/src/r-source/src/appl/lbfgsb.c

Source
/*
 *  R : A Computer Language for Statistical Data Analysis
 *  Copyright (C) 2000-2023 The R 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
 *  https://www.R-project.org/Licenses/
 */
/* l-bfgs-b.f -- translated by f2c (version 19991025).

   This file is included by optim.c.

  From ?optim:
  The code for method ‘"L-BFGS-B"’ is based on Fortran code by Zhu,
  Byrd, Lu-Chen and Nocedal obtained from Netlib (file 'opt/lbfgs_bcm.shar')

  The Fortran files contained no copyright information.

  Byrd, R. H., Lu, P., Nocedal, J. and Zhu, C.  (1995) A limited
  memory algorithm for bound constrained optimization.
  \emph{SIAM J. Scientific Computing}, \bold{16}, 1190--1208.
*/

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

// This has BLAS calls daxpy dcopy ddot dscal

#include <math.h>
#include <float.h> /* for DBL_EPSILON */
#include <string.h>
#include <R_ext/RS.h> /* for F77_CALL */
#include <R_ext/BLAS.h>
#include <R_ext/Linpack.h> /* dpofa, dtrsl  */
//#include <R_ext/Applic.h> // included by optim.c

#include <R_ext/Print.h> /* Rprintf */

#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;

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 *);
static void dcstep(double *, double *,
       double *, double *, double *, double *,
       double *, double *, double *, int *, double *,
       double *);
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 *);
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 *);
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) ========================== */
static
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 *isave)
{
/*     ************

       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.

       *** These are all Fortan 1-based indices.
       *** now only used local to mainlb.
       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;

       *** re-arranged to be of size 21, reduce indices by 21.
       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.

       *** re-arranged to be of size 16, no longer exported.
       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 */
    static int lsnd, ld, lr, lt, lz, lwa, lwn, lss, lws, lwt, lsy, lwy;

    /* make sure csave is initialized */
    csave[0] = '\0';

    /* Parameter adjustments */
    --wa;

    if (strncmp(task, "START", 5) == 0) {
  lws = 1;
  lwy = lws + m * n;
  lsy = lwy + m * n;
  lss = lsy + m * m;
  lwt = lss + m * m;
  lwn = lwt + m * m;
  lsnd = lwn + (m * m << 2);
  lz = lsnd + (m * m << 2);
  lr = lz + n;
  ld = lr + n;
  lt = ld + n;
  lwa = lt + n;
    }

    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, isave);
    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 *isave)
{
/*     ************
       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.

       isave is an integer working array of dimension 21.

       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;
    double d__1, d__2;

    /* Local variables */
    double ddum;
    char word[4]; /* allow for terminator */
    int k = 0; /* -Wall */
    double xstep = 0.0; /* printed before being used */
    double dr, rr;
    int wrk;


    /* Parameter adjustments */
    --indx2;
    --iwhere;
    --indx;
    --t;
    --d;
    --r;
    --z;
    --g;
    --nbd;
    --u;
    --l;
    --x;
    --wa;
    // formerly lsave
    static int prjctd, cnstnd, boxed, updatd;
    // in isave
    static int nintol, iback, nskip, head, col, itail, iter, iupdat,
  nint, nfgv, info, ifun, iword, nfree, nact, ileave, nenter;
    // formerly dsave
    static double theta, fold, tol, dnorm, epsmch, gd, stpmx, sbgnrm,
  stp, gdold, dtd;

    /* Function Body */
    if (strncmp(task, "START", 5) == 0) {
/*    Generate the current machine precision. */
  epsmch = DBL_EPSILON;
  fold = 0.;
  dnorm = 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;
/*       'word' records the status of subspace solutions. */
  strcpy(word, "---");
/*       'info' records the termination information. */
  info = 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 {
/*  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 */
    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_;
  goto L222;
    }
    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 */
/*     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_;
  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_;
  goto L222;
    }
L555:
/* ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc */

/*     Line search and optimality tests. */

/* ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc */
/*     Generate the search direction d:=z-x. */
    for (int i = 1; i <= n; ++i)
  d[i] = z[i] - x[i];
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);
    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");
      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. */
  ++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. */
    for (int i = 1; i <= n; ++i)
  r[i] = g[i] - r[i];
    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) {
  if (iprint >= 0)
      Rprintf("%s\n%s\n",
        "Nonpositive definiteness in Cholesky factorization in formt();",
        "   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:
L1000:
/*     Save local variables. */
//    isave[1-1] = nintol;
//    isave[3-1] = itfile;
//    isave[4-1] = iback;
//    isave[5-1] = nskip;
//    isave[6-1] = head;
//    isave[7-1] = col;
//    isave[8-1] = itail;
//    isave[9-1] = iter;
//    isave[10-1] = iupdat;
//    isave[12-1] = nint;
    isave[13-1] = nfgv;
//    isave[14-1] = info;
//    isave[15-1] = ifun;
//    isave[16-1] = iword;
//    isave[17-1] = nfree;
//    isave[18-1] = nact;
//    isave[19-1] = ileave;
//    isave[20-1] = nenter;

    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;

    /* 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 (int 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 (int 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 i2, k;
    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 (int i = 2; i <= Col; ++i) {
  i2 = *col + i;
  sum = 0.;
  for (int k = 1; k <= i - 1; ++k)
      sum += sy[i + k * sy_dim1] * v[k] / sy[k + k * sy_dim1];
  p[i2] = v[i2] + sum;
    }
/*     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 (int 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 (int i = 1; i <= Col; ++i) {
  p[i] = -p[i] / sqrt(sy[i + i * sy_dim1]);
    }
    for (int 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;
    }
    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 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 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 (int 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 (int 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 (int 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_;
      }
  }
    } /* 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 (int i = 1; i <= n; i++) Rprintf("%g ", xcp[i]);
      Rprintf("\n");
  }
  return;
    }
/*     Initialize c = W'(xcp - x) = 0. */
    for (int 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 (int 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 (int 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 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 (int i = 1; i <= n; ++i)
      r[i] = -g[i];
    }
    else {
  n_f = *nfree;
  for (int 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 (int j = 1; j <= Col; ++j) {
      a1 = wa[j];
      a2 = *theta * wa[Col + j];
      for (int 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.

