lbfgsb.f File Reference

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Functions/Subroutines
subroutine	setulb (n, m, x, l, u, nbd, f, g, factr, pgtol, wa, iwa, task, iprint, csave, lsave, isave, dsave)
subroutine	mainlb (n, m, x, l, u, nbd, f, g, factr, pgtol, ws, wy, sy, ss, wt, wn, snd, z, r, d, t, xp, wa, index, iwhere, indx2, task, iprint, csave, lsave, isave, dsave)
subroutine	active (n, l, u, nbd, x, iwhere, iprint, prjctd, cnstnd, boxed)
subroutine	bmv (m, sy, wt, col, v, p, info)
subroutine	cauchy (n, x, l, u, nbd, g, iorder, iwhere, t, d, xcp, m, wy, ws, sy, wt, theta, col, head, p, c, wbp, v, nseg, iprint, sbgnrm, info, epsmch)
subroutine	cmprlb (n, m, x, g, ws, wy, sy, wt, z, r, wa, index, theta, col, head, nfree, cnstnd, info)
subroutine	errclb (n, m, factr, l, u, nbd, task, info, k)
subroutine	formk (n, nsub, ind, nenter, ileave, indx2, iupdat, updatd, wn, wn1, m, ws, wy, sy, theta, col, head, info)
subroutine	formt (m, wt, sy, ss, col, theta, info)
subroutine	freev (n, nfree, index, nenter, ileave, indx2, iwhere, wrk, updatd, cnstnd, iprint, iter)
subroutine	hpsolb (n, t, iorder, iheap)
subroutine	lnsrlb (n, l, u, nbd, x, f, fold, gd, gdold, g, d, r, t, z, stp, dnorm, dtd, xstep, stpmx, iter, ifun, iback, nfgv, info, task, boxed, cnstnd, csave, isave, dsave)
subroutine	matupd (n, m, ws, wy, sy, ss, d, r, itail, iupdat, col, head, theta, rr, dr, stp, dtd)
subroutine	prn1lb (n, m, l, u, x, iprint, itfile, epsmch)
subroutine	prn2lb (n, x, f, g, iprint, itfile, iter, nfgv, nact, sbgnrm, nseg, word, iword, iback, stp, xstep)
subroutine	prn3lb (n, x, f, task, iprint, info, itfile, iter, nfgv, nintol, nskip, nact, sbgnrm, time, nseg, word, iback, stp, xstep, k, cachyt, sbtime, lnscht)
subroutine	projgr (n, l, u, nbd, x, g, sbgnrm)
subroutine	subsm (n, m, nsub, ind, l, u, nbd, x, d, xp, ws, wy, theta, xx, gg, col, head, iword, wv, wn, iprint, info)
subroutine	dcsrch (f, g, stp, ftol, gtol, xtol, stpmin, stpmax, task, isave, dsave)
subroutine	dcstep (stx, fx, dx, sty, fy, dy, stp, fp, dp, brackt, stpmin, stpmax)

Function/Subroutine Documentation

active()

subroutine active	(	integer	n,
		double precision, dimension(n)	l,
		double precision, dimension(n)	u,
		integer, dimension(n)	nbd,
		double precision, dimension(n)	x,
		integer, dimension(n)	iwhere,
		integer	iprint,
		logical	prjctd,
		logical	cnstnd,
		logical	boxed
	)

      logical          prjctd, cnstnd, boxed
      integer          n, iprint, nbd(n), iwhere(n)
      double precision x(n), l(n), u(n)

c     ************
c
c     Subroutine active
c
c     This subroutine initializes iwhere and projects the initial x to
c       the feasible set if necessary.
c
c     iwhere is an integer array of dimension n.
c       On entry iwhere is unspecified.
c       On exit iwhere(i)=-1  if x(i) has no bounds
c                         3   if l(i)=u(i)
c                         0   otherwise.
c       In cauchy, iwhere is given finer gradations.
c
c
c                           *  *  *
c
c     NEOS, November 1994. (Latest revision June 1996.)
c     Optimization Technology Center.
c     Argonne National Laboratory and Northwestern University.
c     Written by
c                        Ciyou Zhu
c     in collaboration with R.H. Byrd, P. Lu-Chen and J. Nocedal.
c
c
c     ************

      integer          nbdd,i
      double precision zero
      parameter(zero=0.0d0)

c     Initialize nbdd, prjctd, cnstnd and boxed.

      nbdd = 0
      prjctd = .false.
      cnstnd = .false.
      boxed = .true.

c     Project the initial x to the easible set if necessary.

      do 10 i = 1, n
         if (nbd(i) .gt. 0) then
            if (nbd(i) .le. 2 .and. x(i) .le. l(i)) then
               if (x(i) .lt. l(i)) then
                  prjctd = .true.
                  x(i) = l(i)
               endif
               nbdd = nbdd + 1
            else if (nbd(i) .ge. 2 .and. x(i) .ge. u(i)) then
               if (x(i) .gt. u(i)) then
                  prjctd = .true.
                  x(i) = u(i)
               endif
               nbdd = nbdd + 1
            endif
         endif
  10  continue

c     Initialize iwhere and assign values to cnstnd and boxed.

      do 20 i = 1, n
         if (nbd(i) .ne. 2) boxed = .false.
         if (nbd(i) .eq. 0) then
c                                this variable is always free
            iwhere(i) = -1

c           otherwise set x(i)=mid(x(i), u(i), l(i)).
         else
            cnstnd = .true.
            if (nbd(i) .eq. 2 .and. u(i) - l(i) .le. zero) then
c                   this variable is always fixed
               iwhere(i) = 3
            else 
               iwhere(i) = 0
            endif
         endif
  20  continue

      if (iprint .ge. 0) then
         if (prjctd) write (6,*)
     +   'The initial X is infeasible.  Restart with its projection.'
         if (.not. cnstnd)
     +      write (6,*) 'This problem is unconstrained.'
      endif

      if (iprint .gt. 0) write (6,1001) nbdd

 1001 format (/,'At X0 ',i9,' variables are exactly at the bounds') 

      return

bmv()

subroutine bmv	(	integer	m,
		double precision, dimension(m, m)	sy,
		double precision, dimension(m, m)	wt,
		integer	col,
		double precision, dimension(2*col)	v,
		double precision, dimension(2*col)	p,
		integer	info
	)

      integer m, col, info
      double precision sy(m, m), wt(m, m), v(2*col), p(2*col)

c     ************
c
c     Subroutine bmv
c
c     This subroutine computes the product of the 2m x 2m middle matrix 
c       in the compact L-BFGS formula of B and a 2m vector v;  
c       it returns the product in p.
c       
c     m is an integer variable.
c       On entry m is the maximum number of variable metric corrections
c         used to define the limited memory matrix.
c       On exit m is unchanged.
c
c     sy is a double precision array of dimension m x m.
c       On entry sy specifies the matrix S'Y.
c       On exit sy is unchanged.
c
c     wt is a double precision array of dimension m x m.
c       On entry wt specifies the upper triangular matrix J' which is 
c         the Cholesky factor of (thetaS'S+LD^(-1)L').
c       On exit wt is unchanged.
c
c     col is an integer variable.
c       On entry col specifies the number of s-vectors (or y-vectors)
c         stored in the compact L-BFGS formula.
c       On exit col is unchanged.
c
c     v is a double precision array of dimension 2col.
c       On entry v specifies vector v.
c       On exit v is unchanged.
c
c     p is a double precision array of dimension 2col.
c       On entry p is unspecified.
c       On exit p is the product Mv.
c
c     info is an integer variable.
c       On entry info is unspecified.
c       On exit info = 0       for normal return,
c                    = nonzero for abnormal return when the system
c                                to be solved by dtrsl is singular.
c
c     Subprograms called:
c
c       Linpack ... dtrsl.
c
c
c                           *  *  *
c
c     NEOS, November 1994. (Latest revision June 1996.)
c     Optimization Technology Center.
c     Argonne National Laboratory and Northwestern University.
c     Written by
c                        Ciyou Zhu
c     in collaboration with R.H. Byrd, P. Lu-Chen and J. Nocedal.
c
c
c     ************
 
      integer          i,k,i2
      double precision sum
 
      if (col .eq. 0) return
 
c     PART I: solve [  D^(1/2)      O ] [ p1 ] = [ v1 ]
c                   [ -L*D^(-1/2)   J ] [ p2 ]   [ v2 ].

