) > (d__2 = *stp - stpq, abs(
                d__2))) {
            stpf = stpc;
          } else {
            stpf = stpq;
          }
      }

/*     FOURTH CASE. A LOWER FUNCTION VALUE, DERIVATIVES OF THE */
/*     SAME SIGN, AND THE MAGNITUDE OF THE DERIVATIVE DOES */
/*     NOT DECREASE. IF THE MINIMUM IS NOT BRACKETED, THE STEP */
/*     IS EITHER STPMIN OR STPMAX, ELSE THE CUBIC STEP IS TAKEN. */

    }

    else {

      *info = 4;
      bound = FALSE_;
      if (*brackt) {
          theta = (*fp - *fy) * 3 / (*sty - *stp) + *dy + *dp;
/* Computing MAX */
          d__1 = abs(theta), d__2 = abs(*dy), d__1 = max(d__1,d__2), d__2 = 
                abs(*dp);
          s = max(d__1,d__2);
/* Computing 2nd power */
          d__1 = theta / s;
          gamma = s * sqrt(d__1 * d__1 - *dy / s * (*dp / s));
          if (*stp > *sty) {
            gamma = -gamma;
          }
          p = gamma - *dp + theta;
          q = gamma - *dp + gamma + *dy;
          r__ = p / q;
          stpc = *stp + r__ * (*sty - *stp);
          stpf = stpc;
      } else if (*stp > *stx) {
          stpf = *stpmax;
      } else {
          stpf = *stpmin;
      }

    }

/*     UPDATE THE INTERVAL OF UNCERTAINTY. THIS UPDATE DOES NOT */
/*     DEPEND ON THE NEW STEP OR THE CASE ANALYSIS ABOVE. */

    if (*fp > *fx) {
      *sty = *stp;
      *fy = *fp;
      *dy = *dp;
    } else {
      if (sgnd < 0.f) {
          *sty = *stx;
          *fy = *fx;
          *dy = *dx;
      }
      *stx = *stp;
      *fx = *fp;
      *dx = *dp;
    }

/*     COMPUTE THE NEW STEP AND SAFEGUARD IT. */

    stpf = min(*stpmax,stpf);
    stpf = max(*stpmin,stpf);
    *stp = stpf;

    if (*brackt && bound) {
      if (*sty > *stx) {
/* Computing MIN */
          d__1 = *stx + (*sty - *stx) * .66f;
          *stp = min(d__1,*stp);
      } else {
/* Computing MAX */
          d__1 = *stx + (*sty - *stx) * .66f;
          *stp = max(d__1,*stp);
      }
    }
    return 0;

/*     LAST LINE OF SUBROUTINE MCSTEP. */

} /* mcstep_ */


Generated by  Doxygen 1.6.0   Back to index