Logo Search packages:      
Sourcecode: r-base-core-ra version File versions  Download package

lbfgsb.c

/*
 *  R : A Computer Language for Statistical Data Analysis
 *  Copyright (C) 2000-4 The R Development Core Team
 *
 *  This program is free software; you can redistribute it and/or modify
 *  it under the terms of the GNU General Public License as published by
 *  the Free Software Foundation; either version 2 of the License, or
 *  (at your option) any later version.
 *
 *  This program is distributed in the hope that it will be useful,
 *  but WITHOUT ANY WARRANTY; without even the implied warranty of
 *  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
 *  GNU General Public License for more details.
 *
 *  You should have received a copy of the GNU General Public License
 *  along with this program; if not, a copy is available at
 *  http://www.r-project.org/Licenses/
 */
/* l-bfgs-b.f -- translated by f2c (version 19991025).
*/

/* <UTF8> all char uses here are ASCII */

/* Exports setulb()
 *         ======
 * called from lbfgsb() in ../main/optim.c */
#include <math.h>
#include <float.h> /* for DBL_EPSILON */
#include <string.h>
#include <R_ext/RS.h> /* for F77_CALL */
#include <R_ext/Linpack.h>
#include <R_ext/Applic.h>
#include <R_ext/PrtUtil.h> /* Rprintf */

static void timer(double * ttime)
{
    *ttime = 0.0;
}

#define FALSE_ 0
#define TRUE_ 1
#ifndef max
# define max(a, b) (a < b)?(b):(a)
# define min(a, b) (a > b)?(b):(a)
#endif

/* Constants -- needed only as pointer arguments to the BLAS routines
 * --------- Declaring them "const" would give compiler warnings
 * unless the F77_CALL(d....) headers were changed too */
static int c__1 = 1;
static int c__11 = 11;

/*
extern double F77_NAME(ddot)(int *, double *, int *, double *, int *);
extern void F77_NAME(dscal)(int *, double *, double *, int *);
extern void F77_NAME(dcopy)(int *, double *, int *, double *, int *);
extern void F77_NAME(daxpy)(int *, double *, double *, int *, double *, int *);
extern void F77_NAME(dpofa)(double *, int *, int *, int *);
extern void F77_NAME(dtrsl)(double *, int *, int *, double *, int *, int *);
*/

static void active(int, double *, double *, int *, double *, int *,
               int, int *, int *, int *);
static void bmv(int, double *, double *, int *, double *, double *, int *);
static void cauchy(int, double *, double *,
               double *, int *, double *, int *, int *,
               double *, double *, double *, int, double *,
               double *, double *, double *, double *, int * ,
               int *, double *, double *, double *, double *,
               int *, int, double *, int *, double *);
static void cmprlb(int, int, double *,
               double *, double *, double *, double *,
               double *, double *, double *, double *, int *,
               double *, int *, int *, int *, int *,
               int *);
static void dcsrch(double *, double *, double *,
               double, double, double,
               double, double,
               char *, int *, double *);
static void dcstep(double *, double *,
               double *, double *, double *, double *,
               double *, double *, double *, int *, double *,
               double *);
#ifdef NOT_USING_DBL_EPSILON
 static double dpmeps(void);
#endif
static void errclb(int, int, double,
               double *, double *, int *, char *, int *, int *);
static void formk(int, int *, int *, int *, int *, int *, int *,
              int *, double *, double *, int, double *,
              double *, double *, double *, int *, int *, int *);
static void formt(int, double *, double *,
              double *, int *, double *, int *);
static void freev(int, int *, int *,
              int *, int *, int *, int *, int *, int *,
              int *, int, int *);
static void hpsolb(int, double *, int *, int);
static void lnsrlb(int, double *, double *,
               int *, double *, double *, double *, double *,
               double *, double *, double *, double *,
               double *, double *, double *, double *,
               double *, double *, double *, int *, int *,
               int *, int *, int *, char *, int *, int *,
               char *, int *, double *);
static void mainlb(int, int, double *,
               double *, double *, int *, double *, double *,
               double, double *, double *, double *,
               double *, double *, double *, double *,
               double *, double *, double *, double *,
               double *, double *, int *, int *, int *, char *,
               int, char *, int *, int *, double *);
static void matupd(int, int, double *, double *, double *,
               double *, double *, double *, int *, int *,
               int *, int *, double *, double *, double *,
               double *, double *);
static void projgr(int, double *, double *,
               int *, double *, double *, double *);
static void subsm(int, int, int *, int *, double *, double *,
              int *, double *, double *, double *, double *,
              double *, int *, int *, int *, double *,
              double *, int, int *);

static void prn1lb(int n, int m, double *l, double *u, double *x,
               int iprint, double epsmch);
static void prn2lb(int n, double *x, double *f, double *g, int iprint,
               int iter, int nfgv, int nact, double sbgnrm,
               int nint, char *word, int iword, int iback,
               double stp, double xstep);
static void prn3lb(int n, double *x, double *f, char *task, int iprint,
               int info, int iter, int nfgv, int nintol, int nskip,
               int nact, double sbgnrm, int nint,
               char *word, int iback, double stp, double xstep,
               int k);


/* ================    L-BFGS-B (version 2.3)   ========================== */
void setulb(int n, int m, double *x, double *l, double *u, int *nbd,
          double *f, double *g, double factr, double *pgtol,
          double *wa, int * iwa, char *task, int iprint,
          int *lsave, int *isave, double *dsave)
{
/*     ************

       Subroutine setulb

       This subroutine partitions the working arrays wa and iwa, and
       then uses the limited memory BFGS method to solve the bound
       constrained optimization problem by calling mainlb.
       (The direct method will be used in the subspace minimization.)

       n is an integer variable.
       On entry n is the dimension of the problem.
       On exit n is unchanged.

       m is an integer variable.
       On entry m is the maximum number of variable metric corrections
         used to define the limited memory matrix.
       On exit m is unchanged.

       x is a double precision array of dimension n.
       On entry x is an approximation to the solution.
       On exit x is the current approximation.

       l is a double precision array of dimension n.
       On entry l is the lower bound on x.
       On exit l is unchanged.

       u is a double precision array of dimension n.
       On entry u is the upper bound on x.
       On exit u is unchanged.

       nbd is an integer array of dimension n.
       On entry nbd represents the type of bounds imposed on the
         variables, and must be specified as follows:
         nbd(i)=0 if x(i) is unbounded,
              1 if x(i) has only a lower bound,
              2 if x(i) has both lower and upper bounds, and
              3 if x(i) has only an upper bound.
       On exit nbd is unchanged.

       f is a double precision variable.
       On first entry f is unspecified.
       On final exit f is the value of the function at x.

       g is a double precision array of dimension n.
       On first entry g is unspecified.
       On final exit g is the value of the gradient at x.

       factr is a double precision variable.
       On entry factr >= 0 is specified by the user.  The iteration
         will stop when

         (f^k - f^{k+1})/max{|f^k|,|f^{k+1}|,1} <= factr*epsmch

         where epsmch is the machine precision, which is automatically
         generated by the code. Typical values for factr: 1.d+12 for
         low accuracy; 1.d+7 for moderate accuracy; 1.d+1 for extremely
         high accuracy.
       On exit factr is unchanged.

       pgtol is a double precision variable.
       On entry pgtol >= 0 is specified by the user.  The iteration
         will stop when

               max{|proj g_i | i = 1, ..., n} <= pgtol

         where pg_i is the ith component of the projected gradient.
       On exit pgtol is unchanged.

       wa is a double precision working array of length
       (2mmax + 4)nmax + 11mmax^2 + 8mmax.

       iwa is an integer working array of length 3nmax.

       task is a working string of characters of length 60 indicating
       the current job when entering and quitting this subroutine.

       iprint is an integer variable that must be set by the user.
       It controls the frequency and type of output generated:
        iprint<0    no output is generated;
        iprint=0    print only one line at the last iteration;
        0<iprint<99 print also f and |proj g| every iprint iterations;
        iprint=99   print details of every iteration except n-vectors;
        iprint=100  print also the changes of active set and final x;
        iprint>100  print details of every iteration including x and g;
       When iprint > 0, the file iterate.dat will be created to
                    summarize the iteration.

       csave is a working string of characters of length 60.

       lsave is a logical working array of dimension 4.
       On exit with 'task' = NEW_X, the following information is
                                                 available:
         If lsave(1) = .true. then  the initial X has been replaced by
                              its projection in the feasible set;
         If lsave(2) = .true. then  the problem is constrained;
         If lsave(3) = .true. then  each variable has upper and lower
                              bounds;

       isave is an integer working array of dimension 44.
       On exit with 'task' = NEW_X, the following information is
                                                 available:
         isave(22) = the total number of intervals explored in the
                     search of Cauchy points;
         isave(26) = the total number of skipped BFGS updates before
                     the current iteration;
         isave(30) = the number of current iteration;
         isave(31) = the total number of BFGS updates prior the current
                     iteration;
         isave(33) = the number of intervals explored in the search of
                     Cauchy point in the current iteration;
         isave(34) = the total number of function and gradient
                     evaluations;
         isave(36) = the number of function value or gradient
                     evaluations in the current iteration;
         if isave(37) = 0  then the subspace argmin is within the box;
         if isave(37) = 1  then the subspace argmin is beyond the box;
         isave(38) = the number of free variables in the current
                     iteration;
         isave(39) = the number of active constraints in the current
                     iteration;
         n + 1 - isave(40) = the number of variables leaving the set of
                       active constraints in the current iteration;
         isave(41) = the number of variables entering the set of active
                     constraints in the current iteration.

       dsave is a double precision working array of dimension 29.
       On exit with 'task' = NEW_X, the following information is
                                                 available:
         dsave(1) = current 'theta' in the BFGS matrix;
         dsave(2) = f(x) in the previous iteration;
         dsave(3) = factr*epsmch;
         dsave(4) = 2-norm of the line search direction vector;
         dsave(5) = the machine precision epsmch generated by the code;
         dsave(7) = the accumulated time spent on searching for
                                             Cauchy points;
         dsave(8) = the accumulated time spent on
                                       subspace minimization;
         dsave(9) = the accumulated time spent on line search;
         dsave(11) = the slope of the line search function at
                            the current point of line search;
         dsave(12) = the maximum relative step length imposed in
                                               line search;
         dsave(13) = the infinity norm of the projected gradient;
         dsave(14) = the relative step length in the line search;
         dsave(15) = the slope of the line search function at
                           the starting point of the line search;
         dsave(16) = the square of the 2-norm of the line search
                                          direction vector.

       Subprograms called:

       L-BFGS-B Library ... mainlb.


       References:

       [1] R. H. Byrd, P. Lu, J. Nocedal and C. Zhu, ``A limited
       memory algorithm for bound constrained optimization'',
       SIAM J. Scientific Computing 16 (1995), no. 5, pp. 1190--1208.

       [2] C. Zhu, R.H. Byrd, P. Lu, J. Nocedal, ``L-BFGS-B: a
       limited memory FORTRAN code for solving bound constrained
       optimization problems'', Tech. Report, NAM-11, EECS Department,
       Northwestern University, 1994.

       (Postscript files of these papers are available via anonymous
        ftp to ece.nwu.edu in the directory pub/lbfgs/lbfgs_bcm.)

        [Aug 2000: via http://www.ece.nwu.edu/~nocedal/lbfgsb.html]

                       *      *  *

       NEOS, November 1994. (Latest revision April 1997.)
       Optimization Technology Center.
       Argonne National Laboratory and Northwestern University.
       Written by
                    Ciyou Zhu
       in collaboration with R.H. Byrd, P. Lu-Chen and J. Nocedal.

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

    char csave[60];

    /* Local variables */
    int lsnd, l1, l2, l3, ld, lr, lt;
    int lz, lwa, lwn, lss, lws, lwt, lsy, lwy;

    /* make sure csave is initialized */
    csave[0] = '\0';
    
    /* Parameter adjustments */
    --wa;
    --isave;

    /* Function Body */
    if (strncmp(task, "START", 5) == 0) {
      isave[1] = m * n;
      isave[2] = m * m;
      isave[3] = m * m << 2;
      isave[4] = 1;
      isave[5] = isave[4] + isave[1];
      isave[6] = isave[5] + isave[1];
      isave[7] = isave[6] + isave[2];
      isave[8] = isave[7] + isave[2];
      isave[9] = isave[8];
      isave[10] = isave[9] + isave[2];
      isave[11] = isave[10] + isave[3];
      isave[12] = isave[11] + isave[3];
      isave[13] = isave[12] + n;
      isave[14] = isave[13] + n;
      isave[15] = isave[14] + n;
      isave[16] = isave[15] + n;
    }
    l1 = isave[1];
    l2 = isave[2];
    l3 = isave[3];
    lws = isave[4];
    lwy = isave[5];
    lsy = isave[6];
    lss = isave[7];
    lwt = isave[9];
    lwn = isave[10];
    lsnd = isave[11];
    lz = isave[12];
    lr = isave[13];
    ld = isave[14];
    lt = isave[15];
    lwa = isave[16];
    mainlb(n, m, x, l, u, nbd, f, g, factr, pgtol,
         &wa[lws], &wa[lwy], &wa[lsy],&wa[lss], &wa[lwt],&wa[lwn],
         &wa[lsnd], &wa[lz], &wa[lr], &wa[ld], &wa[lt], &wa[lwa],
         iwa, &iwa[n], &iwa[n << 1], task, iprint,
         csave, lsave, &isave[22], dsave);
    return;
} /* setulb */
/* ======================= The end of setulb ============================= */

static void mainlb(int n, int m, double *x,
               double *l, double *u, int *nbd, double *f, double *g,
               double factr, double *pgtol, double *ws, double * wy,
               double *sy, double *ss, double *wt, double *wn,
               double *snd, double *z, double *r, double *d,
               double *t, double *wa, int *indx, int *iwhere,
               int *indx2, char *task, int iprint,
               char *csave, int *lsave, int *isave, double *dsave)
{
/*     ************
       Subroutine mainlb

       This subroutine solves bound constrained optimization problems by
       using the compact formula of the limited memory BFGS updates.

       n is an integer variable.
       On entry n is the number of variables.
       On exit n is unchanged.

       m is an integer variable.
       On entry m is the maximum number of variable metric
          corrections allowed in the limited memory matrix.
       On exit m is unchanged.

       x is a double precision array of dimension n.
       On entry x is an approximation to the solution.
       On exit x is the current approximation.

       l is a double precision array of dimension n.
       On entry l is the lower bound of x.
       On exit l is unchanged.

       u is a double precision array of dimension n.
       On entry u is the upper bound of x.
       On exit u is unchanged.

       nbd is an integer array of dimension n.
       On entry nbd represents the type of bounds imposed on the
         variables, and must be specified as follows:
         nbd(i)=0 if x(i) is unbounded,
              1 if x(i) has only a lower bound,
              2 if x(i) has both lower and upper bounds,
              3 if x(i) has only an upper bound.
       On exit nbd is unchanged.

       f is a double precision variable.
       On first entry f is unspecified.
       On final exit f is the value of the function at x.

       g is a double precision array of dimension n.
       On first entry g is unspecified.
       On final exit g is the value of the gradient at x.

       factr is a double precision variable.
       On entry factr >= 0 is specified by the user.  The iteration
         will stop when

         (f^k - f^{k+1})/max{|f^k|,|f^{k+1}|,1} <= factr*epsmch

         where epsmch is the machine precision, which is automatically
         generated by the code.
       On exit factr is unchanged.

       pgtol is a double precision variable.
       On entry pgtol >= 0 is specified by the user.  The iteration
         will stop when

               max{|proj g_i | i = 1, ..., n} <= pgtol

         where pg_i is the ith component of the projected gradient.
       On exit pgtol is unchanged.

       ws, wy, sy, and wt are double precision working arrays used to
       store the following information defining the limited memory
          BFGS matrix:
          ws, of dimension n x m, stores S, the matrix of s-vectors;
          wy, of dimension n x m, stores Y, the matrix of y-vectors;
          sy, of dimension m x m, stores S'Y;
          ss, of dimension m x m, stores S'S;
          wt, of dimension m x m, stores the Cholesky factorization
                            of (theta*S'S+LD^(-1)L'); see eq.
                            (2.26) in [3].

       wn is a double precision working array of dimension 2m x 2m
       used to store the LEL^T factorization of the indefinite matrix
               K = [-D -Y'ZZ'Y/theta     L_a'-R_z'    ]
                   [L_a -R_z     theta*S'AA'S ]

       where         E = [-I  0]
                   [ 0  I]

       snd is a double precision working array of dimension 2m x 2m
       used to store the lower triangular part of
               N = [Y' ZZ'Y     L_a'+R_z']
                   [L_a +R_z  S'AA'S   ]

       z(n),r(n),d(n),t(n),wa(8*m) are double precision working arrays.
       z is used at different times to store the Cauchy point and
       the Newton point.


       indx is an integer working array of dimension n.
       In subroutine freev, indx is used to store the free and fixed
          variables at the Generalized Cauchy Point (GCP).

       iwhere is an integer working array of dimension n used to record
       the status of the vector x for GCP computation.
       iwhere(i)=0 or -3 if x(i) is free and has bounds,
               1     if x(i) is fixed at l(i), and l(i) .ne. u(i)
               2     if x(i) is fixed at u(i), and u(i) .ne. l(i)
               3     if x(i) is always fixed, i.e.,  u(i)=x(i)=l(i)
              -1     if x(i) is always free, i.e., no bounds on it.

       indx2 is an integer working array of dimension n.
       Within subroutine cauchy, indx2 corresponds to the array iorder.
       In subroutine freev, a list of variables entering and leaving
       the free set is stored in indx2, and it is passed on to
       subroutine formk with this information.

       task is a working string of characters of length 60 indicating
       the current job when entering and leaving this subroutine.

       iprint is an INTEGER variable that must be set by the user.
       It controls the frequency and type of output generated:
        iprint<0    no output is generated;
        iprint=0    print only one line at the last iteration;
        0<iprint<99 print also f and |proj g| every iprint iterations;
        iprint=99   print details of every iteration except n-vectors;
        iprint=100  print also the changes of active set and final x;
        iprint>100  print details of every iteration including x and g;
       When iprint > 0, the file iterate.dat will be created to
                    summarize the iteration.

       csave is a working string of characters of length 60.

       lsave is a logical working array of dimension 4.

       isave is an integer working array of dimension 23.

       dsave is a double precision working array of dimension 29.