c       solve Jp2=v2+LD^(-1)v1.
      p(col + 1) = v(col + 1)
      do 20 i = 2, col
         i2 = col + i
         sum = 0.0d0
         do 10 k = 1, i - 1
            sum = sum + sy(i,k)*v(k)/sy(k,k)
  10     continue
         p(i2) = v(i2) + sum
  20  continue  
c     Solve the triangular system
      call dtrsl(wt,m,col,p(col+1),11,info)
      if (info .ne. 0) return
 
c       solve D^(1/2)p1=v1.
      do 30 i = 1, col
         p(i) = v(i)/sqrt(sy(i,i))
  30  continue 
 
c     PART II: solve [ -D^(1/2)   D^(-1/2)*L'  ] [ p1 ] = [ p1 ]
c                    [  0         J'           ] [ p2 ]   [ p2 ]. 
 
c       solve J^Tp2=p2. 
      call dtrsl(wt,m,col,p(col+1),01,info)
      if (info .ne. 0) return
 
c       compute p1=-D^(-1/2)(p1-D^(-1/2)L'p2)
c                 =-D^(-1/2)p1+D^(-1)L'p2.  
      do 40 i = 1, col
         p(i) = -p(i)/sqrt(sy(i,i))
  40  continue
      do 60 i = 1, col
         sum = 0.d0
         do 50 k = i + 1, col
            sum = sum + sy(k,i)*p(col+k)/sy(i,i)
  50     continue
         p(i) = p(i) + sum
  60  continue

      return

cauchy()

subroutine cauchy	(	integer	n,
		double precision, dimension(n)	x,
		double precision, dimension(n)	l,
		double precision, dimension(n)	u,
		integer, dimension(n)	nbd,
		double precision, dimension(n)	g,
		integer, dimension(n)	iorder,
		integer, dimension(n)	iwhere,
		double precision, dimension(n)	t,
		double precision, dimension(n)	d,
		double precision, dimension(n)	xcp,
		integer	m,
		double precision, dimension(n, col)	wy,
		double precision, dimension(n, col)	ws,
		double precision, dimension(m, m)	sy,
		double precision, dimension(m, m)	wt,
		double precision	theta,
		integer	col,
		integer	head,
		double precision, dimension(2*m)	p,
		double precision, dimension(2*m)	c,
		double precision, dimension(2*m)	wbp,
		double precision, dimension(2*m)	v,
		integer	nseg,
		integer	iprint,
		double precision	sbgnrm,
		integer	info,
		double precision	epsmch
	)

      implicit none
      integer          n, m, head, col, nseg, iprint, info, 
     +                 nbd(n), iorder(n), iwhere(n)
      double precision theta, epsmch,
     +                 x(n), l(n), u(n), g(n), t(n), d(n), xcp(n),
     +                 wy(n, col), ws(n, col), sy(m, m),
     +                 wt(m, m), p(2*m), c(2*m), wbp(2*m), v(2*m)

c     ************
c
c     Subroutine cauchy
c
c     For given x, l, u, g (with sbgnrm > 0), and a limited memory
c       BFGS matrix B defined in terms of matrices WY, WS, WT, and
c       scalars head, col, and theta, this subroutine computes the
c       generalized Cauchy point (GCP), defined as the first local
c       minimizer of the quadratic
c
c                  Q(x + s) = g's + 1/2 s'Bs
c
c       along the projected gradient direction P(x-tg,l,u).
c       The routine returns the GCP in xcp. 
c       
c     n is an integer variable.
c       On entry n is the dimension of the problem.
c       On exit n is unchanged.
c
c     x is a double precision array of dimension n.
c       On entry x is the starting point for the GCP computation.
c       On exit x is unchanged.
c
c     l is a double precision array of dimension n.
c       On entry l is the lower bound of x.
c       On exit l is unchanged.
c
c     u is a double precision array of dimension n.
c       On entry u is the upper bound of x.
c       On exit u is unchanged.
c
c     nbd is an integer array of dimension n.
c       On entry nbd represents the type of bounds imposed on the
c         variables, and must be specified as follows:
c         nbd(i)=0 if x(i) is unbounded,
c                1 if x(i) has only a lower bound,
c                2 if x(i) has both lower and upper bounds, and
c                3 if x(i) has only an upper bound. 
c       On exit nbd is unchanged.
c
c     g is a double precision array of dimension n.
c       On entry g is the gradient of f(x).  g must be a nonzero vector.
c       On exit g is unchanged.
c
c     iorder is an integer working array of dimension n.
c       iorder will be used to store the breakpoints in the piecewise
c       linear path and free variables encountered. On exit,
c         iorder(1),...,iorder(nleft) are indices of breakpoints
c                                which have not been encountered; 
c         iorder(nleft+1),...,iorder(nbreak) are indices of
c                                     encountered breakpoints; and
c         iorder(nfree),...,iorder(n) are indices of variables which
c                 have no bound constraits along the search direction.
c
c     iwhere is an integer array of dimension n.
c       On entry iwhere indicates only the permanently fixed (iwhere=3)
c       or free (iwhere= -1) components of x.
c       On exit iwhere records the status of the current x variables.
c       iwhere(i)=-3  if x(i) is free and has bounds, but is not moved
c                 0   if x(i) is free and has bounds, and is moved
c                 1   if x(i) is fixed at l(i), and l(i) .ne. u(i)
c                 2   if x(i) is fixed at u(i), and u(i) .ne. l(i)
c                 3   if x(i) is always fixed, i.e.,  u(i)=x(i)=l(i)
c                 -1  if x(i) is always free, i.e., it has no bounds.
c
c     t is a double precision working array of dimension n. 
c       t will be used to store the break points.
c
c     d is a double precision array of dimension n used to store
c       the Cauchy direction P(x-tg)-x.
c
c     xcp is a double precision array of dimension n used to return the
c       GCP on exit.
c
c     m is an integer variable.
c       On entry m is the maximum number of variable metric corrections 
c         used to define the limited memory matrix.
c       On exit m is unchanged.
c
c     ws, wy, sy, and wt are double precision arrays.
c       On entry they store information that defines the
c                             limited memory BFGS matrix:
c         ws(n,m) stores S, a set of s-vectors;
c         wy(n,m) stores Y, a set of y-vectors;
c         sy(m,m) stores S'Y;
c         wt(m,m) stores the
c                 Cholesky factorization of (theta*S'S+LD^(-1)L').
c       On exit these arrays are unchanged.
c
c     theta is a double precision variable.
c       On entry theta is the scaling factor specifying B_0 = theta I.
c       On exit theta is unchanged.
c
c     col is an integer variable.
c       On entry col is the actual number of variable metric
c         corrections stored so far.
c       On exit col is unchanged.
c
c     head is an integer variable.
c       On entry head is the location of the first s-vector (or y-vector)
c         in S (or Y).
c       On exit col is unchanged.
c
c     p is a double precision working array of dimension 2m.
c       p will be used to store the vector p = W^(T)d.
c
c     c is a double precision working array of dimension 2m.
c       c will be used to store the vector c = W^(T)(xcp-x).
c
c     wbp is a double precision working array of dimension 2m.
c       wbp will be used to store the row of W corresponding
c         to a breakpoint.
c
c     v is a double precision working array of dimension 2m.
c
c     nseg is an integer variable.
c       On exit nseg records the number of quadratic segments explored
c         in searching for the GCP.
c
c     sg and yg are double precision arrays of dimension m.
c       On entry sg  and yg store S'g and Y'g correspondingly.
c       On exit they are unchanged. 
c 
c     iprint is an INTEGER variable that must be set by the user.
c       It controls the frequency and type of output generated:
c        iprint<0    no output is generated;
c        iprint=0    print only one line at the last iteration;
c        0<iprint<99 print also f and |proj g| every iprint iterations;
c        iprint=99   print details of every iteration except n-vectors;
c        iprint=100  print also the changes of active set and final x;
c        iprint>100  print details of every iteration including x and g;
c       When iprint > 0, the file iterate.dat will be created to
c                        summarize the iteration.
c
c     sbgnrm is a double precision variable.
c       On entry sbgnrm is the norm of the projected gradient at x.
c       On exit sbgnrm is unchanged.
c
c     info is an integer variable.
c       On entry info is 0.
c       On exit info = 0       for normal return,
c                    = nonzero for abnormal return when the the system
c                              used in routine bmv is singular.
c
c     Subprograms called:
c 
c       L-BFGS-B Library ... hpsolb, bmv.
c
c       Linpack ... dscal dcopy, daxpy.
c
c
c     References:
c
c       [1] R. H. Byrd, P. Lu, J. Nocedal and C. Zhu, ``A limited
c       memory algorithm for bound constrained optimization'',
c       SIAM J. Scientific Computing 16 (1995), no. 5, pp. 1190--1208.
c
c       [2] C. Zhu, R.H. Byrd, P. Lu, J. Nocedal, ``L-BFGS-B: FORTRAN
c       Subroutines for Large Scale Bound Constrained Optimization''
c       Tech. Report, NAM-11, EECS Department, Northwestern University,
c       1994.
c
c       (Postscript files of these papers are available via anonymous
c        ftp to eecs.nwu.edu in the directory pub/lbfgs/lbfgs_bcm.)
c
c                           *  *  *
c
c     NEOS, November 1994. (Latest revision June 1996.)
c     Optimization Technology Center.
c     Argonne National Laboratory and Northwestern University.
c     Written by
c                        Ciyou Zhu
c     in collaboration with R.H. Byrd, P. Lu-Chen and J. Nocedal.
c
c
c     ************