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

    /* 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 (int 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;

    /* Local variables */
    int dend, pend;
    int upcl;
    double temp1, temp2, temp3, temp4;
//    int i, k;
    int ipntr, jpntr, k1, m2, dbegin, is, js, iy, 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. */
      for (int jy = 1; jy <= m - 1; ++jy) {
    js = m + jy;
    int 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);
      }
  }
/*      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;
  for (int jy = 1; jy <= *col; ++jy) {
      js = m + jy;
      temp1 = 0.;
      temp2 = 0.;
      temp3 = 0.;
/*         compute element jy of row 'col' of Y'ZZ'Y */
      for (int 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 (int 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;
  }
/*      put new column in block (2,1). */
  int jy = *col;
  jpntr = *head + *col - 1;
  if (jpntr > m) {
      jpntr -= m;
  }
  ipntr = *head;
  for (int i = 1; i <= *col; ++i) {
      is = m + i;
      temp3 = 0.;
/*         compute element i of column 'col' of R_z */
      for (int 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;
  }
  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 (int iy = 1; iy <= upcl; ++iy) {
  is = m + iy;
  jpntr = *head;
  for (int jy = 1; jy <= iy; ++jy) {
      js = m + jy;
      temp1 = 0.;
      temp2 = 0.;
      temp3 = 0.;
      temp4 = 0.;
      for (int 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 (int 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;
  }
  ipntr = ipntr % m + 1;
    }
/*   modify the old parts in block (2,1). */
    ipntr = *head;
    for (int is = m + 1; is <= m + upcl; ++is) {
  jpntr = *head;
  for (int jy = 1; jy <= upcl; ++jy) {
      temp1 = 0.;
      temp3 = 0.;
      for (int k = 1; k <= *nenter; ++k) {
    k1 = indx2[k];
    temp1 += ws[k1 + ipntr * ws_dim1] * wy[k1 + jpntr * wy_dim1];
      }
      for (int 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;
  }
  ipntr = ipntr % m + 1;
    }
/*     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;
    for (int iy = 1; iy <= *col; ++iy) {
  is = *col + iy;
  is1 = m + iy;
  for (int jy = 1; jy <= iy;  ++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;
  }
  for (int jy = 1; jy <= iy - 1; ++jy) {
      wn[jy + is * wn_dim1] = -wn1[is1 + jy * wn1_dim1];
  }
  for (int  jy = iy; jy <= *col; ++jy) {
      wn[jy + is * wn_dim1] = wn1[is1 + jy * wn1_dim1];
  }
  wn[iy + iy * wn_dim1] += sy[iy + iy * sy_dim1];
    }
/*     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 (int 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 (int is = *col + 1; is <= col2; ++is) {
  for (int 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);
  }
    }
/*     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;

    /* Local variables */
    double ddum;
    int k, 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. */
    for (int j = 1; j <= *col; ++j)
  wt[j * wt_dim1 + 1] = *theta * ss[j * ss_dim1 + 1];
    for (int i = 2; i <= *col; ++i) {
  for (int j = i; j <= *col; ++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];
  }
    }
/*     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.

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

    /* Local variables */
    int iact, k;

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

    /* Function Body */
    *nenter = 0;
    *ileave = n + 1;
    if (*iter > 0 && *cnstnd) {/* count the entering and leaving variables. */
  for (int i = 1; i <= *nfree; ++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);
      }
  }
  for (int 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);
      }
   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 (int 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, 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 (int 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;
  }
    }
/*     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)
{
/*     **********

  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 */
    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 (int 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;
      }
        }
    }
      }
  }
    }
    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);
    *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 (int 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;

    /* Local variables */
    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 */
  for (int j = 1; j <= *col - 1; ++j) {
      F77_CALL(dcopy)(&j, &ss[(j + 1) * ss_dim1 + 2], &c__1,
          &ss[j * ss_dim1 + 1], &c__1);
      int 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);
  }
    }
/*    add new information: the last row of SY */
/*                 and the last column of SS: */
    pointr = *head;
    for (int j = 1; j <= *col - 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;
    }
    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.

  ************
*/
    double gi, d__1;

    *sbgnrm = 0.;
    for (int 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 k, m2, js, 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 (int i = 1; i <= *col; ++i) {
  temp1 = 0.;
  temp2 = 0.;
  for (int 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;
    }
/*     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 (int 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 (int jy = 1; jy <= *col; ++jy) {
  js = *col + jy;
  for (int 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;
    }

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

/*     Backtrack to the feasible region. */
    alpha = 1.;
    temp1 = alpha;
    for (int 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;
      }
  }
    }
    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 (int 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)
{
/*  **********

  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)
     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.

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

    /* 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");
  return;
    }
/*      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)
  return;

/*     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");
    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 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)
{
    Rprintf("%s ", title);
    for (int 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("Warning:  more than 10 function and gradient evaluations\n   in the last line search\n"); break;
  case -6: Rprintf("Input nbd(%d) is invalid", k); break;
  case -7: Rprintf("l(%d) > u(%d).  No feasible solution", k, k); 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;
  }
    }
}

Coverage Report

Created: 2026-10-28 06:40