       Subprograms called

       L-BFGS-B Library ... cauchy, subsm, lnsrlb, formk,

        errclb, prn1lb, prn2lb, prn3lb, active, projgr,

        freev, cmprlb, matupd, formt.

       Minpack2 Library ... timer, dpmeps.

       Linpack Library ... dcopy, ddot.


       References:

      [1] R. H. Byrd, P. Lu, J. Nocedal and C. Zhu, ``A limited
      memory algorithm for bound constrained optimization'',
      SIAM J. Scientific Computing 16 (1995), no. 5, pp. 1190--1208.

      [2] C. Zhu, R.H. Byrd, P. Lu, J. Nocedal, ``L-BFGS-B: FORTRAN
      Subroutines for Large Scale Bound Constrained Optimization''
      Tech. Report, NAM-11, EECS Department, Northwestern University,
      1994.

      [3] R. Byrd, J. Nocedal and R. Schnabel "Representations of
      Quasi-Newton Matrices and their use in Limited Memory Methods'',
      Mathematical Programming 63 (1994), no. 4, pp. 129-156.

      (Postscript files of these papers are available via anonymous
       ftp to ece.nwu.edu in the directory pub/lbfgs/lbfgs_bcm.)

                     *  *      *
*/

/*
      NEOS, November 1994. (Latest revision April 1997.)
      Optimization Technology Center.
      Argonne National Laboratory and Northwestern University.
      Written by
                     Ciyou Zhu
      in collaboration with R.H. Byrd, P. Lu-Chen and J. Nocedal.

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

    /* System generated locals */
    int ws_offset=0, wy_offset=0, sy_offset=0, ss_offset=0, wt_offset=0,
      wn_offset=0, snd_offset=0, i__1;
    double d__1, d__2;

    /* Local variables */
    int head;
    double fold;
    int nact;
    double ddum;
    int info;
    double time;
    int nfgv, ifun, iter, nint;
    char word[4]; /* allow for terminator */
    double time1, time2;
    int i, iback, k = 0; /* -Wall */
    double gdold;
    int nfree;
    int boxed;
    int itail;
    double theta;
    double dnorm;
    int nskip, iword;
    double xstep = 0.0, stpmx; /* xstep is printed before being used */
    double gd, dr, rr;
    int ileave;
    int itfile;
    double cachyt, epsmch;
    int updatd;
    double sbtime;
    int prjctd;
    int iupdat;
    int cnstnd;
    double sbgnrm;
    int nenter;
    double lnscht;
    int nintol;
    double dtd;
    int col;
    double tol;
    int wrk;
    double stp, cpu1, cpu2;

    /* Parameter adjustments */
    --indx2;
    --iwhere;
    --indx;
    --t;
    --d;
    --r;
    --z;
    --g;
    --nbd;
    --u;
    --l;
    --x;
    --wa;
    --lsave;
    --isave;
    --dsave;

    /* Function Body */
    if (strncmp(task, "START", 5) == 0) {
      timer(&time1);
/*      Generate the current machine precision. */
#ifdef NOT_USING_DBL_EPSILON
      epsmch = dpmeps();
#else
      epsmch = DBL_EPSILON;
#endif
      fold = 0.;
      dnorm = 0.;
      cpu1 = 0.;
      gd = 0.;
      sbgnrm = 0.;
      stp = 0.;
      xstep = 0.;
      stpmx = 0.;
      gdold = 0.;
      dtd = 0.;
/*      Initialize counters and scalars when task='START'. */
/*         for the limited memory BFGS matrices: */
      col = 0;
      head = 1;
      theta = 1.;
      iupdat = 0;
      updatd = FALSE_;
      iback = 0;
      itail = 0;
      ifun = 0;
      iword = 0;
      nact = 0;
      ileave = 0;
      nenter = 0;
/*         for operation counts: */
      iter = 0;
      nfgv = 0;
      nint = 0;
      nintol = 0;
      nskip = 0;
      nfree = n;
/*         for stopping tolerance: */
      tol = factr * epsmch;
/*         for measuring running time: */
      cachyt = 0.;
      sbtime = 0.;
      lnscht = 0.;
/*         'word' records the status of subspace solutions. */
      strcpy(word, "---");
/*         'info' records the termination information. */
      info = 0;
      itfile = 0;
/*      Check the input arguments for errors. */
      errclb(n, m, factr, &l[1], &u[1], &nbd[1], task, &info, &k);
      if (strncmp(task, "ERROR", 5) == 0) {
          prn3lb(n, x+1, f, task, iprint, info,
               iter, nfgv, nintol, nskip, nact, sbgnrm,
               nint, word, iback, stp, xstep, k);
          return;
      }

      prn1lb(n, m, l+1, u+1, x+1, iprint, epsmch);

/*      Initialize iwhere & project x onto the feasible set. */
      active(n, &l[1], &u[1], &nbd[1], &x[1], &iwhere[1], iprint, &prjctd,
            &cnstnd, &boxed);
/*      The end of the initialization. */
    } else {
/*        restore local variables. */
      prjctd = lsave[1];
      cnstnd = lsave[2];
      boxed = lsave[3];
      updatd = lsave[4];

      nintol = isave[1];
      itfile = isave[3];
      iback = isave[4];
      nskip = isave[5];
      head = isave[6];
      col = isave[7];
      itail = isave[8];
      iter = isave[9];
      iupdat = isave[10];
      nint = isave[12];
      nfgv = isave[13];
      info = isave[14];
      ifun = isave[15];
      iword = isave[16];
      nfree = isave[17];
      nact = isave[18];
      ileave = isave[19];
      nenter = isave[20];

      theta = dsave[1];
      fold = dsave[2];
      tol = dsave[3];
      dnorm = dsave[4];
      epsmch = dsave[5];
      cpu1 = dsave[6];
      cachyt = dsave[7];
      sbtime = dsave[8];
      lnscht = dsave[9];
      time1 = dsave[10];
      gd = dsave[11];
      stpmx = dsave[12];
      sbgnrm = dsave[13];
      stp = dsave[14];
      gdold = dsave[15];
      dtd = dsave[16];
/*    After returning from the driver go to the point where execution */
/*    is to resume. */
      if (strncmp(task, "FG_LN", 5) == 0) goto L666;
      if (strncmp(task, "NEW_X", 5) == 0)     goto L777;
      if (strncmp(task, "FG_ST", 5) == 0)     goto L111;

      if (strncmp(task, "STOP", 4) == 0) {
          if (strncmp(task + 6, "CPU", 3) == 0) {
            /* restore the previous iterate. */
            F77_CALL(dcopy)(&n, &t[1], &c__1, &x[1], &c__1);
            F77_CALL(dcopy)(&n, &r[1], &c__1, &g[1], &c__1);
            *f = fold;
          }
          goto L999;
      }
    }
/*     Compute f0 and g0. */
    strcpy(task, "FG_START");
/*        return to the driver to calculate f and g; reenter at 111. */
    goto L1000;
L111:
    nfgv = 1;
/*     Compute the infinity norm of the (-) projected gradient. */
    projgr(n, &l[1], &u[1], &nbd[1], &x[1], &g[1], &sbgnrm);

    if (iprint >= 1)
      Rprintf("At iterate %5d  f= %12.5g  |proj g|= %12.5g\n",
            iter, *f, sbgnrm);

    if (sbgnrm <= *pgtol) {
/*                        terminate the algorithm. */
      strcpy(task, "CONVERGENCE: NORM OF PROJECTED GRADIENT <= PGTOL");
      goto L999;
    }
/* ----------------- the beginning of the loop -------------------------- */
L222:
    if (iprint >= 99) Rprintf("Iteration %5d\n", iter);
    iword = -1;

    if (! cnstnd && col > 0) {
/*                                  skip the search for GCP. */
      F77_CALL(dcopy)(&n, &x[1], &c__1, &z[1], &c__1);
      wrk = updatd;
      nint = 0;
      goto L333;
    }
/* ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc */

/*     Compute the Generalized Cauchy Point (GCP). */

/* ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc */
    timer(&cpu1);
    cauchy(n, &x[1], &l[1], &u[1], &nbd[1], &g[1], &indx2[1], &iwhere[1], &t[
          1], &d[1], &z[1], m, &wy[wy_offset], &ws[ws_offset], &sy[
          sy_offset], &wt[wt_offset], &theta, &col, &head, &wa[1], &wa[(m
          << 1) + 1], &wa[(m << 2) + 1], &wa[m * 6 + 1], &nint, iprint, &
          sbgnrm, &info, &epsmch);
    if (info != 0) {
/*       singular triangular system detected; refresh the lbfgs memory. */
      if (iprint >= 1)
          Rprintf("%s\n%s\n", "Singular triangular system detected;",
                "   refresh the lbfgs memory and restart the iteration.");
      info = 0;
      col = 0;
      head = 1;
      theta = 1.;
      iupdat = 0;
      updatd = FALSE_;
      timer(&cpu2);
      cachyt = cachyt + cpu2 - cpu1;
      goto L222;
    }
    timer(&cpu2);
    cachyt = cachyt + cpu2 - cpu1;
    nintol += nint;
/*     Count the entering and leaving variables for iter > 0; */
/*     find the index set of free and active variables at the GCP. */
    freev(n, &nfree, &indx[1], &nenter, &ileave, &indx2[1], &iwhere[1], &
          wrk, &updatd, &cnstnd, iprint, &iter);
    nact = n - nfree;
L333:
/*     If there are no free variables or B=theta*I, then */
/*                              skip the subspace minimization. */
    if (nfree == 0 || col == 0) {
      goto L555;
    }
/* ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc */

/*     Subspace minimization. */

/* ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc */
    timer(&cpu1);
/*     Form  the LEL^T factorization of the indefinite */
/*     matrix        K = [-D -Y'ZZ'Y/theta     L_a'-R_z'    ] */
/*                 [L_a -R_z     theta*S'AA'S ] */
/*     where         E = [-I  0] */
/*                 [ 0  I] */
    if (wrk) {
      formk(n, &nfree, &indx[1], &nenter, &ileave, &indx2[1], &iupdat, &
            updatd, &wn[wn_offset], &snd[snd_offset], m, &ws[ws_offset], &
            wy[wy_offset], &sy[sy_offset], &theta, &col, &head, &info);
    }
    if (info != 0) {
/*        nonpositive definiteness in Cholesky factorization; */
/*        refresh the lbfgs memory and restart the iteration. */
      if (iprint >= 0)
          Rprintf("%s\n%s\n",
                "Nonpositive definiteness in Cholesky factorization in formk;",
                "   refresh the lbfgs memory and restart the iteration.");
      info = 0;
      col = 0;
      head = 1;
      theta = 1.;
      iupdat = 0;
      updatd = FALSE_;
      timer(&cpu2);
      sbtime = sbtime + cpu2 - cpu1;
      goto L222;
    }
/*      compute r=-Z'B(xcp-xk)-Z'g (using wa(2m+1)=W'(xcp-x) */
/*                                       from 'cauchy'). */
    cmprlb(n, m, &x[1], &g[1], &ws[ws_offset], &wy[wy_offset], &sy[sy_offset]
          , &wt[wt_offset], &z[1], &r[1], &wa[1], &indx[1], &theta, &
          col, &head, &nfree, &cnstnd, &info);
    if (info != 0) {
      goto L444;
    }
/*     call the direct method. */
    subsm(n, m, &nfree, &indx[1], &l[1], &u[1], &nbd[1], &z[1], &r[1], &
          ws[ws_offset], &wy[wy_offset], &theta, &col, &head, &iword, &wa[1]
          , &wn[wn_offset], iprint, &info);
L444:
    if (info != 0) {
/*        singular triangular system detected; */
/*        refresh the lbfgs memory and restart the iteration. */
      if (iprint >= 1)
          Rprintf("%s\n%s\n", "Singular triangular system detected;",
                "   refresh the lbfgs memory and restart the iteration.");
      info = 0;
      col = 0;
      head = 1;
      theta = 1.;
      iupdat = 0;
      updatd = FALSE_;
      timer(&cpu2);
      sbtime = sbtime + cpu2 - cpu1;
      goto L222;
    }
    timer(&cpu2);
    sbtime = sbtime + cpu2 - cpu1;
L555:
/* ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc */

/*     Line search and optimality tests. */

/* ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc */
/*     Generate the search direction d:=z-x. */
    i__1 = n;
    for (i = 1; i <= i__1; ++i) {
      d[i] = z[i] - x[i];
/* L40: */
    }
    timer(&cpu1);
L666:
    lnsrlb(n, &l[1], &u[1], &nbd[1], &x[1], f, &fold, &gd, &gdold, &g[1], &
          d[1], &r[1], &t[1], &z[1], &stp, &dnorm, &dtd, &xstep, &
          stpmx, &iter, &ifun, &iback, &nfgv, &info, task, &boxed, &cnstnd,
          csave, &isave[22], &dsave[17]);
    if (info != 0 || iback >= 20) {
/*        restore the previous iterate. */
      F77_CALL(dcopy)(&n, &t[1], &c__1, &x[1], &c__1);
      F77_CALL(dcopy)(&n, &r[1], &c__1, &g[1], &c__1);
      *f = fold;
      if (col == 0) {
/*           abnormal termination. */
          if (info == 0) {
            info = -9;
/*            restore the actual number of f and g evaluations etc. */
            --nfgv;
            --ifun;
            --iback;
          }
          strcpy(task, "ERROR: ABNORMAL_TERMINATION_IN_LNSRCH");
          ++iter;
          goto L999;
      } else {
/*           refresh the lbfgs memory and restart the iteration. */
          if (iprint >= 1)
            Rprintf("%s\n%s\n", "Bad direction in the line search;",
                "   refresh the lbfgs memory and restart the iteration.");
          if (info == 0) {
            --nfgv;
          }
          info = 0;
          col = 0;
          head = 1;
          theta = 1.;
          iupdat = 0;
          updatd = FALSE_;
          strcpy(task, "RESTART_FROM_LNSRCH");
          timer(&cpu2);
          lnscht = lnscht + cpu2 - cpu1;
          goto L222;
      }
    } else if (strncmp(task, "FG_LN", 5) == 0) {
/*        return to the driver for calculating f and g; reenter at 666. */
      goto L1000;
    } else {
/*        calculate and print out the quantities related to the new X. */
      timer(&cpu2);
      lnscht = lnscht + cpu2 - cpu1;
      ++iter;
/*      Compute the infinity norm of the projected (-)gradient. */
      projgr(n, &l[1], &u[1], &nbd[1], &x[1], &g[1], &sbgnrm);
/*      Print iteration information. */
      prn2lb(n, x+1, f, g+1, iprint, iter, nfgv, nact,
             sbgnrm, nint, word, iword, iback, stp, xstep);
      goto L1000;
    }
L777:
/*     Test for termination. */
    if (sbgnrm <= *pgtol) {
/*                        terminate the algorithm. */
      strcpy(task, "CONVERGENCE: NORM OF PROJECTED GRADIENT <= PGTOL");
      goto L999;
    }
/* Computing MAX */
    d__1 = fabs(fold), d__2 = fabs(*f), d__1 = max(d__1,d__2);
    ddum = max(d__1,1.);
    if (fold - *f <= tol * ddum) {
/*                              terminate the algorithm. */
      strcpy(task, "CONVERGENCE: REL_REDUCTION_OF_F <= FACTR*EPSMCH");
      if (iback >= 10) info = -5;
/*         i.e., to issue a warning if iback>10 in the line search. */
      goto L999;
    }
/*     Compute d=newx-oldx, r=newg-oldg, rr=y'y and dr=y's. */
    i__1 = n;
    for (i = 1; i <= i__1; ++i) {
      r[i] = g[i] - r[i];
/* L42: */
    }
    rr = F77_CALL(ddot)(&n, &r[1], &c__1, &r[1], &c__1);
    if (stp == 1.) {
      dr = gd - gdold;
      ddum = -gdold;
    } else {
      dr = (gd - gdold) * stp;
      F77_CALL(dscal)(&n, &stp, &d[1], &c__1);
      ddum = -gdold * stp;
    }
    if (dr <= epsmch * ddum) {
/*                      skip the L-BFGS update. */
      ++nskip;
      updatd = FALSE_;
      if (iprint >= 1)
          Rprintf("ys=%10.3e  -gs=%10.3e, BFGS update SKIPPED\n", dr, ddum);
      goto L888;
    }
/* ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc */

/*     Update the L-BFGS matrix. */

/* ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc */
    updatd = TRUE_;
    ++iupdat;
/*     Update matrices WS and WY and form the middle matrix in B. */
    matupd(n, m, &ws[ws_offset], &wy[wy_offset], &sy[sy_offset], &ss[
          ss_offset], &d[1], &r[1], &itail, &iupdat, &col, &head, &
          theta, &rr, &dr, &stp, &dtd);
/*     Form the upper half of the pds T = theta*SS + L*D^(-1)*L'; */
/*      Store T in the upper triangular of the array wt; */
/*      Cholesky factorize T to J*J' with */
/*         J' stored in the upper triangular of wt. */
    formt(m, &wt[wt_offset], &sy[sy_offset], &ss[ss_offset], &col, &theta, &
          info);
    if (info != 0) {
/*        nonpositive definiteness in Cholesky factorization; */
/*        refresh the lbfgs memory and restart the iteration. */
      if (iprint >= 0)
          Rprintf("%s\n%s\n",
                "Nonpositive definiteness in Cholesky factorization in formk;",
                "   refresh the lbfgs memory and restart the iteration.");
      info = 0;
      col = 0;
      head = 1;
      theta = 1.;
      iupdat = 0;
      updatd = FALSE_;
      goto L222;
    }
/*     Now the inverse of the middle matrix in B is */
/*     [  D^(1/2)  O ] [ -D^(1/2)    D^(-1/2)*L' ] */
/*     [ -L*D^(-1/2)     J ] [      0      J'        ] */
L888:
/* -------------------- the end of the loop ----------------------------- */
    goto L222;
L999:
    timer(&time2);
    time = time2 - time1;
L1000:
/*     Save local variables. */
    lsave[1] = prjctd;
    lsave[2] = cnstnd;
    lsave[3] = boxed;
    lsave[4] = updatd;
    isave[1] = nintol;
    isave[3] = itfile;
    isave[4] = iback;
    isave[5] = nskip;
    isave[6] = head;
    isave[7] = col;
    isave[8] = itail;
    isave[9] = iter;
    isave[10] = iupdat;
    isave[12] = nint;
    isave[13] = nfgv;
    isave[14] = info;
    isave[15] = ifun;
    isave[16] = iword;
    isave[17] = nfree;
    isave[18] = nact;
    isave[19] = ileave;
    isave[20] = nenter;
    dsave[1] = theta;
    dsave[2] = fold;
    dsave[3] = tol;
    dsave[4] = dnorm;
    dsave[5] = epsmch;
    dsave[6] = cpu1;
    dsave[7] = cachyt;
    dsave[8] = sbtime;
    dsave[9] = lnscht;
    dsave[10] = time1;
    dsave[11] = gd;
    dsave[12] = stpmx;
    dsave[13] = sbgnrm;
    dsave[14] = stp;
    dsave[15] = gdold;
    dsave[16] = dtd;
    prn3lb(n, x+1, f, task, iprint, info,
         iter, nfgv, nintol, nskip, nact, sbgnrm,
         nint, word, iback, stp, xstep, k);
    return;
} /* mainlb */
/* ======================= The end of mainlb ============================= */

static void active(int n, double *l, double *u,
               int *nbd, double *x, int *iwhere, int iprint,
               int *prjctd, int *cnstnd, int *boxed)
{
/*    ************

      Subroutine active

      This subroutine initializes iwhere and projects the initial x to
        the feasible set if necessary.

      iwhere is an integer array of dimension n.
        On entry iwhere is unspecified.
        On exit iwhere(i)=-1  if x(i) has no bounds
                      3 if l(i)=u(i)
                      0 otherwise.
        In cauchy, iwhere is given finer gradations.


                        *      *  *

      NEOS, November 1994. (Latest revision June 1996.)
      Optimization Technology Center.
      Argonne National Laboratory and Northwestern University.
      Written by
                     Ciyou Zhu
      in collaboration with R.H. Byrd, P. Lu-Chen and J. Nocedal.

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

    /* Local variables */
    int nbdd, i;

    /* Parameter adjustments */
    --iwhere;
    --x;
    --nbd;
    --u;
    --l;

    /* Function Body */

/*Initialize nbdd, prjctd, cnstnd and boxed. */
    nbdd = 0;
    *prjctd = FALSE_;
    *cnstnd = FALSE_;
    *boxed = TRUE_;
/*     Project the initial x to the easible set if necessary. */
    for (i = 1; i <= n; ++i) {
      if (nbd[i] > 0) {
          if (nbd[i] <= 2 && x[i] <= l[i]) {
            if (x[i] < l[i]) {
                *prjctd = TRUE_;
                x[i] = l[i];
            }
            ++nbdd;
          } else if (nbd[i] >= 2 && x[i] >= u[i]) {
            if (x[i] > u[i]) {
                *prjctd = TRUE_;
                x[i] = u[i];
            }
            ++nbdd;
          }
      }
    }

/*     Initialize iwhere and assign values to cnstnd and boxed. */
    for (i = 1; i <= n; ++i) {
      if (nbd[i] != 2) {
          *boxed = FALSE_;
      }
      if (nbd[i] == 0) {
/*                        this variable is always free */
          iwhere[i] = -1;
/*         otherwise set x(i)=mid(x(i), u(i), l(i)). */
      } else {
          *cnstnd = TRUE_;
          if (nbd[i] == 2 && u[i] - l[i] <= 0.) {
/*               this variable is always fixed */
            iwhere[i] = 3;
          } else {
            iwhere[i] = 0;
          }
      }
    }
    if (iprint >= 0) {
      if (*prjctd)
          Rprintf("The initial X is infeasible.  Restart with its projection.\n");
      if (!*cnstnd) Rprintf("This problem is unconstrained.\n");
    }
    if (iprint > 0)
      Rprintf("At X0, %d variables are exactly at the bounds\n", nbdd);

    return;
} /* active */
/* ======================= The end of active ============================= */

static void bmv(int m, double *sy, double *wt,
            int *col, double *v, double *p, int *info)
{
/*     ************

 *     Subroutine bmv

 *     This subroutine computes the product of the 2m x 2m middle matrix
 *     in the compact L-BFGS formula of B and a 2m vector v;
 *     it returns the product in p.

 *     m is an integer variable.
 *     On entry m is the maximum number of variable metric corrections
 *       used to define the limited memory matrix.
 *     On exit m is unchanged.

 *     sy is a double precision array of dimension m x m.
 *     On entry sy specifies the matrix S'Y.
 *     On exit sy is unchanged.

 *     wt is a double precision array of dimension m x m.
 *     On entry wt specifies the upper triangular matrix J' which is
 *       the Cholesky factor of (thetaS'S+LD^(-1)L').
 *     On exit wt is unchanged.

 *     col is an integer variable.
 *     On entry col specifies the number of s-vectors (or y-vectors)
 *       stored in the compact L-BFGS formula.
 *     On exit col is unchanged.

 *     v is a double precision array of dimension 2col.
 *     On entry v specifies vector v.
 *     On exit v is unchanged.

 *     p is a double precision array of dimension 2col.
 *     On entry p is unspecified.
 *     On exit p is the product Mv.

 *     info is an integer variable.
 *     On entry info is unspecified.
 *     On exit info = 0 for normal return,
 *                = nonzero for abnormal return when the system
 *                        to be solved by dtrsl is singular.

 *     Subprograms called:

 *     Linpack ... dtrsl.


 *                     *      *  *

 *     NEOS, November 1994. (Latest revision June 1996.)
 *     Optimization Technology Center.
 *     Argonne National Laboratory and Northwestern University.
 *     Written by
 *                  Ciyou Zhu
 *     in collaboration with R.H. Byrd, P. Lu-Chen and J. Nocedal.

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

    /* System generated locals */
    int sy_dim1, sy_offset, wt_dim1, wt_offset, Col;


    /* Local variables */
    int i, k;
    int i2;
    double sum;

    /* Parameter adjustments */
    wt_dim1 = m;
    wt_offset = 1 + wt_dim1 * 1;
    wt -= wt_offset;
    sy_dim1 = m;
    sy_offset = 1 + sy_dim1 * 1;
    sy -= sy_offset;
    --p;
    --v;

    /* Function Body */
    if (*col == 0) {
      return;
    }
/*    PART I: solve [    D^(1/2)      O ] [ p1 ] = [ v1 ]
 *                [ -L*D^(-1/2)   J ] [ p2 ]   [ v2 ].
 *    solve Jp2=v2+LD^(-1)v1.
 */
    Col = *col;
    p[*col + 1] = v[*col + 1];
    for (i = 2; i <= Col; ++i) {
      i2 = *col + i;
      sum = 0.;
      for (k = 1; k <= i - 1; ++k) {
          sum += sy[i + k * sy_dim1] * v[k] / sy[k + k * sy_dim1];
      }
      p[i2] = v[i2] + sum;
/* L20: */
    }
/*     Solve the triangular system */
    F77_CALL(dtrsl)(&wt[wt_offset], &m, col, &p[*col + 1], &c__11, info);
    if (*info != 0) {
      return;
    }
/*     solve D^(1/2)p1=v1. */
    for (i = 1; i <= Col; ++i) {
      p[i] = v[i] / sqrt(sy[i + i * sy_dim1]);
    }

/*    PART II: solve [ -D^(1/2)   D^(-1/2)*L'    ] [ p1 ] = [ p1 ]
 *                 [  0     J'             ] [ p2 ]   [ p2 ].
 *    solve J^Tp2=p2.
 */
    F77_CALL(dtrsl)(&wt[wt_offset], &m, col, &p[*col + 1], &c__1, info);
    if (*info != 0) {
      return;
    }
/*     compute p1=-D^(-1/2)(p1-D^(-1/2)L'p2) */
/*             =-D^(-1/2)p1 + D^(-1)L'p2. */
    for (i = 1; i <= Col; ++i) {
      p[i] = -p[i] / sqrt(sy[i + i * sy_dim1]);
    }
    for (i = 1; i <= Col; ++i) {
      sum = 0.;
      for (k = i + 1; k <= Col; ++k) {
          sum += sy[k + i * sy_dim1] * p[*col + k] / sy[i + i * sy_dim1];
      }
      p[i] += sum;
/* L60: */
    }
    return;
} /* bmv */
/* ======================== The end of bmv =============================== */

static void cauchy(int n, double *x, double *l, double *u, int *nbd,
               double *g, int *iorder, int * iwhere, double *t,
               double *d, double *xcp, int m,
               double *wy, double *ws, double *sy, double *wt,
               double *theta, int *col, int *head, double *p,
               double *c, double *wbp, double *v, int *nint,
               int iprint, double *sbgnrm, int *info, double * epsmch)
{
/*     ************
       Subroutine cauchy

       For given x, l, u, g (with sbgnrm > 0), and a limited memory
       BFGS matrix B defined in terms of matrices WY, WS, WT, and
       scalars head, col, and theta, this subroutine computes the
       generalized Cauchy point (GCP), defined as the first local
       minimizer of the quadratic

                Q(x + s) = g's + 1/2 s'Bs

       along the projected gradient direction P(x-tg,l,u).
       The routine returns the GCP in xcp.

       n is an integer variable.
       On entry n is the dimension of the problem.
       On exit n is unchanged.

       x is a double precision array of dimension n.
       On entry x is the starting point for the GCP computation.
       On exit x is unchanged.

       l is a double precision array of dimension n.
       On entry l is the lower bound of x.
       On exit l is unchanged.

       u is a double precision array of dimension n.
       On entry u is the upper bound of x.
       On exit u is unchanged.

       nbd is an integer array of dimension n.
       On entry nbd represents the type of bounds imposed on the
         variables, and must be specified as follows:
         nbd(i)=0 if x(i) is unbounded,
              1 if x(i) has only a lower bound,
              2 if x(i) has both lower and upper bounds, and
              3 if x(i) has only an upper bound.
       On exit nbd is unchanged.

       g is a double precision array of dimension n.
       On entry g is the gradient of f(x).  g must be a nonzero vector.
       On exit g is unchanged.

       iorder is an integer working array of dimension n.
       iorder will be used to store the breakpoints in the piecewise
       linear path and free variables encountered. On exit,
         iorder(1),...,iorder(nleft) are indices of breakpoints
                          which have not been encountered;
         iorder(nleft+1),...,iorder(nbreak) are indices of
                               encountered breakpoints; and
         iorder(nfree),...,iorder(n) are indices of variables which
               have no bound constraits along the search direction.

       iwhere is an integer array of dimension n.
       On entry iwhere indicates only the permanently fixed (iwhere=3)
       or free (iwhere= -1) components of x.
       On exit iwhere records the status of the current x variables.
       iwhere(i)=-3  if x(i) is free and has bounds, but is not moved
               0   if x(i) is free and has bounds, and is moved
               1   if x(i) is fixed at l(i), and l(i) .ne. u(i)
               2   if x(i) is fixed at u(i), and u(i) .ne. l(i)
               3   if x(i) is always fixed, i.e.,  u(i)=x(i)=l(i)
               -1  if x(i) is always free, i.e., it has no bounds.

       t is a double precision working array of dimension n.
       t will be used to store the break points.

       d is a double precision array of dimension n used to store
       the Cauchy direction P(x-tg)-x.

       xcp is a double precision array of dimension n used to return the
       GCP on exit.

       m is an integer variable.
       On entry m is the maximum number of variable metric corrections
         used to define the limited memory matrix.
       On exit m is unchanged.

       ws, wy, sy, and wt are double precision arrays.
       On entry they store information that defines the
                         limited memory BFGS matrix:
         ws(n,m) stores S, a set of s-vectors;
         wy(n,m) stores Y, a set of y-vectors;
         sy(m,m) stores S'Y;
         wt(m,m) stores the
               Cholesky factorization of (theta*S'S+LD^(-1)L').
       On exit these arrays are unchanged.

       theta is a double precision variable.
       On entry theta is the scaling factor specifying B_0 = theta I.
       On exit theta is unchanged.

       col is an integer variable.
       On entry col is the actual number of variable metric
         corrections stored so far.
       On exit col is unchanged.

       head is an integer variable.
       On entry head is the location of the first s-vector
         (or y-vector) in S (or Y).
       On exit col is unchanged.

       p is a double precision working array of dimension 2m.
       p will be used to store the vector p = W^(T)d.

       c is a double precision working array of dimension 2m.
       c will be used to store the vector c = W^(T)(xcp-x).

       wbp is a double precision working array of dimension 2m.
       wbp will be used to store the row of W corresponding
         to a breakpoint.

       v is a double precision working array of dimension 2m.

       nint is an integer variable.
       On exit nint records the number of quadratic segments explored
         in searching for the GCP.

       iprint is an INTEGER variable that must be set by the user.
       It controls the frequency and type of output generated:
        iprint<0    no output is generated;
        iprint=0    print only one line at the last iteration;
        0<iprint<99 print also f and |proj g| every iprint iterations;
        iprint=99   print details of every iteration except n-vectors;
        iprint=100  print also the changes of active set and final x;
        iprint>100  print details of every iteration including x and g;
       When iprint > 0, the file iterate.dat will be created to
                    summarize the iteration.

       sbgnrm is a double precision variable.
       On entry sbgnrm is the norm of the projected gradient at x.
       On exit sbgnrm is unchanged.

       info is an integer variable.
       On entry info is 0.
       On exit info = 0 for normal return,
                  = nonzero for abnormal return when the the system
                        used in routine bmv is singular.