      logical          xlower,xupper,bnded
      integer          i,j,col2,nfree,nbreak,pointr,
     +                 ibp,nleft,ibkmin,iter
      double precision f1,f2,dt,dtm,tsum,dibp,zibp,dibp2,bkmin,
     +                 tu,tl,wmc,wmp,wmw,ddot,tj,tj0,neggi,sbgnrm,
     +                 f2_org
      double precision one,zero
      parameter(one=1.0d0,zero=0.0d0)
 
c     Check the status of the variables, reset iwhere(i) if necessary;
c       compute the Cauchy direction d and the breakpoints t; initialize
c       the derivative f1 and the vector p = W'd (for theta = 1).
 
      if (sbgnrm .le. zero) then
         if (iprint .ge. 0) write (6,*) 'Subgnorm = 0.  GCP = X.'
         call dcopy(n,x,1,xcp,1)
         return
      endif 
      bnded = .true.
      nfree = n + 1
      nbreak = 0
      ibkmin = 0
      bkmin = zero
      col2 = 2*col
      f1 = zero
      if (iprint .ge. 99) write (6,3010)

c     We set p to zero and build it up as we determine d.

      do 20 i = 1, col2
         p(i) = zero
  20  continue 

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

      do 50 i = 1, n 
         neggi = -g(i)      
         if (iwhere(i) .ne. 3 .and. iwhere(i) .ne. -1) then
c             if x(i) is not a constant and has bounds,
c             compute the difference between x(i) and its bounds.
            if (nbd(i) .le. 2) tl = x(i) - l(i)
            if (nbd(i) .ge. 2) tu = u(i) - x(i)

c           If a variable is close enough to a bound
c             we treat it as at bound.
            xlower = nbd(i) .le. 2 .and. tl .le. zero
            xupper = nbd(i) .ge. 2 .and. tu .le. zero

c              reset iwhere(i).
            iwhere(i) = 0
            if (xlower) then
               if (neggi .le. zero) iwhere(i) = 1
            else if (xupper) then
               if (neggi .ge. zero) iwhere(i) = 2
            else
               if (abs(neggi) .le. zero) iwhere(i) = -3
            endif
         endif 
         pointr = head
         if (iwhere(i) .ne. 0 .and. iwhere(i) .ne. -1) then
            d(i) = zero
         else
            d(i) = neggi
            f1 = f1 - neggi*neggi
c             calculate p := p - W'e_i* (g_i).
            do 40 j = 1, col
               p(j) = p(j) +  wy(i,pointr)* neggi
               p(col + j) = p(col + j) + ws(i,pointr)*neggi
               pointr = mod(pointr,m) + 1
  40        continue 
            if (nbd(i) .le. 2 .and. nbd(i) .ne. 0
     +                        .and. neggi .lt. zero) then
c                                 x(i) + d(i) is bounded; compute t(i).
               nbreak = nbreak + 1
               iorder(nbreak) = i
               t(nbreak) = tl/(-neggi)
               if (nbreak .eq. 1 .or. t(nbreak) .lt. bkmin) then
                  bkmin = t(nbreak)
                  ibkmin = nbreak
               endif
            else if (nbd(i) .ge. 2 .and. neggi .gt. zero) then
c                                 x(i) + d(i) is bounded; compute t(i).
               nbreak = nbreak + 1
               iorder(nbreak) = i
               t(nbreak) = tu/neggi
               if (nbreak .eq. 1 .or. t(nbreak) .lt. bkmin) then
                  bkmin = t(nbreak)
                  ibkmin = nbreak
               endif
            else
c                x(i) + d(i) is not bounded.
               nfree = nfree - 1
               iorder(nfree) = i
               if (abs(neggi) .gt. zero) bnded = .false.
            endif
         endif
  50  continue 
 
c     The indices of the nonzero components of d are now stored
c       in iorder(1),...,iorder(nbreak) and iorder(nfree),...,iorder(n).
c       The smallest of the nbreak breakpoints is in t(ibkmin)=bkmin.
 
      if (theta .ne. one) then
c                   complete the initialization of p for theta not= one.
         call dscal(col,theta,p(col+1),1)
      endif
 
c     Initialize GCP xcp = x.

      call dcopy(n,x,1,xcp,1)

      if (nbreak .eq. 0 .and. nfree .eq. n + 1) then
c                  is a zero vector, return with the initial xcp as GCP.
         if (iprint .gt. 100) write (6,1010) (xcp(i), i = 1, n)
         return
      endif    
 
c     Initialize c = W'(xcp - x) = 0.
  
      do 60 j = 1, col2
         c(j) = zero
  60  continue 
 
c     Initialize derivative f2.
 
      f2 =  -theta*f1 
      f2_org  =  f2
      if (col .gt. 0) then
         call bmv(m,sy,wt,col,p,v,info)
         if (info .ne. 0) return
         f2 = f2 - ddot(col2,v,1,p,1)
      endif
      dtm = -f1/f2
      tsum = zero
      nseg = 1
      if (iprint .ge. 99) 
     +   write (6,*) 'There are ',nbreak,'  breakpoints '
 
c     If there are no breakpoints, locate the GCP and return. 
 
      if (nbreak .eq. 0) goto 888
             
      nleft = nbreak
      iter = 1
 
 
      tj = zero
 
c------------------- the beginning of the loop -------------------------
 
 777  continue
 
c     Find the next smallest breakpoint;
c       compute dt = t(nleft) - t(nleft + 1).
 
      tj0 = tj
      if (iter .eq. 1) then
c         Since we already have the smallest breakpoint we need not do
c         heapsort yet. Often only one breakpoint is used and the
c         cost of heapsort is avoided.
         tj = bkmin
         ibp = iorder(ibkmin)
      else
         if (iter .eq. 2) then
c             Replace the already used smallest breakpoint with the
c             breakpoint numbered nbreak > nlast, before heapsort call.
            if (ibkmin .ne. nbreak) then
               t(ibkmin) = t(nbreak)
               iorder(ibkmin) = iorder(nbreak)
            endif 
c        Update heap structure of breakpoints
c           (if iter=2, initialize heap).
         endif
         call hpsolb(nleft,t,iorder,iter-2)
         tj = t(nleft)
         ibp = iorder(nleft)  
      endif 
         
      dt = tj - tj0
 
      if (dt .ne. zero .and. iprint .ge. 100) then
         write (6,4011) nseg,f1,f2
         write (6,5010) dt
         write (6,6010) dtm
      endif          
 
c     If a minimizer is within this interval, locate the GCP and return. 
 
      if (dtm .lt. dt) goto 888
 
c     Otherwise fix one variable and
c       reset the corresponding component of d to zero.
    
      tsum = tsum + dt
      nleft = nleft - 1
      iter = iter + 1
      dibp = d(ibp)
      d(ibp) = zero
      if (dibp .gt. zero) then
         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
      endif
      if (iprint .ge. 100) write (6,*) 'Variable  ',ibp,'  is fixed.'
      if (nleft .eq. 0 .and. nbreak .eq. n) then
c                                             all n variables are fixed,
c                                                return with xcp as GCP.
         dtm = dt
         goto 999
      endif
 
c     Update the derivative information.
 
      nseg = nseg + 1
      dibp2 = dibp**2
 
c     Update f1 and f2.
 
c        temporarily set f1 and f2 for col=0.
      f1 = f1 + dt*f2 + dibp2 - theta*dibp*zibp
      f2 = f2 - theta*dibp2

      if (col .gt. 0) then
c                          update c = c + dt*p.
         call daxpy(col2,dt,p,1,c,1)
 
c           choose wbp,
c           the row of W corresponding to the breakpoint encountered.
         pointr = head
         do 70 j = 1,col
            wbp(j) = wy(ibp,pointr)
            wbp(col + j) = theta*ws(ibp,pointr)
            pointr = mod(pointr,m) + 1
  70     continue 
 
c           compute (wbp)Mc, (wbp)Mp, and (wbp)M(wbp)'.
         call bmv(m,sy,wt,col,wbp,v,info)
         if (info .ne. 0) return
         wmc = ddot(col2,c,1,v,1)
         wmp = ddot(col2,p,1,v,1) 
         wmw = ddot(col2,wbp,1,v,1)
 
c           update p = p - dibp*wbp. 
         call daxpy(col2,-dibp,wbp,1,p,1)
 
c           complete updating f1 and f2 while col > 0.
         f1 = f1 + dibp*wmc
         f2 = f2 + 2.0d0*dibp*wmp - dibp2*wmw
      endif

      f2 = max(epsmch*f2_org,f2)
      if (nleft .gt. 0) then
         dtm = -f1/f2
         goto 777
c                 to repeat the loop for unsearched intervals. 
      else if(bnded) then
         f1 = zero
         f2 = zero
         dtm = zero
      else
         dtm = -f1/f2
      endif 

c------------------- the end of the loop -------------------------------
 
 888  continue
      if (iprint .ge. 99) then
         write (6,*)
         write (6,*) 'GCP found in this segment'
         write (6,4010) nseg,f1,f2
         write (6,6010) dtm
      endif 
      if (dtm .le. zero) dtm = zero
      tsum = tsum + dtm
 
c     Move free variables (i.e., the ones w/o breakpoints) and 
c       the variables whose breakpoints haven't been reached.
 
      call daxpy(n,tsum,d,1,xcp,1)
 
 999  continue
 
c     Update c = c + dtm*p = W'(x^c - x) 
c       which will be used in computing r = Z'(B(x^c - x) + g).
 
      if (col .gt. 0) call daxpy(col2,dtm,p,1,c,1)
      if (iprint .gt. 100) write (6,1010) (xcp(i),i = 1,n)
      if (iprint .ge. 99) write (6,2010)

 1010 format ('Cauchy X =  ',/,(4x,1p,6(1x,d11.4)))
 2010 format (/,'---------------- exit CAUCHY----------------------',/)
 3010 format (/,'---------------- CAUCHY entered-------------------')
 4010 format ('Piece    ',i3,' --f1, f2 at start point ',1p,2(1x,d11.4))
 4011 format (/,'Piece    ',i3,' --f1, f2 at start point ',
     +        1p,2(1x,d11.4))
 5010 format ('Distance to the next break point =  ',1p,d11.4)
 6010 format ('Distance to the stationary point =  ',1p,d11.4) 
 
      return
 

cmprlb()

subroutine cmprlb	(	integer	n,
		integer	m,
		double precision, dimension(n)	x,
		double precision, dimension(n)	g,
		double precision, dimension(n, m)	ws,
		double precision, dimension(n, m)	wy,
		double precision, dimension(m, m)	sy,
		double precision, dimension(m, m)	wt,
		double precision, dimension(n)	z,
		double precision, dimension(n)	r,
		double precision, dimension(4*m)	wa,
		integer, dimension(n)	index,
		double precision	theta,
		integer	col,
		integer	head,
		integer	nfree,
		logical	cnstnd,
		integer	info
	)

 
      logical          cnstnd
      integer          n, m, col, head, nfree, info, index(n)
      double precision theta, 
     +                 x(n), g(n), z(n), r(n), wa(4*m), 
     +                 ws(n, m), wy(n, m), sy(m, m), wt(m, m)