       Subprograms called:

       L-BFGS-B Library ... hpsolb, bmv.

       Linpack ... dscal dcopy, daxpy.


       References:

      [1] R. H. Byrd, P. Lu, J. Nocedal and C. Zhu, ``A limited
      memory algorithm for bound constrained optimization'',
      SIAM J. Scientific Computing 16 (1995), no. 5, pp. 1190--1208.

      [2] C. Zhu, R.H. Byrd, P. Lu, J. Nocedal, ``L-BFGS-B: FORTRAN
      Subroutines for Large Scale Bound Constrained Optimization''
      Tech. Report, NAM-11, EECS Department, Northwestern University, 1994.

      (Postscript files of these papers are available via anonymous
       ftp to ece.nwu.edu in the directory pub/lbfgs/lbfgs_bcm.)

                       *      *  *

      NEOS, November 1994. (Latest revision April 1997.)
      Optimization Technology Center.
      Argonne National Laboratory and Northwestern University.
      Written by
                      Ciyou Zhu
      in collaboration with R.H. Byrd, P. Lu-Chen and J. Nocedal.

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

    /* System generated locals */
    int wy_dim1, wy_offset, ws_dim1, ws_offset, sy_dim1, sy_offset,
          wt_dim1, wt_offset, i__2;
    double d__1;

    /* Local variables */
    double bkmin, dibp, dibp2, zibp, neggi, tsum;
    double f1, f2, f2_org__, dt, tj, tj0, tl= 0.0, tu=0.0, dtm, wmc, wmp, wmw;

    int i, j, ibp, iter, bnded, nfree, nleft, nbreak, ibkmin, pointr;
    int xlower, xupper, col2;

    /* Parameter adjustments */
    --xcp;
    --d;
    --t;
    --iwhere;
    --iorder;
    --g;
    --nbd;
    --u;
    --l;
    --x;
    --v;
    --wbp;
    --c;
    --p;
    wt_dim1 = m;    wt_offset = 1 + wt_dim1 * 1;    wt -= wt_offset;
    sy_dim1 = m;    sy_offset = 1 + sy_dim1 * 1;    sy -= sy_offset;
    ws_dim1 = n;    ws_offset = 1 + ws_dim1 * 1;    ws -= ws_offset;
    wy_dim1 = n;    wy_offset = 1 + wy_dim1 * 1;    wy -= wy_offset;

    /* Function Body */

/*     Check the status of the variables, reset iwhere(i) if necessary;
 *     compute the Cauchy direction d and the breakpoints t; initialize
 *     the derivative f1 and the vector p = W'd (for theta = 1).
 */

    if (*sbgnrm <= 0.) {
      if (iprint >= 0) Rprintf("Subgnorm = 0.  GCP = X.\n");
      F77_CALL(dcopy)(&n, &x[1], &c__1, &xcp[1], &c__1);
      return;
    }
    bnded = TRUE_;
    nfree = n + 1;
    nbreak = 0;
    ibkmin = 0;
    bkmin = 0.;
    col2 = *col << 1;
    f1 = 0.;
    if (iprint >= 99)
      Rprintf("\n---------------- CAUCHY entered-------------------\n\n");

/*     We set p to zero and build it up as we determine d. */
    for (i = 1; i <= col2; ++i)
      p[i] = 0.;

/*     In the following loop we determine for each variable its bound */
/*      status and its breakpoint, and update p accordingly. */
/*      Smallest breakpoint is identified. */

    for (i = 1; i <= n; ++i) {
      neggi = -g[i];
      if (iwhere[i] != 3 && iwhere[i] != -1) {
/*           if x(i) is not a constant and has bounds, */
/*           compute the difference between x(i) and its bounds. */
          if (nbd[i] <= 2) {
            tl = x[i] - l[i];
          }
          if (nbd[i] >= 2) {
            tu = u[i] - x[i];
          }
/*         If a variable is close enough to a bound */
/*           we treat it as at bound. */
          xlower = nbd[i] <= 2 && tl <= 0.;
          xupper = nbd[i] >= 2 && tu <= 0.;
/*          reset iwhere(i). */
          iwhere[i] = 0;
          if (xlower) {
            if (neggi <= 0.) {
                iwhere[i] = 1;
            }
          } else if (xupper) {
            if (neggi >= 0.) {
                iwhere[i] = 2;
            }
          } else {
            if (fabs(neggi) <= 0.) {
                iwhere[i] = -3;
            }
          }
      }
      pointr = *head;
      if (iwhere[i] != 0 && iwhere[i] != -1) {
          d[i] = 0.;
      } else {
          d[i] = neggi;
          f1 -= neggi * neggi;
/*           calculate p := p - W'e_i* (g_i). */
          i__2 = *col;
          for (j = 1; j <= i__2; ++j) {
            p[j] += wy[i + pointr * wy_dim1] * neggi;
            p[*col + j] += ws[i + pointr * ws_dim1] * neggi;
            pointr = pointr % m + 1;
          }
          if (nbd[i] <= 2 && nbd[i] != 0 && neggi < 0.) {
/*                         x(i) + d(i) is bounded; compute t(i). */
            ++nbreak;
            iorder[nbreak] = i;
            t[nbreak] = tl / (-neggi);
            if (nbreak == 1 || t[nbreak] < bkmin) {
                bkmin = t[nbreak];
                ibkmin = nbreak;
            }
          } else if (nbd[i] >= 2 && neggi > 0.) {
/*                         x(i) + d(i) is bounded; compute t(i). */
            ++nbreak;
            iorder[nbreak] = i;
            t[nbreak] = tu / neggi;
            if (nbreak == 1 || t[nbreak] < bkmin) {
                bkmin = t[nbreak];
                ibkmin = nbreak;
            }
          } else {/*            x(i) + d(i) is not bounded. */
            --nfree;
            iorder[nfree] = i;
            if (fabs(neggi) > 0.)
                bnded = FALSE_;
          }
      }
/* L50: */
    } /* for(i = 1:n) */

/*     The indices of the nonzero components of d are now stored */
/*     in iorder(1),...,iorder(nbreak) and iorder(nfree),...,iorder(n). */
/*     The smallest of the nbreak breakpoints is in t(ibkmin)=bkmin. */
    if (*theta != 1.) {
/*               complete the initialization of p for theta not= one. */
      F77_CALL(dscal)(col, theta, &p[*col + 1], &c__1);
    }
/*     Initialize GCP xcp = x. */
    F77_CALL(dcopy)(&n, &x[1], &c__1, &xcp[1], &c__1);
    if (nbreak == 0 && nfree == n + 1) {
/*              is a zero vector, return with the initial xcp as GCP. */
      if (iprint > 100) {
          Rprintf("Cauchy X =  ");
          for(i = 1; i <= n; i++) Rprintf("%g ", xcp[i]);
          Rprintf("\n");
      }
      return;
    }
/*     Initialize c = W'(xcp - x) = 0. */
    for (j = 1; j <= col2; ++j)
      c[j] = 0.;

/*     Initialize derivative f2. */
    f2 = -(*theta) * f1;
    f2_org__ = f2;
    if (*col > 0) {
      bmv(m, &sy[sy_offset], &wt[wt_offset], col, &p[1], &v[1], info);
      if (*info != 0) {
          return;
      }
      f2 -= F77_CALL(ddot)(&col2, &v[1], &c__1, &p[1], &c__1);
    }
    dtm = -f1 / f2;
    tsum = 0.;
    *nint = 1;
    if (iprint >= 99) Rprintf("There are %d  breakpoints\n", nbreak);

/*     If there are no breakpoints, locate the GCP and return. */
    if (nbreak == 0) {
      goto L888;
    }
    nleft = nbreak;
    iter = 1;
    tj = 0.;
/* ------------------- the beginning of the loop ------------------------- */
L777:
/*     Find the next smallest breakpoint; */
/*     compute dt = t(nleft) - t(nleft + 1). */
    tj0 = tj;
    if (iter == 1) {
/*       Since we already have the smallest breakpoint we need not do */
/*       heapsort yet. Often only one breakpoint is used and the */
/*       cost of heapsort is avoided. */
      tj = bkmin;
      ibp = iorder[ibkmin];
    } else {
      if (iter == 2) {
          /* Replace the already used smallest breakpoint with the */
          /* breakpoint numbered nbreak > nlast, before heapsort call. */
          if (ibkmin != nbreak) {
            t[ibkmin] = t[nbreak];
            iorder[ibkmin] = iorder[nbreak];
          }
      }
      /* Update heap structure of breakpoints */
      /* (if iter=2, initialize heap). */
      hpsolb(nleft, &t[1], &iorder[1], iter - 2);
      tj = t[nleft];
      ibp = iorder[nleft];
    }
    dt = tj - tj0;

    if (dt != 0 && iprint >=  100) {
      Rprintf("\nPiece    %3i f1, f2 at start point %11.4e %11.4e\n",
            *nint, f1, f2);
      Rprintf("Distance to the next break point =  %11.4e\n", dt);
      Rprintf("Distance to the stationary point =  %11.4e\n", dtm);
    }

/*     If a minimizer is within this interval, */
/*     locate the GCP and return. */
    if (dtm < dt) {
      goto L888;
    }
/*     Otherwise fix one variable and */
/*     reset the corresponding component of d to zero. */
    tsum += dt;
    --nleft;
    ++iter;
    dibp = d[ibp];
    d[ibp] = 0.;
    if (dibp > 0.) {
      zibp = u[ibp] - x[ibp];
      xcp[ibp] = u[ibp];
      iwhere[ibp] = 2;
    } else {
      zibp = l[ibp] - x[ibp];
      xcp[ibp] = l[ibp];
      iwhere[ibp] = 1;
    }
    if (iprint >= 100) Rprintf("Variable  %d  is fixed.\n", ibp);
    if (nleft == 0 && nbreak == n) {
/*                                   all n variables are fixed, */
/*                                    return with xcp as GCP. */
      dtm = dt;
      goto L999;
    }
/*     Update the derivative information. */
    ++(*nint);
    dibp2 = dibp * dibp;
/*     Update f1 and f2. */
/*      temporarily set f1 and f2 for col=0. */
    f1 += dt * f2 + dibp2 - *theta * dibp * zibp;
    f2 -= *theta * dibp2;
    if (*col > 0) {
/*                    update c = c + dt*p. */
      F77_CALL(daxpy)(&col2, &dt, &p[1], &c__1, &c[1], &c__1);
/*         choose wbp, */
/*         the row of W corresponding to the breakpoint encountered. */
      pointr = *head;
      for (j = 1; j <= *col; ++j) {
          wbp[j] = wy[ibp + pointr * wy_dim1];
          wbp[*col + j] = *theta * ws[ibp + pointr * ws_dim1];
          pointr = pointr % m + 1;
      }
/*         compute (wbp)Mc, (wbp)Mp, and (wbp)M(wbp)'. */
      bmv(m, &sy[sy_offset], &wt[wt_offset], col, &wbp[1], &v[1], info);
      if (*info != 0) {
          return;
      }
      wmc = F77_CALL(ddot)(&col2,  &c[1], &c__1, &v[1], &c__1);
      wmp = F77_CALL(ddot)(&col2,  &p[1], &c__1, &v[1], &c__1);
      wmw = F77_CALL(ddot)(&col2,&wbp[1], &c__1, &v[1], &c__1);
/*         update p = p - dibp*wbp. */
      d__1 = -dibp;
      F77_CALL(daxpy)(&col2, &d__1, &wbp[1], &c__1, &p[1], &c__1);
/*         complete updating f1 and f2 while col > 0. */
      f1 += dibp * wmc;
      f2 += (2. * dibp * wmp - dibp2 * wmw);
    }
    if(f2 < (d__1 = *epsmch * f2_org__)) f2 = d__1;
    if (nleft > 0) {
      dtm = -f1 / f2;
      goto L777;
/*             to repeat the loop for unsearched intervals. */
    } else if (bnded) {
      f1 = 0.;
      f2 = 0.;
      dtm = 0.;
    } else {
      dtm = -f1 / f2;
    }
/* ------------------- the end of the loop ------------------------------- */
L888:
    if (iprint >= 99) {
      Rprintf("\nGCP found in this segment\n");
      Rprintf("Piece    %3i f1, f2 at start point %11.4e %11.4e\n",
            *nint,f1,f2);
      Rprintf("Distance to the stationary point =  %11.4e\n", dtm);
    }

    if (dtm <= 0.) {
      dtm = 0.;
    }
    tsum += dtm;
/*     Move free variables (i.e., the ones w/o breakpoints) and */
/*     the variables whose breakpoints haven't been reached. */
    F77_CALL(daxpy)(&n, &tsum, &d[1], &c__1, &xcp[1], &c__1);
L999:
/*     Update c = c + dtm*p = W'(x^c - x) */
/*     which will be used in computing r = Z'(B(x^c - x) + g). */
    if (*col > 0) {
      F77_CALL(daxpy)(&col2, &dtm, &p[1], &c__1, &c[1], &c__1);
    }
    if (iprint >= 100) {
      Rprintf("Cauchy X =  ");
      for(i = 1; i <= n; i++) Rprintf("%g ", xcp[i]);
      Rprintf("\n");
    }

    if (iprint >= 99)
      Rprintf("\n---------------- exit CAUCHY----------------------\n\n");
    return;
} /* cauchy */
/* ====================== The end of cauchy ============================== */

static void cmprlb(int n, int m, double *x,
               double *g, double *ws, double *wy, double *sy,
               double *wt, double *z, double *r, double *wa,
               int *indx, double *theta, int *col, int *head,
               int *nfree, int *cnstnd, int *info)
{
/*    ************

      Subroutine cmprlb

      This subroutine computes r=-Z'B(xcp-xk)-Z'g by using
      wa(2m+1)=W'(xcp-x) from subroutine cauchy.

      Subprograms called:

        L-BFGS-B Library ... bmv.


                        *      *  *

      NEOS, November 1994. (Latest revision June 1996.)
      Optimization Technology Center.
      Argonne National Laboratory and Northwestern University.
      Written by
                     Ciyou Zhu
      in collaboration with R.H. Byrd, P. Lu-Chen and J. Nocedal.

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

    /* System generated locals */
    int ws_dim1, ws_offset, wy_dim1, wy_offset, sy_dim1, sy_offset,
          wt_dim1, wt_offset, Col, n_f;

    /* Local variables */
    int i, j, k;
    double a1, a2;
    int pointr;

    /* Parameter adjustments */
    --indx;
    --r;
    --z;
    --g;
    --x;
    --wa;
    wt_dim1 = m;
    wt_offset = 1 + wt_dim1 * 1;
    wt -= wt_offset;
    sy_dim1 = m;
    sy_offset = 1 + sy_dim1 * 1;
    sy -= sy_offset;
    wy_dim1 = n;
    wy_offset = 1 + wy_dim1 * 1;
    wy -= wy_offset;
    ws_dim1 = n;
    ws_offset = 1 + ws_dim1 * 1;
    ws -= ws_offset;

    /* Function Body */
    Col = *col;
    if (! (*cnstnd) && Col > 0) {
      for (i = 1; i <= n; ++i)
          r[i] = -g[i];
    }
    else {
      n_f = *nfree;
      for (i = 1; i <= n_f; ++i) {
          k = indx[i];
          r[i] = -(*theta) * (z[k] - x[k]) - g[k];
      }
      bmv(m, &sy[sy_offset], &wt[wt_offset], col,
          &wa[(m << 1) + 1], &wa[1], info);
      if (*info != 0) {
          *info = -8;
          return;
      }
      pointr = *head;
      for (j = 1; j <= Col; ++j) {
          a1 = wa[j];
          a2 = *theta * wa[Col + j];
          for (i = 1; i <= n_f; ++i) {
            k = indx[i];
            r[i] += wy[k + pointr * wy_dim1] * a1 +
                  ws[k + pointr * ws_dim1] * a2;
          }
          pointr = pointr % m + 1;
      }
    }
    return;
} /* cmprlb */
/* ======================= The end of cmprlb ============================= */

static void errclb(int n, int m, double factr, double *l, double *u,
               int *nbd, char *task, int *info, int *k)
{
/*    ************
      Subroutine errclb

      This subroutine checks the validity of the input data.