c     ************
c
c     Subroutine cmprlb 
c
c       This subroutine computes r=-Z'B(xcp-xk)-Z'g by using 
c         wa(2m+1)=W'(xcp-x) from subroutine cauchy.
c
c     Subprograms called:
c
c       L-BFGS-B Library ... bmv.
c
c
c                           *  *  *
c
c     NEOS, November 1994. (Latest revision June 1996.)
c     Optimization Technology Center.
c     Argonne National Laboratory and Northwestern University.
c     Written by
c                        Ciyou Zhu
c     in collaboration with R.H. Byrd, P. Lu-Chen and J. Nocedal.
c
c
c     ************
 
      integer          i,j,k,pointr
      double precision a1,a2

      if (.not. cnstnd .and. col .gt. 0) then 
         do 26 i = 1, n
            r(i) = -g(i)
  26     continue
      else
         do 30 i = 1, nfree
            k = index(i)
            r(i) = -theta*(z(k) - x(k)) - g(k)
  30     continue
         call bmv(m,sy,wt,col,wa(2*m+1),wa(1),info)
         if (info .ne. 0) then
            info = -8
            return
         endif
         pointr = head 
         do 34 j = 1, col
            a1 = wa(j)
            a2 = theta*wa(col + j)
            do 32 i = 1, nfree
               k = index(i)
               r(i) = r(i) + wy(k,pointr)*a1 + ws(k,pointr)*a2
  32        continue
            pointr = mod(pointr,m) + 1
  34     continue
      endif

      return

dcsrch()

subroutine dcsrch	(	double precision	f,
		double precision	g,
		double precision	stp,
		double precision	ftol,
		double precision	gtol,
		double precision	xtol,
		double precision	stpmin,
		double precision	stpmax,
		character*(*)	task,
		integer, dimension(2)	isave,
		double precision, dimension(13)	dsave
	)

      character*(*) task
      integer isave(2)
      double precision f,g,stp,ftol,gtol,xtol,stpmin,stpmax
      double precision dsave(13)
c     **********
c
c     Subroutine dcsrch
c
c     this subroutine finds a step that satisfies a sufficient
c     decrease condition and a curvature condition.
c
c     each call of the subroutine updates an interval with 
c     endpoints stx and sty. the interval is initially chosen 
c     so that it contains a minimizer of the modified function
c
c           psi(stp) = f(stp) - f(0) - ftol*stp*f'(0).
c
c     If psi(stp) <= 0 and f'(stp) >= 0 for some step, then the
c     interval is chosen so that it contains a minimizer of f. 
c
c     the algorithm is designed to find a step that satisfies 
c     the sufficient decrease condition 
c
c           f(stp) <= f(0) + ftol*stp*f'(0),
c
c     and the curvature condition
c
c           abs(f'(stp)) <= gtol*abs(f'(0)).
c
c     If ftol is less than gtol and if, for example, the function
c     is bounded below, then there is always a step which satisfies
c     both conditions. 
c
c     If no step can be found that satisfies both conditions, then 
c     the algorithm stops with a warning. In this case stp only 
c     satisfies the sufficient decrease condition.
c
c     A typical invocation of dcsrch has the following outline:
c
c     task = 'start'
c  10 continue
c        call dcsrch( ... )
.eq.c        if (task  'fg') then
c           Evaluate the function and the gradient at stp 
c           goto 10
c           end if
c
c     NOTE: The user must no alter work arrays between calls.
c
c     The subroutine statement is
c
c        subroutine dcsrch(f,g,stp,ftol,gtol,xtol,stpmin,stpmax,
c                          task,isave,dsave)
c     where
c
c       f is a double precision variable.
c         On initial entry f is the value of the function at 0.
c            On subsequent entries f is the value of the 
c            function at stp.
c         On exit f is the value of the function at stp.
c
c       g is a double precision variable.
c         On initial entry g is the derivative of the function at 0.
c            On subsequent entries g is the derivative of the 
c            function at stp.
c         On exit g is the derivative of the function at stp.
c
c       stp is a double precision variable. 
c         On entry stp is the current estimate of a satisfactory 
c            step. On initial entry, a positive initial estimate 
c            must be provided. 
c         On exit stp is the current estimate of a satisfactory step
c            if task = 'fg'. If task = 'conv' then stp satisfies
c            the sufficient decrease and curvature condition.
c
c       ftol is a double precision variable.
c         On entry ftol specifies a nonnegative tolerance for the 
c            sufficient decrease condition.
c         On exit ftol is unchanged.
c
c       gtol is a double precision variable.
c         On entry gtol specifies a nonnegative tolerance for the 
c            curvature condition. 
c         On exit gtol is unchanged.
c
c       xtol is a double precision variable.
c         On entry xtol specifies a nonnegative relative tolerance
c            for an acceptable step. The subroutine exits with a
c            warning if the relative difference between sty and stx
c            is less than xtol.
c         On exit xtol is unchanged.
c
c       stpmin is a double precision variable.
c         On entry stpmin is a nonnegative lower bound for the step.
c         On exit stpmin is unchanged.
c
c       stpmax is a double precision variable.
c         On entry stpmax is a nonnegative upper bound for the step.
c         On exit stpmax is unchanged.
c
c       task is a character variable of length at least 60.
c         On initial entry task must be set to 'start'.
c         On exit task indicates the required action:
c
c            If task(1:2) = 'fg' then evaluate the function and 
c            derivative at stp and call dcsrch again.
c
c            If task(1:4) = 'conv' then the search is successful.
c
c            If task(1:4) = 'warn' then the subroutine is not able
c            to satisfy the convergence conditions. The exit value of
c            stp contains the best point found during the search.
c
c            If task(1:5) = 'error' then there is an error in the
c            input arguments.
c
c         On exit with convergence, a warning or an error, the
c            variable task contains additional information.
c
c       isave is an integer work array of dimension 2.
c         
c       dsave is a double precision work array of dimension 13.
c
c     Subprograms called
c
c       MINPACK-2 ... dcstep
c
c     MINPACK-1 Project. June 1983.
c     Argonne National Laboratory. 
c     Jorge J. More' and david j. thuente.
c
c     minpack-2 project. october 1993.
c     argonne national laboratory and university of minnesota. 
c     brett m. averick, richard g. carter, and jorge j. more'. 
c
c     **********
      double precision zero,p5,p66
      parameter(zero=0.0d0,p5=0.5d0,p66=0.66d0)
      double precision xtrapl,xtrapu
      parameter(xtrapl=1.1d0,xtrapu=4.0d0)

      logical brackt
      integer stage
      double precision finit,ftest,fm,fx,fxm,fy,fym,ginit,gtest,
     +       gm,gx,gxm,gy,gym,stx,sty,stmin,stmax,width,width1

c     Initialization block.

.eq.      if (task(1:5)  'start') then

c        Check the input arguments for errors.

.lt.         if (stp  stpmin) task = 'error: stp .LT. stpmin'
.gt.         if (stp  stpmax) task = 'error: stp .GT. stpmax'
.ge.         if (g  zero) task = 'error: initial g .GE. zero'
.lt.         if (ftol  zero) task = 'error: ftol .LT. zero'
.lt.         if (gtol  zero) task = 'error: gtol .LT. zero'
.lt.         if (xtol  zero) task = 'error: xtol .LT. zero'
.lt.         if (stpmin  zero) task = 'error: stpmin .LT. zero'
.lt.         if (stpmax  stpmin) task = 'error: stpmax .LT. stpmin'

c        Exit if there are errors on input.