                        *      *  *

      NEOS, November 1994. (Latest revision April 1997.)
      Optimization Technology Center.
      Argonne National Laboratory and Northwestern University.
      Written by
                     Ciyou Zhu
      in collaboration with R.H. Byrd, P. Lu-Chen and J. Nocedal.

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

    /* Local variables */
    int i;

    /* Parameter adjustments */
    --nbd;
    --u;
    --l;

    /* Function Body */
/*     Check the input arguments for errors. */
    if (n <= 0)
      strcpy(task, "ERROR: N .LE. 0");
    if (m <= 0)
      strcpy(task, "ERROR: M .LE. 0");
    if (factr < 0.)
      strcpy(task, "ERROR: FACTR .LT. 0");

/*     Check the validity of the arrays nbd(i), u(i), and l(i). */
    for (i = 1; i <= n; ++i) {
      if (nbd[i] < 0 || nbd[i] > 3) {
/*                                       return */
          strcpy(task, "ERROR: INVALID NBD");
          *info = -6;
          *k = i;
      }
      if (nbd[i] == 2) {
          if (l[i] > u[i]) {
/*                            return */
            strcpy(task, "ERROR: NO FEASIBLE SOLUTION");
            *info = -7;
            *k = i;
          }
      }
    }
    return;
} /* errclb */
/* ======================= The end of errclb ============================= */

static void formk(int n, int *nsub, int *ind, int * nenter, int *ileave,
              int *indx2, int *iupdat, int * updatd, double *wn,
              double *wn1, int m, double *ws, double *wy, double *sy,
              double *theta, int *col, int *head, int *info)
{
/*     ************

     Subroutine formk

     This subroutine forms  the LEL^T factorization of the indefinite

       matrix      K = [-D -Y'ZZ'Y/theta     L_a'-R_z'  ]
                 [L_a -R_z           theta*S'AA'S ]
                                        where E = [-I  0]
                                                [ 0  I]
     The matrix K can be shown to be equal to the matrix M^[-1]N
       occurring in section 5.1 of [1], as well as to the matrix
       Mbar^[-1] Nbar in section 5.3.

     n is an integer variable.
       On entry n is the dimension of the problem.
       On exit n is unchanged.

     nsub is an integer variable
       On entry nsub is the number of subspace variables in free set.
       On exit nsub is not changed.

     ind is an integer array of dimension nsub.
       On entry ind specifies the indices of subspace variables.
       On exit ind is unchanged.

     nenter is an integer variable.
       On entry nenter is the number of variables entering the
       free set.
       On exit nenter is unchanged.

     ileave is an integer variable.
       On entry indx2(ileave),...,indx2(n) are the variables leaving
       the free set.
       On exit ileave is unchanged.

     indx2 is an integer array of dimension n.
       On entry indx2(1),...,indx2(nenter) are the variables entering
       the free set, while indx2(ileave),...,indx2(n) are the
       variables leaving the free set.
       On exit indx2 is unchanged.
p
     iupdat is an integer variable.
       On entry iupdat is the total number of BFGS updates made so far.
       On exit iupdat is unchanged.

     updatd is a logical variable.
       On entry 'updatd' is true if the L-BFGS matrix is updatd.
       On exit 'updatd' is unchanged.

     wn is a double precision array of dimension 2m x 2m.
       On entry wn is unspecified.
       On exit the upper triangle of wn stores the LEL^T factorization
       of the 2*col x 2*col indefinite matrix
                 [-D -Y'ZZ'Y/theta     L_a'-R_z'  ]
                 [L_a -R_z           theta*S'AA'S ]

     wn1 is a double precision array of dimension 2m x 2m.
       On entry wn1 stores the lower triangular part of
                 [Y' ZZ'Y     L_a'+R_z']
                 [L_a+R_z     S'AA'S       ]
       in the previous iteration.
       On exit wn1 stores the corresponding updated matrices.
       The purpose of wn1 is just to store these inner products
       so they can be easily updated and inserted into wn.

     m is an integer variable.
       On entry m is the maximum number of variable metric corrections
       used to define the limited memory matrix.
       On exit m is unchanged.

     ws, wy, sy, and wtyy are double precision arrays;
     theta is a double precision variable;
     col is an integer variable;
     head is an integer variable.
       On entry they store the information defining the
                                limited memory BFGS matrix:
       ws(n,m) stores S, a set of s-vectors;
       wy(n,m) stores Y, a set of y-vectors;
       sy(m,m) stores S'Y;
       wtyy(m,m) stores the Cholesky factorization
                           of (theta*S'S+LD^(-1)L')
       theta is the scaling factor specifying B_0 = theta I;
       col is the number of variable metric corrections stored;
       head is the location of the 1st s- (or y-) vector in S (or Y).
       On exit they are unchanged.

     info is an integer variable.
       On entry info is unspecified.
       On exit info =  0 for normal return;
                = -1 when the 1st Cholesky factorization failed;
                = -2 when the 2st Cholesky factorization failed.

     Subprograms called:

       Linpack ... dcopy, dpofa, dtrsl.


     References:
       [1] R. H. Byrd, P. Lu, J. Nocedal and C. Zhu, ``A limited
       memory algorithm for bound constrained optimization'',
       SIAM J. Scientific Computing 16 (1995), no. 5, pp. 1190--1208.

       [2] C. Zhu, R.H. Byrd, P. Lu, J. Nocedal, ``L-BFGS-B: a
       limited memory FORTRAN code for solving bound constrained
       optimization problems'', Tech. Report, NAM-11, EECS Department,
       Northwestern University, 1994.

       (Postscript files of these papers are available via anonymous
      ftp to ece.nwu.edu in the directory pub/lbfgs/lbfgs_bcm.)

                     *  *      *

     NEOS, November 1994. (Latest revision April 1997.)
     Optimization Technology Center.
     Argonne National Laboratory and Northwestern University.
     Written by
                  Ciyou Zhu
     in collaboration with R.H. Byrd, P. Lu-Chen and J. Nocedal.

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

    /* System generated locals */
    int wn_dim1, wn_offset, wn1_dim1, wn1_offset, ws_dim1, ws_offset,
          wy_dim1, wy_offset, sy_dim1, sy_offset, i__1, i__2;

    /* Local variables */
    int dend, pend;
    int upcl;
    double temp1, temp2, temp3, temp4;
    int i, k;
    int ipntr, jpntr, k1, m2, dbegin, is, js, iy, jy, pbegin, is1, js1,
          col2;

    /* Parameter adjustments */
    --indx2;
    --ind;
    sy_dim1 = m;
    sy_offset = 1 + sy_dim1 * 1;
    sy -= sy_offset;
    wy_dim1 = n;
    wy_offset = 1 + wy_dim1 * 1;
    wy -= wy_offset;
    ws_dim1 = n;
    ws_offset = 1 + ws_dim1 * 1;
    ws -= ws_offset;
    wn1_dim1 = 2 * m;
    wn1_offset = 1 + wn1_dim1 * 1;
    wn1 -= wn1_offset;
    wn_dim1 = 2 * m;
    wn_offset = 1 + wn_dim1 * 1;
    wn -= wn_offset;

    /* Function Body */

/*     Form the lower triangular part of */
/*           WN1 = [Y' ZZ'Y     L_a'+R_z'] */
/*                 [L_a+R_z     S'AA'S   ] */
/*      where L_a is the strictly lower triangular part of S'AA'Y */
/*          R_z is the upper triangular part of S'ZZ'Y. */

    if (*updatd) {
      if (*iupdat > m) {/*          shift old part of WN1. */
          i__1 = m - 1;
          for (jy = 1; jy <= i__1; ++jy) {
            js = m + jy;
            i__2 = m - jy;
            F77_CALL(dcopy)(&i__2, &wn1[jy + 1 + (jy + 1)* wn1_dim1], &c__1,
                               &wn1[jy + jy * wn1_dim1], &c__1);
            F77_CALL(dcopy)(&i__2, &wn1[js + 1 + (js + 1)* wn1_dim1], &c__1,
                               &wn1[js + js * wn1_dim1], &c__1);
            i__2 = m - 1;
            F77_CALL(dcopy)(&i__2, &wn1[m + 2 + (jy + 1) * wn1_dim1], &c__1,
                               &wn1[m + 1 + jy * wn1_dim1], &c__1);
/* L10: */
          }
      }
/*        put new rows in blocks (1,1), (2,1) and (2,2). */
      pbegin = 1;
      pend = *nsub;
      dbegin = *nsub + 1;
      dend = n;
      iy = *col;
      is = m + *col;
      ipntr = *head + *col - 1;
      if (ipntr > m) {
          ipntr -= m;
      }
      jpntr = *head;
      i__1 = *col;
      for (jy = 1; jy <= i__1; ++jy) {
          js = m + jy;
          temp1 = 0.;
          temp2 = 0.;
          temp3 = 0.;
/*           compute element jy of row 'col' of Y'ZZ'Y */
          for (k = pbegin; k <= pend; ++k) {
            k1 = ind[k];
            temp1 += wy[k1 + ipntr * wy_dim1] * wy[k1 + jpntr * wy_dim1];
          }
/*           compute elements jy of row 'col' of L_a and S'AA'S */
          for (k = dbegin; k <= dend; ++k) {
            k1 = ind[k];
            temp2 += ws[k1 + ipntr * ws_dim1] * ws[k1 + jpntr * ws_dim1];
            temp3 += ws[k1 + ipntr * ws_dim1] * wy[k1 + jpntr * wy_dim1];
          }
          wn1[iy + jy * wn1_dim1] = temp1;
          wn1[is + js * wn1_dim1] = temp2;
          wn1[is + jy * wn1_dim1] = temp3;
          jpntr = jpntr % m + 1;
/* L20: */
      }
/*        put new column in block (2,1). */
      jy = *col;
      jpntr = *head + *col - 1;
      if (jpntr > m) {
          jpntr -= m;
      }
      ipntr = *head;
      i__1 = *col;
      for (i = 1; i <= i__1; ++i) {
          is = m + i;
          temp3 = 0.;
/*           compute element i of column 'col' of R_z */
          for (k = pbegin; k <= pend; ++k) {
            k1 = ind[k];
            temp3 += ws[k1 + ipntr * ws_dim1] * wy[k1 + jpntr * wy_dim1];
          }
          ipntr = ipntr % m + 1;
          wn1[is + jy * wn1_dim1] = temp3;
/* L30: */
      }
      upcl = *col - 1;
    } else {
      upcl = *col;
    }
/*     modify the old parts in blocks (1,1) and (2,2) due to changes */
/*     in the set of free variables. */
    ipntr = *head;
    for (iy = 1; iy <= upcl; ++iy) {
      is = m + iy;
      jpntr = *head;
      for (jy = 1; jy <= iy; ++jy) {
          js = m + jy;
          temp1 = 0.;
          temp2 = 0.;
          temp3 = 0.;
          temp4 = 0.;
          for (k = 1; k <= *nenter; ++k) {
            k1 = indx2[k];
            temp1 += wy[k1 + ipntr * wy_dim1] * wy[k1 + jpntr * wy_dim1];
            temp2 += ws[k1 + ipntr * ws_dim1] * ws[k1 + jpntr * ws_dim1];
          }
          for (k = *ileave; k <= n; ++k) {
            k1 = indx2[k];
            temp3 += wy[k1 + ipntr * wy_dim1] * wy[k1 + jpntr * wy_dim1];
            temp4 += ws[k1 + ipntr * ws_dim1] * ws[k1 + jpntr * ws_dim1];
          }
          wn1[iy + jy * wn1_dim1] = wn1[iy + jy * wn1_dim1] + temp1 - temp3;
          wn1[is + js * wn1_dim1] = wn1[is + js * wn1_dim1] - temp2 + temp4;
          jpntr = jpntr % m + 1;
/* L40: */
      }
      ipntr = ipntr % m + 1;
/* L45: */
    }
/*     modify the old parts in block (2,1). */
    ipntr = *head;
    for (is = m + 1; is <= m + upcl; ++is) {
      jpntr = *head;
      for (jy = 1; jy <= upcl; ++jy) {
          temp1 = 0.;
          temp3 = 0.;
          for (k = 1; k <= *nenter; ++k) {
            k1 = indx2[k];
            temp1 += ws[k1 + ipntr * ws_dim1] * wy[k1 + jpntr * wy_dim1];
          }
          for (k = *ileave; k <= n; ++k) {
            k1 = indx2[k];
            temp3 += ws[k1 + ipntr * ws_dim1] * wy[k1 + jpntr * wy_dim1];
          }
          if (is <= jy + m) {
            wn1[is + jy * wn1_dim1] +=  temp1 - temp3;
          } else {
            wn1[is + jy * wn1_dim1] += -temp1 + temp3;
          }
          jpntr = jpntr % m + 1;
/* L55: */
      }
      ipntr = ipntr % m + 1;
/* L60: */
    }
/*     Form the upper triangle of WN = [D+Y' ZZ'Y/theta       -L_a'+R_z' ] */
/*                             [-L_a +R_z  S'AA'S*theta] */
    m2 = m << 1;
    i__1 = *col;
    for (iy = 1; iy <= i__1; ++iy) {
      is = *col + iy;
      is1 = m + iy;
      i__2 = iy;
      for (jy = 1; jy <= i__2; ++jy) {
          js = *col + jy;
          js1 = m + jy;
          wn[jy + iy * wn_dim1] = wn1[iy + jy * wn1_dim1] / *theta;
          wn[js + is * wn_dim1] = wn1[is1 + js1 * wn1_dim1] * *theta;
/* L65: */
      }
      i__2 = iy - 1;
      for (jy = 1; jy <= i__2; ++jy) {
          wn[jy + is * wn_dim1] = -wn1[is1 + jy * wn1_dim1];
      }
      i__2 = *col;
      for (jy = iy; jy <= i__2; ++jy) {
          wn[jy + is * wn_dim1] = wn1[is1 + jy * wn1_dim1];
      }
      wn[iy + iy * wn_dim1] += sy[iy + iy * sy_dim1];
/* L70: */
    }
/*     Form the upper triangle of */
/*        WN= [  LL'            L^-1(-L_a'+R_z')] */
/*          [(-L_a +R_z)L'^-1   S'AA'S*theta  ] */
/*      first Cholesky factor (1,1) block of wn to get LL' */
/*                    with L' stored in the upper triangle of wn. */
    F77_CALL(dpofa)(&wn[wn_offset], &m2, col, info);
    if (*info != 0) {
      *info = -1;
      return;
    }
/*      then form L^-1(-L_a'+R_z') in the (1,2) block. */
    col2 = *col << 1;
    for (js = *col + 1; js <= col2; ++js) {
      F77_CALL(dtrsl)(&wn[wn_offset], &m2, col,
                  &wn[js * wn_dim1 + 1], &c__11, info);
    }
/*     Form S'AA'S*theta + (L^-1(-L_a'+R_z'))'L^-1(-L_a'+R_z') in the */
/*      upper triangle of (2,2) block of wn. */
    for (is = *col + 1; is <= col2; ++is) {
      for (js = is; js <= col2; ++js) {
          wn[is + js * wn_dim1] +=
            F77_CALL(ddot)(col, &wn[is * wn_dim1 + 1], &c__1,
                         &wn[js * wn_dim1 + 1], &c__1);
      }
/* L72: */
    }
/*     Cholesky factorization of (2,2) block of wn. */
    F77_CALL(dpofa)(&wn[*col + 1 + (*col + 1) * wn_dim1], &m2, col, info);
    if (*info != 0) {
      *info = -2;
      return;
    }
    return;
} /* formk */
/* ======================= The end of formk ============================== */

static void formt(int m, double *wt, double *sy, double *ss,
              int *col, double *theta, int *info)
{
/*     ************

     Subroutine formt

       This subroutine forms the upper half of the pos. def. and symm.
       T = theta*SS + L*D^(-1)*L', stores T in the upper triangle
       of the array wt, and performs the Cholesky factorization of T
       to produce J*J', with J' stored in the upper triangle of wt.

     Subprograms called:

       Linpack ... dpofa.


                     *  *      *

     NEOS, November 1994. (Latest revision June 1996.)
     Optimization Technology Center.
     Argonne National Laboratory and Northwestern University.
     Written by
                  Ciyou Zhu
     in collaboration with R.H. Byrd, P. Lu-Chen and J. Nocedal.

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

    /* System generated locals */
    int wt_dim1, wt_offset, sy_dim1, sy_offset, ss_dim1, ss_offset, i__1;

    /* Local variables */
    double ddum;
    int i, j, k;
    int k1;

    /* Parameter adjustments */
    ss_dim1 = m;
    ss_offset = 1 + ss_dim1 * 1;
    ss -= ss_offset;
    sy_dim1 = m;
    sy_offset = 1 + sy_dim1 * 1;
    sy -= sy_offset;
    wt_dim1 = m;
    wt_offset = 1 + wt_dim1 * 1;
    wt -= wt_offset;

    /* Function Body */

/*     Form the upper half of  T = theta*SS + L*D^(-1)*L', */
/*      store T in the upper triangle of the array wt. */
    i__1 = *col;
    for (j = 1; j <= i__1; ++j) {
      wt[j * wt_dim1 + 1] = *theta * ss[j * ss_dim1 + 1];
    }
    for (i = 2; i <= i__1; ++i) {
      for (j = i; j <= i__1; ++j) {
          k1 = min(i,j) - 1;
          ddum = 0.;
          for (k = 1; k <= k1; ++k) {
            ddum += sy[i + k * sy_dim1] * sy[j + k * sy_dim1] / sy[k +
                  k * sy_dim1];
          }
          wt[i + j * wt_dim1] = ddum + *theta * ss[i + j * ss_dim1];
      }
/* L55: */
    }
/*     Cholesky factorize T to J*J' with */
/*      J' stored in the upper triangle of wt. */
    F77_CALL(dpofa)(&wt[wt_offset], &m, col, info);
    if (*info != 0) {
      *info = -3;
    }
    return;
} /* formt */
/* ======================= The end of formt ============================== */

static void freev(int n, int *nfree, int *indx,
              int *nenter, int *ileave, int *indx2, int *iwhere,
              int *wrk, int *updatd, int *cnstnd, int iprint,
              int *iter)
{
/*    ************

      Subroutine freev

      This subroutine counts the entering and leaving variables when
        iter > 0, and finds the index set of free and active variables
        at the GCP.

      cnstnd is a int variable indicating whether bounds are present

      indx is an int array of dimension n
        for i=1,...,nfree, indx(i) are the indices of free variables
        for i=nfree+1,...,n, indx(i) are the indices of bound variables
        On entry after the first iteration, indx gives
          the free variables at the previous iteration.
        On exit it gives the free variables based on the determination
          in cauchy using the array iwhere.

      indx2 is an int array of dimension n
        On entry indx2 is unspecified.
        On exit with iter>0, indx2 indicates which variables
           have changed status since the previous iteration.
        For i= 1,...,nenter, indx2(i) have changed from bound to free.
        For i= ileave+1,...,n, indx2(i) have changed from free to bound.


                        *      *  *

      NEOS, November 1994. (Latest revision June 1996.)
      Optimization Technology Center.
      Argonne National Laboratory and Northwestern University.
      Written by
                     Ciyou Zhu
      in collaboration with R.H. Byrd, P. Lu-Chen and J. Nocedal.

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

    /* System generated locals */
    int i__1;

    /* Local variables */
    int iact, i, k;

    /* Parameter adjustments */
    --iwhere;
    --indx2;
    --indx;

    /* Function Body */
    *nenter = 0;
    *ileave = n + 1;
    if (*iter > 0 && *cnstnd) {/* count the entering and leaving variables. */
      i__1 = *nfree;
      for (i = 1; i <= i__1; ++i) {
          k = indx[i];
          if (iwhere[k] > 0) {
            --(*ileave);
            indx2[*ileave] = k;
            if (iprint >= 100)
                Rprintf("Variable %d leaves the set of free variables\n",
                      k);
          }
/* L20: */
      }
      for (i = *nfree + 1; i <= n; ++i) {
          k = indx[i];
          if (iwhere[k] <= 0) {
            ++(*nenter);
            indx2[*nenter] = k;
            if (iprint >= 100)
                Rprintf("Variable %d enters the set of free variables\n",
                      k);
          }
/* L22: */
         if (iprint >= 100)
           Rprintf("%d variables leave; %d variables enter\n",
                 n + 1 - *ileave, *nenter);
      }
    }
    *wrk = *ileave < n + 1 || *nenter > 0 || *updatd;
/*     Find the index set of free and active variables at the GCP. */
    *nfree = 0;
    iact = n + 1;
    for (i = 1; i <= n; ++i) {
      if (iwhere[i] <= 0) {
          ++(*nfree);
          indx[*nfree] = i;
      } else {
          --iact;
          indx[iact] = i;
      }
    }
    if (iprint >= 99)
      Rprintf("%d  variables are free at GCP on iteration %d\n",
            *nfree, *iter + 1);
    return;
} /* freev */
/* ======================= The end of freev ============================== */

static void hpsolb(int n, double *t, int *iorder, int iheap)
{
/*    ************

      Subroutine hpsolb

      This subroutine sorts out the least element of t, and puts the
        remaining elements of t in a heap.

      n is an int variable.
        On entry n is the dimension of the arrays t and iorder.
        On exit n is unchanged.

      t is a double precision array of dimension n.
        On entry t stores the elements to be sorted,
        On exit t(n) stores the least elements of t, and t(1) to t(n-1)
          stores the remaining elements in the form of a heap.

      iorder is an int array of dimension n.
        On entry iorder(i) is the index of t(i).
        On exit iorder(i) is still the index of t(i), but iorder may be
          permuted in accordance with t.

      iheap is an int variable specifying the task.
        On entry iheap should be set as follows:
          iheap .eq. 0 if t(1) to t(n) is not in the form of a heap,
          iheap .ne. 0 if otherwise.
        On exit iheap is unchanged.


      References:
        Algorithm 232 of CACM (J. W. J. Williams): HEAPSORT.

                        *      *  *

      NEOS, November 1994. (Latest revision June 1996.)
      Optimization Technology Center.
      Argonne National Laboratory and Northwestern University.
      Written by
                     Ciyou Zhu
      in collaboration with R.H. Byrd, P. Lu-Chen and J. Nocedal.

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

    /* Local variables */
    double ddum;
    int i, j, k, indxin, indxou;
    double out;

    /* Parameter adjustments */
    --iorder;
    --t;

    /* Function Body */
    if (iheap == 0) {
/*      Rearrange the elements t(1) to t(n) to form a heap. */
      for (k = 2; k <= n; ++k) {
          ddum = t[k];
          indxin = iorder[k];
/*         Add ddum to the heap. */
          i = k;
h_loop:
          if (i > 1) {
            j = i / 2;
            if (ddum < t[j]) {
                t[i] = t[j];
                iorder[i] = iorder[j];
                i = j;
                goto h_loop;
            }
          }
          t[i] = ddum;
          iorder[i] = indxin;
/* L20: */
      }
    }
/*     Assign to 'out' the value of t(1), the least member of the heap, */
/*      and rearrange the remaining members to form a heap as */
/*      elements 1 to n-1 of t. */
    if (n > 1) {
      i = 1;
      out = t[1];
      indxou = iorder[1];
      ddum = t[n];
      indxin = iorder[n];
/*      Restore the heap */
Loop:
      j = i + i;
      if (j <= n - 1) {
          if (t[j + 1] < t[j]) {
            ++j;
          }
          if (t[j] < ddum) {
            t[i] = t[j];
            iorder[i] = iorder[j];
            i = j;
            goto Loop;
          }
      }
      t[i] = ddum;
      iorder[i] = indxin;
/*     Put the least member in t(n). */
      t[n] = out;
      iorder[n] = indxou;
    }
    return;
} /* hpsolb */
/* ====================== The end of hpsolb ============================== */

static void lnsrlb(int n, double *l, double *u,
               int *nbd, double *x, double *f, double *fold,
               double *gd, double *gdold, double *g, double *d,
               double *r, double *t, double *z, double *stp,
               double *dnorm, double *dtd, double *xstep,
               double *stpmx, int *iter, int *ifun, int *iback, int *nfgv,
               int *info, char *task, int *boxed, int *cnstnd,
               char *csave, int *isave, double *dsave)
{
/*     **********

      Subroutine lnsrlb

      This subroutine calls subroutine dcsrch from the Minpack2 library
        to perform the line search.  Subroutine dscrch is safeguarded so
        that all trial points lie within the feasible region.

      Subprograms called:

        Minpack2 Library ... dcsrch.

        Linpack ... dtrsl, ddot.


                        *      *  *

      NEOS, November 1994. (Latest revision June 1996.)
      Optimization Technology Center.
      Argonne National Laboratory and Northwestern University.
      Written by
                     Ciyou Zhu
      in collaboration with R.H. Byrd, P. Lu-Chen and J. Nocedal.

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

    /* For dcsrch(): */
    const double stpmin = 0.;
    const double ftol = .001;
    const double gtol = .9;
    const double xtol = .1;

    /* System generated locals */
    double d1;

    /* Local variables */
    int i;
    double a1, a2;

    /* Parameter adjustments */
    --z;
    --t;
    --r;
    --d;
    --g;
    --x;
    --nbd;
    --u;
    --l;

    /* Function Body */
    if (strncmp(task, "FG_LN", 5) == 0) {
      goto L556;
    }
    *dtd = F77_CALL(ddot)(&n, &d[1], &c__1, &d[1], &c__1);
    *dnorm = sqrt(*dtd);
/*     Determine the maximum step length. */
    *stpmx = 1e10;
    if (*cnstnd) {
      if (*iter == 0) {
          *stpmx = 1.;
      } else {
          for (i = 1; i <= n; ++i) {
            a1 = d[i];
            if (nbd[i] != 0) {
                if (a1 < 0. && nbd[i] <= 2) {
                  a2 = l[i] - x[i];
                  if (a2 >= 0.) {
                      *stpmx = 0.;
                  } else if (a1 * *stpmx < a2) {
                      *stpmx = a2 / a1;
                  }
                } else if (a1 > 0. && nbd[i] >= 2) {
                  a2 = u[i] - x[i];
                  if (a2 <= 0.) {
                      *stpmx = 0.;
                  } else if (a1 * *stpmx > a2) {
                      *stpmx = a2 / a1;
                  }
                }
            }
/* L43: */
          }
      }
    }
    if (*iter == 0 && ! (*boxed)) {
      d1 = 1. / *dnorm;
      *stp = min(d1,*stpmx);
    } else {
      *stp = 1.;
    }
    F77_CALL(dcopy)(&n, &x[1], &c__1, &t[1], &c__1);
    F77_CALL(dcopy)(&n, &g[1], &c__1, &r[1], &c__1);
    *fold = *f;
    *ifun = 0;
    *iback = 0;
    strcpy(csave, "START");
L556:
    *gd = F77_CALL(ddot)(&n, &g[1], &c__1, &d[1], &c__1);
    if (*ifun == 0) {
      *gdold = *gd;
      if (*gd >= 0.) {
/*                       the directional derivative >=0. */
/*                       Line search is impossible. */
          *info = -4;
          return;
      }
    }
    dcsrch(f, gd, stp,
         ftol, gtol, xtol,
         stpmin, *stpmx,
         csave, isave, dsave);
    *xstep = *stp * *dnorm;
    if (strncmp(csave, "CONV", 4) != 0 && strncmp(csave, "WARN", 4) != 0) {
      strcpy(task, "FG_LNSRCH");
      ++(*ifun);
      ++(*nfgv);
      *iback = *ifun - 1;
      if (*stp == 1.) {
          F77_CALL(dcopy)(&n, &z[1], &c__1, &x[1], &c__1);
      } else {
          for (i = 1; i <= n; ++i) {
            x[i] = *stp * d[i] + t[i];
          }
      }
    } else {
      strcpy(task, "NEW_X");
    }
    return;
} /* lnsrlb */
/* ======================= The end of lnsrlb ============================= */

static void matupd(int n, int m, double *ws,
               double *wy, double *sy, double *ss, double *d,
               double *r, int *itail, int *iupdat, int *col,
               int *head, double *theta, double *rr, double *dr,
               double *stp, double *dtd)
{
/*    ************

      Subroutine matupd

        This subroutine updates matrices WS and WY, and forms the
          middle matrix in B.

      Subprograms called:

        Linpack ... dcopy, ddot.


                        *      *  *

      NEOS, November 1994. (Latest revision June 1996.)
      Optimization Technology Center.
      Argonne National Laboratory and Northwestern University.
      Written by
                     Ciyou Zhu
      in collaboration with R.H. Byrd, P. Lu-Chen and J. Nocedal.

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

    /* System generated locals */
    int ws_dim1, ws_offset, wy_dim1, wy_offset, sy_dim1, sy_offset,
          ss_dim1, ss_offset, i__1, i__2;

    /* Local variables */
    int j;
    int pointr;

    /* Parameter adjustments */
    --r;
    --d;
    ss_dim1 = m;
    ss_offset = 1 + ss_dim1 * 1;
    ss -= ss_offset;
    sy_dim1 = m;
    sy_offset = 1 + sy_dim1 * 1;
    sy -= sy_offset;
    wy_dim1 = n;
    wy_offset = 1 + wy_dim1 * 1;
    wy -= wy_offset;
    ws_dim1 = n;
    ws_offset = 1 + ws_dim1 * 1;
    ws -= ws_offset;

    /* Function Body */

/*     Set pointers for matrices WS and WY. */
    if (*iupdat <= m) {
      *col = *iupdat;
      *itail = (*head + *iupdat - 2) % m + 1;
    } else {
      *itail = *itail % m + 1;
      *head = *head % m + 1;
    }
/*     Update matrices WS and WY. */
    F77_CALL(dcopy)(&n, &d[1], &c__1, &ws[*itail * ws_dim1 + 1], &c__1);
    F77_CALL(dcopy)(&n, &r[1], &c__1, &wy[*itail * wy_dim1 + 1], &c__1);
/*     Set theta=yy/ys. */
    *theta = *rr / *dr;
/*     Form the middle matrix in B. */
/*      update the upper triangle of SS, */
/*                               and the lower triangle of SY: */
    if (*iupdat > m) {
/*                      move old information */
      i__1 = *col - 1;
      for (j = 1; j <= i__1; ++j) {
          F77_CALL(dcopy)(&j, &ss[(j + 1) * ss_dim1 + 2], &c__1,
                      &ss[j * ss_dim1 + 1], &c__1);
          i__2 = *col - j;
          F77_CALL(dcopy)(&i__2, &sy[j + 1 + (j + 1) * sy_dim1], &c__1,
                      &sy[j + j * sy_dim1], &c__1);
/* L50: */
      }
    }
/*      add new information: the last row of SY */
/*                                   and the last column of SS: */
    pointr = *head;
    i__1 = *col - 1;
    for (j = 1; j <= i__1; ++j) {
      sy[*col + j * sy_dim1] =
          F77_CALL(ddot)(&n, &d[1], &c__1, &wy[pointr * wy_dim1 + 1], &c__1);
      ss[j + *col * ss_dim1] =
          F77_CALL(ddot)(&n, &ws[pointr * ws_dim1 + 1], &c__1, &d[1], &c__1);
      pointr = pointr % m + 1;
/* L51: */
    }
    if (*stp == 1.) {
      ss[*col + *col * ss_dim1] = *dtd;
    } else {
      ss[*col + *col * ss_dim1] = *stp * *stp * *dtd;
    }
    sy[*col + *col * sy_dim1] = *dr;
    return;
} /* matupd */
/* ======================= The end of matupd ============================= */

static void projgr(int n, double *l, double *u,
               int *nbd, double *x, double *g, double *sbgnrm)
{
/*    ************

      Subroutine projgr

      This subroutine computes the infinity norm of the projected gradient.


                        *      *  *

      NEOS, November 1994. (Latest revision April 1997.)
      Optimization Technology Center.
      Argonne National Laboratory and Northwestern University.
      Written by
                     Ciyou Zhu
      in collaboration with R.H. Byrd, P. Lu-Chen and J. Nocedal.