.eq.         if (task(1:5)  'error') return

c        Initialize local variables.

         brackt = .false.
         stage = 1
         finit = f
         ginit = g
         gtest = ftol*ginit
         width = stpmax - stpmin
         width1 = width/p5

c        The variables stx, fx, gx contain the values of the step, 
c        function, and derivative at the best step. 
c        The variables sty, fy, gy contain the value of the step, 
c        function, and derivative at sty.
c        The variables stp, f, g contain the values of the step, 
c        function, and derivative at stp.

         stx = zero
         fx = finit
         gx = ginit
         sty = zero
         fy = finit
         gy = ginit
         stmin = zero
         stmax = stp + xtrapu*stp
         task = 'fg'

         goto 1000

      else

c        Restore local variables.

.eq.         if (isave(1)  1) then
            brackt = .true.
         else
            brackt = .false.
         endif
         stage = isave(2) 
         ginit = dsave(1) 
         gtest = dsave(2) 
         gx = dsave(3) 
         gy = dsave(4) 
         finit = dsave(5) 
         fx = dsave(6) 
         fy = dsave(7) 
         stx = dsave(8) 
         sty = dsave(9) 
         stmin = dsave(10) 
         stmax = dsave(11) 
         width = dsave(12) 
         width1 = dsave(13) 

      endif

c     If psi(stp) <= 0 and f'(stp) >= 0 for some step, then the
c     algorithm enters the second stage.

      ftest = finit + stp*gtest
      if (stage .eq. 1 .and. f .le. ftest .and. g .ge. zero) 
     +   stage = 2

c     test for warnings.

      if (brackt .and. (stp .le. stmin .or. stp .ge. stmax))
     +   task = 'WARNING: ROUNDING ERRORS PREVENT PROGRESS'
      if (brackt .and. stmax - stmin .le. xtol*stmax) 
     +   task = 'WARNING: XTOL TEST SATISFIED'
      if (stp .eq. stpmax .and. f .le. ftest .and. g .le. gtest) 
     +   task = 'WARNING: STP = STPMAX'
      if (stp .eq. stpmin .and. (f .gt. ftest .or. g .ge. gtest)) 
     +   task = 'WARNING: STP = STPMIN'

c     test for convergence.

      if (f .le. ftest .and. abs(g) .le. gtol*(-ginit)) 
     +   task = 'CONVERGENCE'

c     test for termination.

      if (task(1:4) .eq. 'WARN' .or. task(1:4) .eq. 'CONV') goto 1000

c     a modified function is used to predict the step during the
c     first stage if a lower function value has been obtained but 
c     the decrease is not sufficient.

      if (stage .eq. 1 .and. f .le. fx .and. f .gt. ftest) then

c        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

c        Call dcstep to update stx, sty, and to compute the new step.

         call dcstep(stx,fxm,gxm,sty,fym,gym,stp,fm,gm,
     +               brackt,stmin,stmax)

c        reset the function and derivative values for f.

         fx = fxm + stx*gtest
         fy = fym + sty*gtest
         gx = gxm + gtest
         gy = gym + gtest

      else

c       Call dcstep to update stx, sty, and to compute the new step.

        call dcstep(stx,fx,gx,sty,fy,gy,stp,f,g,
     +              brackt,stmin,stmax)

      endif

c     decide if a bisection step is needed.

      if (brackt) then
         if (abs(sty-stx) .ge. p66*width1) stp = stx + p5*(sty - stx)
         width1 = width
         width = abs(sty-stx)
      endif

c     set the minimum and maximum steps allowed for stp.

      if (brackt) then
         stmin = min(stx,sty)
         stmax = max(stx,sty)
      else
         stmin = stp + xtrapl*(stp - stx)
         stmax = stp + xtrapu*(stp - stx)
      endif
 
c     force the step to be within the bounds stpmax and stpmin.
 
      stp = max(stp,stpmin)
      stp = min(stp,stpmax)

c     If further progress is not possible, let stp be the best
c     point obtained during the search.

      if (brackt .and. (stp .le. stmin .or. stp .ge. stmax)
     +   .or. (brackt .and. stmax-stmin .le. xtol*stmax)) stp = stx

c     obtain another function and derivative.

      task = 'FG'

 1000 continue

c     Save local variables.

      if (brackt) then
         isave(1) = 1
      else
         isave(1) = 0
      endif
      isave(2) = stage
      dsave(1) =  ginit
      dsave(2) =  gtest
      dsave(3) =  gx
      dsave(4) =  gy
      dsave(5) =  finit
      dsave(6) =  fx
      dsave(7) =  fy
      dsave(8) =  stx
      dsave(9) =  sty
      dsave(10) = stmin
      dsave(11) = stmax
      dsave(12) = width
      dsave(13) = width1

      return

dcstep()

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

      logical brackt
      double precision stx,fx,dx,sty,fy,dy,stp,fp,dp,stpmin,stpmax
c     **********
c
c     Subroutine dcstep
c
c     This subroutine computes a safeguarded step for a search
c     procedure and updates an interval that contains a step that
c     satisfies a sufficient decrease and a curvature condition.
c
c     The parameter stx contains the step with the least function
c     value. If brackt is set to .true. then a minimizer has
c     been bracketed in an interval with endpoints stx and sty.
c     The parameter stp contains the current step. 
c     The subroutine assumes that if brackt is set to .true. then
c
c           min(stx,sty) < stp < max(stx,sty),
c
c     and that the derivative at stx is negative in the direction 
c     of the step.
c
c     The subroutine statement is
c
c       subroutine dcstep(stx,fx,dx,sty,fy,dy,stp,fp,dp,brackt,
c                         stpmin,stpmax)
c
c     where
c
c       stx is a double precision variable.
c         On entry stx is the best step obtained so far and is an
c            endpoint of the interval that contains the minimizer. 
c         On exit stx is the updated best step.
c
c       fx is a double precision variable.
c         On entry fx is the function at stx.
c         On exit fx is the function at stx.
c
c       dx is a double precision variable.
c         On entry dx is the derivative of the function at 
c            stx. The derivative must be negative in the direction of 
c            the step, that is, dx and stp - stx must have opposite 
c            signs.
c         On exit dx is the derivative of the function at stx.
c
c       sty is a double precision variable.
c         On entry sty is the second endpoint of the interval that 
c            contains the minimizer.
c         On exit sty is the updated endpoint of the interval that 
c            contains the minimizer.
c
c       fy is a double precision variable.
c         On entry fy is the function at sty.
c         On exit fy is the function at sty.
c
c       dy is a double precision variable.
c         On entry dy is the derivative of the function at sty.
c         On exit dy is the derivative of the function at the exit sty.
c
c       stp is a double precision variable.
c         On entry stp is the current step. If brackt is set to .true.
c            then on input stp must be between stx and sty. 
c         On exit stp is a new trial step.
c
c       fp is a double precision variable.
c         On entry fp is the function at stp
c         On exit fp is unchanged.
c
c       dp is a double precision variable.
c         On entry dp is the the derivative of the function at stp.
c         On exit dp is unchanged.
c
c       brackt is an logical variable.
c         On entry brackt specifies if a minimizer has been bracketed.
c            Initially brackt must be set to .false.
c         On exit brackt specifies if a minimizer has been bracketed.
c            When a minimizer is bracketed brackt is set to .true.
c
c       stpmin is a double precision variable.
c         On entry stpmin is a lower bound for the step.
c         On exit stpmin is unchanged.
c
c       stpmax is a double precision variable.
c         On entry stpmax is an upper bound for the step.
c         On exit stpmax is unchanged.
c
c     MINPACK-1 Project. June 1983
c     Argonne National Laboratory. 
c     Jorge J. More' and David J. Thuente.
c
c     MINPACK-2 Project. October 1993.
c     Argonne National Laboratory and University of Minnesota. 
c     Brett M. Averick and Jorge J. More'.
c
c     **********
      double precision zero,p66,two,three
      parameter(zero=0.0d0,p66=0.66d0,two=2.0d0,three=3.0d0)
   
      double precision gamma,p,q,r,s,sgnd,stpc,stpf,stpq,theta

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

c     First case: A higher function value. The minimum is bracketed. 
c     If the cubic step is closer to stx than the quadratic step, the 
c     cubic step is taken, otherwise the average of the cubic and 
c     quadratic steps is taken.