      ************
*/
    int i;
    double gi, d__1;

    *sbgnrm = 0.;
    for (i = 0; i < n; ++i) {
      gi = g[i];
      if (nbd[i] != 0) {
          if (gi < 0.) {
            if (nbd[i] >= 2) {
                if(gi < (d__1 = x[i] - u[i]))
                  gi = d__1;
            }
          } else {
            if (nbd[i] <= 2) {
                if(gi > (d__1 = x[i] - l[i]))
                  gi = d__1;
            }
          }
      }
      if(*sbgnrm < (d__1 = fabs(gi))) *sbgnrm = d__1;
    }
    return;
} /* projgr */


static void subsm(int n, int m, int *nsub, int *ind,
              double *l, double *u, int *nbd, double *x,
              double *d, double *ws, double *wy, double *theta,
              int *col, int *head, int *iword, double *wv,
              double *wn, int iprint, int *info)
{
/*    ************

      Subroutine subsm

      Given xcp, l, u, r, an index set that specifies
      the active set at xcp, and an l-BFGS matrix B
      (in terms of WY, WS, SY, WT, head, col, and theta),
      this subroutine computes an approximate solution
      of the subspace problem

      (P)   min Q(x) = r'(x-xcp) + 1/2 (x-xcp)' B (x-xcp)

            subject to l <= x <= u
                   x_i = xcp_i   for all i in A(xcp)

      along the subspace unconstrained Newton direction

         d = -(Z'BZ)^(-1) r.

      The formula for the Newton direction, given the L-BFGS matrix
      and the Sherman-Morrison formula, is

         d = (1/theta)r + (1/theta*2) Z'WK^(-1)W'Z r.

      where
              K = [-D -Y'ZZ'Y/theta     L_a'-R_z'  ]
                  [L_a -R_z           theta*S'AA'S ]

      Note that this procedure for computing d differs
      from that described in [1]. One can show that the matrix K is
      equal to the matrix M^[-1]N in that paper.

      n is an integer variable.
      On entry n is the dimension of the problem.
      On exit n is unchanged.

      m is an integer variable.
      On entry m is the maximum number of variable metric corrections
        used to define the limited memory matrix.
      On exit m is unchanged.

      nsub is an integer variable.
      On entry nsub is the number of free variables.
      On exit nsub is unchanged.

      ind is an integer array of dimension nsub.
      On entry ind specifies the coordinate indices of free variables.
      On exit ind is unchanged.

      l is a double precision array of dimension n.
      On entry l is the lower bound of x.
      On exit l is unchanged.

      u is a double precision array of dimension n.
      On entry u is the upper bound of x.
      On exit u is unchanged.

      nbd is a integer array of dimension n.
      On entry nbd represents the type of bounds imposed on the
        variables, and must be specified as follows:
        nbd(i)=0 if x(i) is unbounded,
             1 if x(i) has only a lower bound,
             2 if x(i) has both lower and upper bounds, and
             3 if x(i) has only an upper bound.
      On exit nbd is unchanged.

      x is a double precision array of dimension n.
      On entry x specifies the Cauchy point xcp.
      On exit x(i) is the minimizer of Q over the subspace of
               free variables.

      d is a double precision array of dimension n.
      On entry d is the reduced gradient of Q at xcp.
      On exit d is the Newton direction of Q.

      ws and wy are double precision arrays;
      theta is a double precision variable;
      col is an integer variable;
      head is an integer variable.
      On entry they store the information defining the
                                 limited memory BFGS matrix:
        ws(n,m) stores S, a set of s-vectors;
        wy(n,m) stores Y, a set of y-vectors;
        theta is the scaling factor specifying B_0 = theta I;
        col is the number of variable metric corrections stored;
        head is the location of the 1st s- (or y-) vector in S (or Y).
      On exit they are unchanged.

      iword is an integer variable.
      On entry iword is unspecified.
      On exit iword specifies the status of the subspace solution.
        iword = 0 if the solution is in the box,
              1 if some bound is encountered.

      wv is a double precision working array of dimension 2m.

      wn is a double precision array of dimension 2m x 2m.
      On entry the upper triangle of wn stores the LEL^T factorization
        of the indefinite matrix

             K = [-D -Y'ZZ'Y/theta   L_a'-R_z'  ]
               [L_a -R_z             theta*S'AA'S ]
                                         where E = [-I  0]
                                                 [ 0  I]
      On exit wn is unchanged.

      iprint is an INTEGER variable that must be set by the user.
      It controls the frequency and type of output generated:
       iprint<0    no output is generated;
       iprint=0    print only one line at the last iteration;
       0<iprint<99 print also f and |proj g| every iprint iterations;
       iprint=99   print details of every iteration except n-vectors;
       iprint=100  print also the changes of active set and final x;
       iprint>100  print details of every iteration including x and g;
      When iprint > 0, the file iterate.dat will be created to
                   summarize the iteration.

      info is an integer variable.
      On entry info is unspecified.
      On exit info = 0       for normal return,
                 = nonzero for abnormal return
                           when the matrix K is ill-conditioned.

      Subprograms called:

      Linpack dtrsl.


      References:

      [1] R. H. Byrd, P. Lu, J. Nocedal and C. Zhu, ``A limited
      memory algorithm for bound constrained optimization'',
      SIAM J. Scientific Computing 16 (1995), no. 5, pp. 1190--1208.



                      *  *  *

      NEOS, November 1994. (Latest revision June 1996.)
      Optimization Technology Center.
      Argonne National Laboratory and Northwestern University.
      Written by
                     Ciyou Zhu
      in collaboration with R.H. Byrd, P. Lu-Chen and J. Nocedal.

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

    /* System generated locals */
    int ws_offset, wn_dim1, wn_offset;

    /* Local variables */
    double alpha, dk, temp1, temp2;
    int i, j, k, m2, js, jy, pointr, ibd = 0, col2, ns;

    /* Parameter adjustments */
    --d;
    --u;
    --l;
    --x;
    --ind;
    --nbd;
    --wv;
    wn_dim1 = 2 * m;
    wn_offset = 1 + wn_dim1 * 1;
    wn -= wn_offset;
    /* ws[] and wy[] are both  [n x m ] :*/
    ws_offset = 1 + n * 1;
    ws -= ws_offset;
    wy -= ws_offset;

    ns = *nsub;
    if (ns <= 0)
      return;

/*     Compute wv = W'Zd. */
    pointr = *head;
    for (i = 1; i <= *col; ++i) {
      temp1 = 0.;
      temp2 = 0.;
      for (j = 1; j <= ns; ++j) {
          k = ind[j];
          temp1 += wy[k + pointr * n] * d[j];
          temp2 += ws[k + pointr * n] * d[j];
      }
      wv[i] = temp1;
      wv[*col + i] = *theta * temp2;
      pointr = pointr % m + 1;
/* L20: */
    }
/*     Compute wv:=K^(-1)wv. */
    m2 = m << 1;
    col2 = *col << 1;
    F77_CALL(dtrsl)(&wn[wn_offset], &m2, &col2, &wv[1], &c__11, info);
    if (*info != 0) {
      return;
    }
    for (i = 1; i <= *col; ++i)
      wv[i] = -wv[i];

    F77_CALL(dtrsl)(&wn[wn_offset], &m2, &col2, &wv[1], &c__1, info);
    if (*info != 0) {
      return;
    }
/*     Compute d = (1/theta)d + (1/theta**2)Z'W wv. */
    pointr = *head;
    for (jy = 1; jy <= *col; ++jy) {
      js = *col + jy;
      for (i = 1; i <= ns; ++i) {
          k = ind[i];
          d[i] += (wy[k + pointr * n] * wv[jy] / *theta +
                   ws[k + pointr * n] * wv[js]);
      }
      pointr = pointr % m + 1;
/* L40: */
    }

    for (i = 1; i <= ns; ++i)
      d[i] /= *theta;

/*     Backtrack to the feasible region. */
    alpha = 1.;
    temp1 = alpha;
    for (i = 1; i <= ns; ++i) {
      k = ind[i];
      dk = d[i];
      if (nbd[k] != 0) {
          if (dk < 0. && nbd[k] <= 2) {
            temp2 = l[k] - x[k];
            if (temp2 >= 0.) {
                temp1 = 0.;
            } else if (dk * alpha < temp2) {
                temp1 = temp2 / dk;
            }
          } else if (dk > 0. && nbd[k] >= 2) {
            temp2 = u[k] - x[k];
            if (temp2 <= 0.) {
                temp1 = 0.;
            } else if (dk * alpha > temp2) {
                temp1 = temp2 / dk;
            }
          }
          if (temp1 < alpha) {
            alpha = temp1;
            ibd = i;
          }
      }
/* L60: */
    }
    if (alpha < 1.) {
      dk = d[ibd];
      k = ind[ibd];
      if (dk > 0.) {
          x[k] = u[k];
          d[ibd] = 0.;
      } else if (dk < 0.) {
          x[k] = l[k];
          d[ibd] = 0.;
      }
    }
    for (i = 1; i <= ns; ++i)
      x[ind[i]] += alpha * d[i];

    *iword = (alpha < 1.) ? 1 : 0;

    return;
} /* subsm */
/* ====================== The end of subsm =============================== */

static void dcsrch(double *f, double *g, double *stp,
               /*Chgd: the next five are no longer pointers:*/
               double ftol, double gtol, double xtol,
               double stpmin, double stpmax,
               char *task, int *isave, double *dsave)
{
/*    **********

      Subroutine dcsrch

      This subroutine finds a step that satisfies a sufficient
      decrease condition and a curvature condition.

      Each call of the subroutine updates an interval with
      endpoints stx and sty. The interval is initially chosen
      so that it contains a minimizer of the modified function

            psi(stp) = f(stp) - f(0) - ftol*stp*f'(0).

      If psi(stp) <= 0 and f'(stp) >= 0 for some step, then the
      interval is chosen so that it contains a minimizer of f.

      The algorithm is designed to find a step that satisfies
      the sufficient decrease condition

            f(stp) <= f(0) + ftol*stp*f'(0),

      and the curvature condition

            abs(f'(stp)) <= gtol*abs(f'(0)).

      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 that satisfies both conditions, then
      the algorithm stops with a warning. In this case stp only
      satisfies the sufficient decrease condition.

      A typical invocation of dcsrch has the following outline:

      task = 'START'
     10 continue
         call dcsrch( ... )
         if (task .eq. 'FG') then
            Evaluate the function and the gradient at stp
            goto 10
            end if

      NOTE: The user must no alter work arrays between calls.

     The subroutine statement is

      subroutine dcsrch(f,g,stp,ftol,gtol,xtol,stpmin,stpmax,
                    task,isave,dsave)
     where

      f is a double precision variable.
        On initial entry f is the value of the function at 0.
           On subsequent entries f is the value of the
           function at stp.
        On exit f is the value of the function at stp.

      g is a double precision variable.
        On initial entry g is the derivative of the function at 0.
           On subsequent entries g is the derivative of the
           function at stp.
        On exit g is the derivative of the function at stp.

      stp is a double precision variable.
        On entry stp is the current estimate of a satisfactory
           step. On initial entry, a positive initial estimate
           must be provided.
        On exit stp is the current estimate of a satisfactory step
           if task = 'FG'. If task = 'CONV' then stp satisfies
           the sufficient decrease and curvature condition.

      ftol is a double precision variable.
        On entry ftol specifies a nonnegative tolerance for the
           sufficient decrease condition.
        On exit ftol is unchanged.

      gtol is a double precision variable.
        On entry gtol specifies a nonnegative tolerance for the
           curvature condition.
        On exit gtol is unchanged.

      xtol is a double precision variable.
        On entry xtol specifies a nonnegative relative tolerance
           for an acceptable step. The subroutine exits with a
           warning if the relative difference between sty and stx
           is less than xtol.
        On exit xtol is unchanged.

      stpmin is a double precision variable.
        On entry stpmin is a nonnegative lower bound for the step.
        On exit stpmin is unchanged.

      stpmax is a double precision variable.
        On entry stpmax is a nonnegative upper bound for the step.
        On exit stpmax is unchanged.

      task is a character variable of length at least 60.
        On initial entry task must be set to 'START'.
        On exit task indicates the required action:

           If task(1:2) = 'FG' then evaluate the function and
           derivative at stp and call dcsrch again.

           If task(1:4) = 'CONV' then the search is successful.

           If task(1:4) = 'WARN' then the subroutine is not able
           to satisfy the convergence conditions. The exit value of
           stp contains the best point found during the search.

           If task(1:5) = 'ERROR' then there is an error in the
           input arguments.

        On exit with convergence, a warning or an error, the
           variable task contains additional information.

      isave is an integer work array of dimension 2.

      dsave is a double precision work array of dimension 13.

     Subprograms called

      MINPACK-2 ... dcstep


      MINPACK-1 Project. June 1983.
      Argonne National Laboratory.
      Jorge J. More' and David J. Thuente.

      MINPACK-2 Project. October 1993.
      Argonne National Laboratory and University of Minnesota.
      Brett M. Averick, Richard G. Carter, and Jorge J. More'.

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

    /* Local variables */
    int stage;
    double finit, ginit, width, ftest, gtest, stmin, stmax, width1, fm,
          gm, fx, fy, gx, gy;
    int brackt;
    double fxm, fym, gxm, gym, stx, sty;

    /* Parameter adjustments */
    --dsave;
    --isave;

    /* Function Body */

/*     Initialization block. */
    if (strncmp(task, "START", 5) == 0) {
/*      Check the input arguments for errors. */
      if (*stp < stpmin)      strcpy(task, "ERROR: STP .LT. STPMIN");
      if (*stp > stpmax)      strcpy(task, "ERROR: STP .GT. STPMAX");
      if (*g >= 0.)           strcpy(task, "ERROR: INITIAL G .GE. ZERO");
      if (ftol < 0.)          strcpy(task, "ERROR: FTOL .LT. ZERO");
      if (gtol < 0.)          strcpy(task, "ERROR: GTOL .LT. ZERO");
      if (xtol < 0.)          strcpy(task, "ERROR: XTOL .LT. ZERO");
      if (stpmin < 0.)  strcpy(task, "ERROR: STPMIN .LT. ZERO");
      if (stpmax < stpmin)    strcpy(task, "ERROR: STPMAX .LT. STPMIN");

/*      Exit if there are errors on input. */
      if (strncmp(task, "ERROR", 5) == 0) {
          return;
      }
/*      Initialize local variables. */
      brackt = FALSE_;
      stage = 1;
      finit = *f;
      ginit = *g;
      gtest = ftol * ginit;
      width = stpmax - stpmin;
      width1 = width / .5;
/*      The variables stx, fx, gx contain the values of the step, */
/*      function, and derivative at the best step. */
/*      The variables sty, fy, gy contain the value of the step, */
/*      function, and derivative at sty. */
/*      The variables stp, f, g contain the values of the step, */
/*      function, and derivative at stp. */
      stx = 0.;   fx = finit; gx = ginit;
      sty = 0.;   fy = finit; gy = ginit;
      stmin = 0.;
      stmax = *stp + *stp * 4.;
      strcpy(task, "FG");
      goto L1000;
    } else {
/*      Restore local variables. */
      if (isave[1] == 1) {
          brackt = TRUE_;
      } else {
          brackt = FALSE_;
      }
      stage = isave[2];
      ginit = dsave[1];
      gtest = dsave[2];
      gx = dsave[3];
      gy = dsave[4];
      finit = dsave[5];
      fx = dsave[6];
      fy = dsave[7];
      stx = dsave[8];
      sty = dsave[9];
      stmin = dsave[10];
      stmax = dsave[11];
      width = dsave[12];
      width1 = dsave[13];
    }
/*      If psi(stp) <= 0 and f'(stp) >= 0 for some step, then the */
/*      algorithm enters the second stage. */
    ftest = finit + *stp * gtest;
    if (stage == 1 && *f <= ftest && *g >= 0.) {
      stage = 2;
    }
/*    Test for warnings. */
    if (brackt && (*stp <= stmin || *stp >= stmax))
      strcpy(task, "WARNING: ROUNDING ERRORS PREVENT PROGRESS");
    if (brackt && stmax - stmin <= xtol * stmax)
      strcpy(task, "WARNING: XTOL TEST SATISFIED");
    if (*stp == stpmax && *f <= ftest && *g <= gtest)
      strcpy(task, "WARNING: STP = STPMAX");
    if (*stp == stpmin && (*f > ftest || *g >= gtest))
      strcpy(task, "WARNING: STP = STPMIN");
/*    Test for convergence. */
    if (*f <= ftest && fabs(*g) <= gtol * (-ginit))
      strcpy(task, "CONVERGENCE");
/*    Test for termination. */
    if (strncmp(task, "WARN", 4) == 0 || strncmp(task, "CONV", 4) == 0)
      goto L1000;