      if (fp .gt. fx) then
         theta = three*(fx - fp)/(stp - stx) + dx + dp
         s = max(abs(theta),abs(dx),abs(dp))
         gamma = s*sqrt((theta/s)**2 - (dx/s)*(dp/s))
         if (stp .lt. stx) gamma = -gamma
         p = (gamma - dx) + theta
         q = ((gamma - dx) + gamma) + dp
         r = p/q
         stpc = stx + r*(stp - stx)
         stpq = stx + ((dx/((fx - fp)/(stp - stx) + dx))/two)*
     +                                                       (stp - stx)
         if (abs(stpc-stx) .lt. abs(stpq-stx)) then
            stpf = stpc
         else
            stpf = stpc + (stpq - stpc)/two
         endif
         brackt = .true.

c     Second case: A lower function value and derivatives of opposite 
c     sign. The minimum is bracketed. If the cubic step is farther from 
c     stp than the secant step, the cubic step is taken, otherwise the 
c     secant step is taken.

      else if (sgnd .lt. zero) then
         theta = three*(fx - fp)/(stp - stx) + dx + dp
         s = max(abs(theta),abs(dx),abs(dp))
         gamma = s*sqrt((theta/s)**2 - (dx/s)*(dp/s))
         if (stp .gt. stx) gamma = -gamma
         p = (gamma - dp) + theta
         q = ((gamma - dp) + gamma) + dx
         r = p/q
         stpc = stp + r*(stx - stp)
         stpq = stp + (dp/(dp - dx))*(stx - stp)
         if (abs(stpc-stp) .gt. abs(stpq-stp)) then
            stpf = stpc
         else
            stpf = stpq
         endif
         brackt = .true.

c     Third case: A lower function value, derivatives of the same sign,
c     and the magnitude of the derivative decreases.

      else if (abs(dp) .lt. abs(dx)) then

c        The cubic step is computed only if the cubic tends to infinity 
c        in the direction of the step or if the minimum of the cubic
c        is beyond stp. Otherwise the cubic step is defined to be the 
c        secant step.

         theta = three*(fx - fp)/(stp - stx) + dx + dp
         s = max(abs(theta),abs(dx),abs(dp))

c        The case gamma = 0 only arises if the cubic does not tend
c        to infinity in the direction of the step.

         gamma = s*sqrt(max(zero,(theta/s)**2-(dx/s)*(dp/s)))
         if (stp .gt. stx) gamma = -gamma
         p = (gamma - dp) + theta
         q = (gamma + (dx - dp)) + gamma
         r = p/q
         if (r .lt. zero .and. gamma .ne. zero) then
            stpc = stp + r*(stx - stp)
         else if (stp .gt. stx) then
            stpc = stpmax
         else
            stpc = stpmin
         endif
         stpq = stp + (dp/(dp - dx))*(stx - stp)

         if (brackt) then

c           A minimizer has been bracketed. If the cubic step is 
c           closer to stp than the secant step, the cubic step is 
c           taken, otherwise the secant step is taken.

            if (abs(stpc-stp) .lt. abs(stpq-stp)) then
               stpf = stpc
            else
               stpf = stpq
            endif
            if (stp .gt. stx) then
               stpf = min(stp+p66*(sty-stp),stpf)
            else
               stpf = max(stp+p66*(sty-stp),stpf)
            endif
         else

c           A minimizer has not been bracketed. If the cubic step is 
c           farther from stp than the secant step, the cubic step is 
c           taken, otherwise the secant step is taken.

            if (abs(stpc-stp) .gt. abs(stpq-stp)) then
               stpf = stpc
            else
               stpf = stpq
            endif
            stpf = min(stpmax,stpf)
            stpf = max(stpmin,stpf)
         endif

c     Fourth case: A lower function value, derivatives of the same sign, 
c     and the magnitude of the derivative does not decrease. If the 
c     minimum is not bracketed, the step is either stpmin or stpmax, 
c     otherwise the cubic step is taken.

      else
         if (brackt) then
            theta = three*(fp - fy)/(sty - stp) + dy + dp
            s = max(abs(theta),abs(dy),abs(dp))
            gamma = s*sqrt((theta/s)**2 - (dy/s)*(dp/s))
            if (stp .gt. sty) gamma = -gamma
            p = (gamma - dp) + theta
            q = ((gamma - dp) + gamma) + dy
            r = p/q
            stpc = stp + r*(sty - stp)
            stpf = stpc
         else if (stp .gt. stx) then
            stpf = stpmax
         else
            stpf = stpmin
         endif
      endif

c     Update the interval which contains a minimizer.

      if (fp .gt. fx) then
         sty = stp
         fy = fp
         dy = dp
      else
         if (sgnd .lt. zero) then
            sty = stx
            fy = fx
            dy = dx
         endif
         stx = stp
         fx = fp
         dx = dp
      endif

c     Compute the new step.

      stp = stpf

      return

errclb()

subroutine errclb	(	integer	n,
		integer	m,
		double precision	factr,
		double precision, dimension(n)	l,
		double precision, dimension(n)	u,
		integer, dimension(n)	nbd,
		character*60	task,
		integer	info,
		integer	k
	)

 
      character*60     task
      integer          n, m, info, k, nbd(n)
      double precision factr, l(n), u(n)

c     ************
c
c     Subroutine errclb
c
c     this subroutine checks the validity of the input data.
c
c
c                           *  *  *
c
c     neos, november 1994. (latest revision june 1996.)
c     optimization technology center.
c     argonne national laboratory and northwestern university.
c     written by
c                        ciyou zhu
c     in collaboration with r.h. byrd, p. lu-chen and j. nocedal.
c
c
c     ************

      integer          i
      double precision one,zero
      parameter(one=1.0d0,zero=0.0d0)

c     check the input arguments for errors.

      if (n .le. 0) task = .LE.'ERROR: N  0'
      if (m .le. 0) task = .LE.'ERROR: M  0'
      if (factr .lt. zero) task = .LT.'ERROR: FACTR  0'

c     check the validity of the arrays nbd(i), u(i), and l(i).