/*     A modified function is used to predict the step during the */
/*     first stage if a lower function value has been obtained but */
/*     the decrease is not sufficient. */
    if (stage == 1 && *f <= fx && *f > ftest) {
/*      Define the modified function and derivative values. */
      fm = *f - *stp * gtest;
      fxm = fx - stx * gtest;
      fym = fy - sty * gtest;
      gm = *g - gtest;
      gxm = gx - gtest;
      gym = gy - gtest;
/*      Call dcstep to update stx, sty, and to compute the new step. */
      dcstep(&stx, &fxm, &gxm, &sty, &fym, &gym, stp, &fm, &gm, &brackt, &
            stmin, &stmax);
/*      Reset the function and derivative values for f. */
      fx = fxm + stx * gtest;
      fy = fym + sty * gtest;
      gx = gxm + gtest;
      gy = gym + gtest;
    } else {
/*     Call dcstep to update stx, sty, and to compute the new step. */
      dcstep(&stx, &fx, &gx, &sty, &fy, &gy, stp, f, g, &brackt, &stmin, &
            stmax);
    }
/*     Decide if a bisection step is needed. */
    if (brackt) {
      if (fabs(sty - stx) >= width1 * .66) {
          *stp = stx + (sty - stx) * .5;
      }
      width1 = width;
      width = fabs(sty - stx);
    }
/*     Set the minimum and maximum steps allowed for stp. */
    if (brackt) {
      stmin = min(stx,sty);
      stmax = max(stx,sty);
    } else {
      stmin = *stp + (*stp - stx) * 1.1;
      stmax = *stp + (*stp - stx) * 4.;
    }
/*     Force the step to be within the bounds stpmax and stpmin. */
    if(*stp < stpmin) *stp = stpmin;
    if(*stp > stpmax) *stp = stpmax;

/*     If further progress is not possible, let stp be the best */
/*     point obtained during the search. */
    if ((brackt && (*stp <= stmin || *stp >= stmax)) ||
      (brackt && (stmax - stmin <= xtol * stmax))) {
      *stp = stx;
    }
/*     Obtain another function and derivative. */
    strcpy(task, "FG");
L1000:
/*     Save local variables. */
    if (brackt) {
      isave[1] = 1;
    } else {
      isave[1] = 0;
    }
    isave[2] = stage;
    dsave[1] = ginit;
    dsave[2] = gtest;
    dsave[3] = gx;
    dsave[4] = gy;
    dsave[5] = finit;
    dsave[6] = fx;
    dsave[7] = fy;
    dsave[8] = stx;
    dsave[9] = sty;
    dsave[10] = stmin;
    dsave[11] = stmax;
    dsave[12] = width;
    dsave[13] = width1;
    return;
} /* dcsrch */
/* ====================== The end of dcsrch ============================== */

static void dcstep(double *stx, double *fx, double *dx,
               double *sty, double *fy, double *dy, double *stp,
               double *fp, double *dp, int *brackt, double *stpmin,
               double *stpmax)
{
/*    **********

      Subroutine dcstep

      This subroutine computes a safeguarded step for a search
      procedure and updates an interval that contains a step that
      satisfies a sufficient decrease and a curvature condition.

      The parameter stx contains the step with the least function
      value. If brackt is set to .true. then a minimizer has
      been bracketed in an interval with endpoints stx and sty.
      The parameter stp contains the current step.
      The subroutine assumes that if brackt is set to .true. then

            min(stx,sty) < stp < max(stx,sty),

      and that the derivative at stx is negative in the direction
      of the step.

      The subroutine statement is

      subroutine dcstep(stx,fx,dx,sty,fy,dy,stp,fp,dp,brackt,
                    stpmin,stpmax)

      where

      stx is a double precision variable.
        On entry stx is the best step obtained so far and is an
           endpoint of the interval that contains the minimizer.
        On exit stx is the updated best step.

      fx is a double precision variable.
        On entry fx is the function at stx.
        On exit fx is the function at stx.

      dx is a double precision variable.
        On entry dx is the derivative of the function at
           stx. The derivative must be negative in the direction of
           the step, that is, dx and stp - stx must have opposite
           signs.
        On exit dx is the derivative of the function at stx.

      sty is a double precision variable.
        On entry sty is the second endpoint of the interval that
           contains the minimizer.
        On exit sty is the updated endpoint of the interval that
           contains the minimizer.

      fy is a double precision variable.
        On entry fy is the function at sty.
        On exit fy is the function at sty.

      dy is a double precision variable.
        On entry dy is the derivative of the function at sty.
        On exit dy is the derivative of the function at the exit sty.

      stp is a double precision variable.
        On entry stp is the current step. If brackt is set to .true.
           then on input stp must be between stx and sty.
        On exit stp is a new trial step.

      fp is a double precision variable.
        On entry fp is the function at stp
        On exit fp is unchanged.

      dp is a double precision variable.
        On entry dp is the the derivative of the function at stp.
        On exit dp is unchanged.

      brackt is an logical variable.
        On entry brackt specifies if a minimizer has been bracketed.
           Initially brackt must be set to .false.
        On exit brackt specifies if a minimizer has been bracketed.
           When a minimizer is bracketed brackt is set to .true.

      stpmin is a double precision variable.
        On entry stpmin is a lower bound for the step.
        On exit stpmin is unchanged.

      stpmax is a double precision variable.
        On entry stpmax is an upper bound for the step.
        On exit stpmax is unchanged.

      MINPACK-1 Project. June 1983
      Argonne National Laboratory.
      Jorge J. More' and David J. Thuente.

      MINPACK-2 Project. October 1993.
      Argonne National Laboratory and University of Minnesota.
      Brett M. Averick and Jorge J. More'.

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

    /* System generated locals */
    double d__1, d__2;

    /* Local variables */
    double sgnd, stpc, stpf, stpq, p, q, gamm, r__, s, theta;

    sgnd = *dp * (*dx / fabs(*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, otherwise the average of the cubic and */
/*     quadratic steps is taken. */
    if (*fp > *fx) {
      theta = (*fx - *fp) * 3. / (*stp - *stx) + *dx + *dp;
/* Computing MAX */
      d__1 = fabs(theta), d__2 = fabs(*dx),
          d__1 = max(d__1,d__2), d__2 = fabs(*dp);
      s = max(d__1,d__2);
/* Computing 2nd power */
      d__1 = theta / s;
      gamm = s * sqrt(d__1 * d__1 - *dx / s * (*dp / s));
      if (*stp < *stx) {
          gamm = -gamm;
      }
      p = gamm - *dx + theta;
      q = gamm - *dx + gamm + *dp;
      r__ = p / q;
      stpc = *stx + r__ * (*stp - *stx);
      stpq = *stx + *dx / ((*fx - *fp) / (*stp - *stx) + *dx) / 2. * (*stp
            - *stx);
      if (fabs(stpc - *stx) < fabs(stpq - *stx)) {
          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 farther from */
/*     stp than the secant step, the cubic step is taken, otherwise the */
/*     secant step is taken. */
    } else if (sgnd < 0.) {
      theta = (*fx - *fp) * 3. / (*stp - *stx) + *dx + *dp;
/* Computing MAX */
      d__1 = fabs(theta), d__2 = fabs(*dx),
          d__1 = max(d__1,d__2), d__2 = fabs(*dp);
      s = max(d__1,d__2);
/* Computing 2nd power */
      d__1 = theta / s;
      gamm = s * sqrt(d__1 * d__1 - *dx / s * (*dp / s));
      if (*stp > *stx) {
          gamm = -gamm;
      }
      p = gamm - *dp + theta;
      q = gamm - *dp + gamm + *dx;
      r__ = p / q;
      stpc = *stp + r__ * (*stx - *stp);
      stpq = *stp + *dp / (*dp - *dx) * (*stx - *stp);
      if (fabs(stpc - *stp) > fabs(stpq - *stp)) {
          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. */
    } else if (fabs(*dp) < fabs(*dx)) {
/*      The cubic step is computed only 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 the */
/*      secant step. */
      theta = (*fx - *fp) * 3. / (*stp - *stx) + *dx + *dp;
/* Computing MAX */
      d__1 = fabs(theta), d__2 = fabs(*dx),
          d__1 = max(d__1,d__2), d__2 = fabs(*dp);
      s = max(d__1,d__2);
/*      The case gamm = 0 only arises if the cubic does not tend */
/*      to infinity in the direction of the step. */
/* Computing MAX */
/* Computing 2nd power */
      d__1 = theta / s;
      d__1 = d__1 * d__1 - *dx / s * (*dp / s);
      gamm = d__1 < 0 ? 0. : s * sqrt(d__1);
      if (*stp > *stx) {
          gamm = -gamm;
      }
      p = gamm - *dp + theta;
      q = gamm + (*dx - *dp) + gamm;
      r__ = p / q;
      if (r__ < 0. && gamm != 0.) {
          stpc = *stp + r__ * (*stx - *stp);
      } else if (*stp > *stx) {
          stpc = *stpmax;
      } else {
          stpc = *stpmin;
      }
      stpq = *stp + *dp / (*dp - *dx) * (*stx - *stp);
      if (*brackt) {
/*         A minimizer has been bracketed. If the cubic step is */
/*         closer to stp than the secant step, the cubic step is */
/*         taken, otherwise the secant step is taken. */
          if (fabs(stpc - *stp) < fabs(stpq - *stp)) {
            stpf = stpc;
          } else {
            stpf = stpq;
          }
          d__1 = *stp + (*sty - *stp) * .66;
          if (*stp > *stx) {
            stpf = min(d__1,stpf);
          } else {
            stpf = max(d__1,stpf);
          }
      } else {
/*         A minimizer has not been bracketed. If the cubic step is */
/*         farther from stp than the secant step, the cubic step is */
/*         taken, otherwise the secant step is taken. */
          if (fabs(stpc - *stp) > fabs(stpq - *stp)) {
            stpf = stpc;
          } else {
            stpf = stpq;
          }
          stpf = min(*stpmax,stpf);
          stpf = max(*stpmin,stpf);
      }
/*     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, otherwise the cubic step is taken. */
    } else {
      if (*brackt) {
          theta = (*fp - *fy) * 3. / (*sty - *stp) + *dy + *dp;
/* Computing MAX */
          d__1 = fabs(theta), d__2 = fabs(*dy), d__1 = max(d__1,d__2), d__2 =
                fabs(*dp);
          s = max(d__1,d__2);
/* Computing 2nd power */
          d__1 = theta / s;
          gamm = s * sqrt(d__1 * d__1 - *dy / s * (*dp / s));
          if (*stp > *sty) {
            gamm = -gamm;
          }
          p = gamm - *dp + theta;
          q = gamm - *dp + gamm + *dy;
          r__ = p / q;
          stpc = *stp + r__ * (*sty - *stp);
          stpf = stpc;
      } else if (*stp > *stx) {
          stpf = *stpmax;
      } else {
          stpf = *stpmin;
      }
    }
/*     Update the interval which contains a minimizer. */
    if (*fp > *fx) {
      *sty = *stp;
      *fy = *fp;
      *dy = *dp;
    } else {
      if (sgnd < 0.) {
          *sty = *stx;
          *fy = *fx;
          *dy = *dx;
      }
      *stx = *stp;
      *fx = *fp;
      *dx = *dp;
    }
/*     Compute the new step. */
    *stp = stpf;
    return;
} /* dcstep */
/* ====================== The end of dcstep ============================== */

static void pvector(char *title, double *x, int n)
{
    int i;
    Rprintf("%s ", title);
    for (i = 0; i < n; i++) Rprintf("%g ", x[i]);
    Rprintf("\n");
}


static void prn1lb(int n, int m, double *l, double *u, double *x,
               int iprint, double epsmch)
{
    if (iprint >=  0) {
      Rprintf("N = %d, M = %d machine precision = %g\n", n, m, epsmch);
      if (iprint >= 100){
          pvector("L =", l, n);
          pvector("X0 =",x, n);
          pvector("U =", u, n);
      }
    }
}


static void prn2lb(int n, double *x, double *f, double *g, int iprint,
               int iter, int nfgv, int nact, double sbgnrm,
               int nint, char *word, int iword, int iback,
               double stp, double xstep)
{
    if (iprint >=  99) {
      Rprintf("LINE SEARCH %d times; norm of step = %g\n", iback, xstep);
      if (iprint > 100) {
          pvector("X =", x, n);
          pvector("G =", g, n);
      }
    } else if (iprint > 0 && iter%iprint == 0) {
      Rprintf("At iterate %5d  f = %12.5g  |proj g|=  %12.5g\n",
            iter, *f, sbgnrm);
    }
}

static void prn3lb(int n, double *x, double *f, char *task, int iprint,
               int info, int iter, int nfgv, int nintol, int nskip,
               int nact, double sbgnrm, int nint,
               char *word, int iback, double stp, double xstep,
               int k)
{
    if(strncmp(task, "CONV", 4) == 0) {
      if (iprint >= 0) {
          Rprintf("\niterations %d\nfunction evaluations %d\nsegments explored during Cauchy searches %d\nBFGS updates skipped %d\nactive bounds at final generalized Cauchy point %d\nnorm of the final projected gradient %g\nfinal function value %g\n\n", iter, nfgv, nintol, nskip, nact, sbgnrm, *f);
      }
      if (iprint >= 100) pvector("X =", x, n);
      if (iprint >= 1) Rprintf("F = %g\n", *f);
    }
    if (iprint >= 0) {
      switch(info) {
      case -1: Rprintf("Matrix in 1st Cholesky factorization in formk is not Pos. Def."); break;
      case -2: Rprintf("Matrix in 2st Cholesky factorization in formk is not Pos. Def."); break;
      case -3: Rprintf("Matrix in the Cholesky factorization in formt is not Pos. Def."); break;
      case -4: Rprintf("Derivative >= 0, backtracking line search impossible."); break;
      case -5: Rprintf("l(%d) > u(%d).  No feasible solution", k, k); break;
      case -6: Rprintf("Input nbd(%d) is invalid", k); break;
      case -7: Rprintf("Warning:  more than 10 function and gradient evaluations\n   in the last line search\n"); break;
      case -8: Rprintf("The triangular system is singular."); break;
      case -9: Rprintf("%s\n%s\n", "Line search cannot locate an adequate point after 20 function", "and gradient evaluations"); break;
      default: break;
      }
    }
}




#ifdef NOT_USING_DBL_EPSILON

static double dpmeps(void)
{
/*     **********

     Subroutine dpeps

     This subroutine computes the machine precision parameter
     dpmeps as the smallest floating point number such that
     1 + dpmeps differs from 1.

     This subroutine is based on the subroutine machar described in

     W. J. Cody,
     MACHAR: A subroutine to dynamically determine machine parameters,
     ACM Trans. Math. Soft., 14, 1988, pages 303-311.

     The subroutine statement is:

       subroutine dpeps(dpmeps)

     where

       dpmeps is a double precision variable.
       On entry dpmeps need not be specified.
       On exit dpmeps is the machine precision.

     MINPACK-2 Project. February 1991.
     Argonne National Laboratory and University of Minnesota.
     Brett M. Averick.

     ******* */

    /* Initialized data */

    static double zero = 0.;
    static double one = 1.;
    static double two = 2.;

    /* System generated locals */
    int i__1;
    double ret_val;

    /* Local variables */
    double beta;
    int irnd;
    double temp, temp1, a, b;
    int i;
    double betah;
    int ibeta, negep;
    double tempa;
    int itemp, it;
    double betain;

/*     determine ibeta, beta ala malcolm. */
    a = one;
    b = one;
L10:
    a += a;
    temp = a + one;
    temp1 = temp - a;
    if (temp1 - one == zero) {
      goto L10;
    }
L20:
    b += b;
    temp = a + b;
    itemp = (int) (temp - a);
    if (itemp == 0) {
      goto L20;
    }
    ibeta = itemp;
    beta = (double) ibeta;
/*     determine it, irnd. */
    it = 0;
    b = one;
L30:
    ++it;
    b *= beta;
    temp = b + one;
    temp1 = temp - b;
    if (temp1 - one == zero) {
      goto L30;
    }
    irnd = 0;
    betah = beta / two;
    temp = a + betah;
    if (temp - a != zero) {
      irnd = 1;
    }
    tempa = a + beta;
    temp = tempa + betah;
    if (irnd == 0 && temp - tempa != zero) {
      irnd = 2;
    }
/*     determine dpmeps. */
    negep = it + 3;
    betain = one / beta;
    a = one;
    i__1 = negep;
    for (i = 1; i <= i__1; ++i) {
      a *= betain;
    }
L50:
    temp = one + a;
    if (temp - one != zero) {
      goto L60;
    }
    a *= beta;
    goto L50;
L60:
    ret_val = a;
    if (ibeta == 2 || irnd == 0) {
      goto L70;
    }
    a = a * (one + a) / two;
    temp = one + a;
    if (temp - one != zero) {
      ret_val = a;
    }
L70:
    return ret_val;
} /* dpmeps */

#endif

Generated by  Doxygen 1.6.0   Back to index