      do 10 i = 1, n
         if (nbd(i) .lt. 0 .or. nbd(i) .gt. 3) then
c                                                   return
            task = 'ERROR: INVALID NBD'
            info = -6
            k = i
         endif
         if (nbd(i) .eq. 2) then
            if (l(i) .gt. u(i)) then
c                                    return
               task = 'ERROR: NO FEASIBLE SOLUTION'
               info = -7
               k = i
            endif
         endif
  10  continue

      return

formk()

subroutine formk	(	integer	n,
		integer	nsub,
		integer, dimension(n)	ind,
		integer	nenter,
		integer	ileave,
		integer, dimension(n)	indx2,
		integer	iupdat,
		logical	updatd,
		double precision, dimension(2*m, 2*m)	wn,
		double precision, dimension(2*m, 2*m)	wn1,
		integer	m,
		double precision, dimension(n, m)	ws,
		double precision, dimension(n, m)	wy,
		double precision, dimension(m, m)	sy,
		double precision	theta,
		integer	col,
		integer	head,
		integer	info
	)

      integer          n, nsub, m, col, head, nenter, ileave, iupdat,
     +                 info, ind(n), indx2(n)
      double precision theta, wn(2*m, 2*m), wn1(2*m, 2*m),
     +                 ws(n, m), wy(n, m), sy(m, m)
      logical          updatd

c     ************
c
c     Subroutine formk 
c
c     This subroutine forms  the LEL^T factorization of the indefinite
c
c       matrix    K = [-D -Y'ZZ'Y/theta     L_a'-R_z'  ]
c                     [L_a -R_z           theta*S'AA'S ]
c                                                    where E = [-I  0]
c                                                              [ 0  I]
c     The matrix K can be shown to be equal to the matrix M^[-1]N
c       occurring in section 5.1 of [1], as well as to the matrix
c       Mbar^[-1] Nbar in section 5.3.
c
c     n is an integer variable.
c       On entry n is the dimension of the problem.
c       On exit n is unchanged.
c
c     nsub is an integer variable
c       On entry nsub is the number of subspace variables in free set.
c       On exit nsub is not changed.
c
c     ind is an integer array of dimension nsub.
c       On entry ind specifies the indices of subspace variables.
c       On exit ind is unchanged. 
c
c     nenter is an integer variable.
c       On entry nenter is the number of variables entering the 
c         free set.
c       On exit nenter is unchanged. 
c
c     ileave is an integer variable.
c       On entry indx2(ileave),...,indx2(n) are the variables leaving
c         the free set.
c       On exit ileave is unchanged. 
c
c     indx2 is an integer array of dimension n.
c       On entry indx2(1),...,indx2(nenter) are the variables entering
c         the free set, while indx2(ileave),...,indx2(n) are the
c         variables leaving the free set.
c       On exit indx2 is unchanged. 
c
c     iupdat is an integer variable.
c       On entry iupdat is the total number of BFGS updates made so far.
c       On exit iupdat is unchanged. 
c
c     updatd is a logical variable.
c       On entry 'updatd' is true if the L-BFGS matrix is updatd.
c       On exit 'updatd' is unchanged. 
c
c     wn is a double precision array of dimension 2m x 2m.
c       On entry wn is unspecified.
c       On exit the upper triangle of wn stores the LEL^T factorization
c         of the 2*col x 2*col indefinite matrix
c                     [-D -Y'ZZ'Y/theta     L_a'-R_z'  ]
c                     [L_a -R_z           theta*S'AA'S ]
c
c     wn1 is a double precision array of dimension 2m x 2m.
c       On entry wn1 stores the lower triangular part of 
c                     [Y' ZZ'Y   L_a'+R_z']
c                     [L_a+R_z   S'AA'S   ]
c         in the previous iteration.
c       On exit wn1 stores the corresponding updated matrices.
c       The purpose of wn1 is just to store these inner products
c       so they can be easily updated and inserted into wn.
c
c     m is an integer variable.
c       On entry m is the maximum number of variable metric corrections
c         used to define the limited memory matrix.
c       On exit m is unchanged.
c
c     ws, wy, sy, and wtyy are double precision arrays;
c     theta is a double precision variable;
c     col is an integer variable;
c     head is an integer variable.
c       On entry they store the information defining the
c                                          limited memory BFGS matrix:
c         ws(n,m) stores S, a set of s-vectors;
c         wy(n,m) stores Y, a set of y-vectors;
c         sy(m,m) stores S'Y;
c         wtyy(m,m) stores the Cholesky factorization
c                                   of (theta*S'S+LD^(-1)L')
c         theta is the scaling factor specifying B_0 = theta I;
c         col is the number of variable metric corrections stored;
c         head is the location of the 1st s- (or y-) vector in S (or Y).
c       On exit they are unchanged.
c
c     info is an integer variable.
c       On entry info is unspecified.
c       On exit info =  0 for normal return;
c                    = -1 when the 1st Cholesky factorization failed;
c                    = -2 when the 2st Cholesky factorization failed.
c
c     Subprograms called:
c
c       Linpack ... dcopy, dpofa, dtrsl.
c
c
c     References:
c       [1] R. H. Byrd, P. Lu, J. Nocedal and C. Zhu, ``A limited
c       memory algorithm for bound constrained optimization'',
c       SIAM J. Scientific Computing 16 (1995), no. 5, pp. 1190--1208.
c
c       [2] C. Zhu, R.H. Byrd, P. Lu, J. Nocedal, ``L-BFGS-B: a
c       limited memory FORTRAN code for solving bound constrained
c       optimization problems'', Tech. Report, NAM-11, EECS Department,
c       Northwestern University, 1994.
c
c       (Postscript files of these papers are available via anonymous
c        ftp to eecs.nwu.edu in the directory pub/lbfgs/lbfgs_bcm.)
c
c                           *  *  *
c
c     NEOS, November 1994. (Latest revision June 1996.)
c     Optimization Technology Center.
c     Argonne National Laboratory and Northwestern University.
c     Written by
c                        Ciyou Zhu
c     in collaboration with R.H. Byrd, P. Lu-Chen and J. Nocedal.
c
c
c     ************

      integer          m2,ipntr,jpntr,iy,is,jy,js,is1,js1,k1,i,k,
     +                 col2,pbegin,pend,dbegin,dend,upcl
      double precision ddot,temp1,temp2,temp3,temp4
      double precision one,zero
      parameter(one=1.0d0,zero=0.0d0)

c     Form the lower triangular part of
c               WN1 = [Y' ZZ'Y   L_a'+R_z'] 
c                     [L_a+R_z   S'AA'S   ]
c        where L_a is the strictly lower triangular part of S'AA'Y
c              R_z is the upper triangular part of S'ZZ'Y.
      
      if (updatd) then
         if (iupdat .gt. m) then 
c                                 shift old part of WN1.
            do 10 jy = 1, m - 1
               js = m + jy
               call dcopy(m-jy,wn1(jy+1,jy+1),1,wn1(jy,jy),1)
               call dcopy(m-jy,wn1(js+1,js+1),1,wn1(js,js),1)
               call dcopy(m-1,wn1(m+2,jy+1),1,wn1(m+1,jy),1)
  10        continue
         endif
 
c          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 .gt. m) ipntr = ipntr - m    
         jpntr = head
         do 20 jy = 1, col
            js = m + jy
            temp1 = zero
            temp2 = zero
            temp3 = zero
c             compute element jy of row 'col' of Y'ZZ'Y
            do 15 k = pbegin, pend
               k1 = ind(k)
               temp1 = temp1 + wy(k1,ipntr)*wy(k1,jpntr)
  15        continue
c             compute elements jy of row 'col' of L_a and S'AA'S
            do 16 k = dbegin, dend
               k1 = ind(k)
               temp2 = temp2 + ws(k1,ipntr)*ws(k1,jpntr)
               temp3 = temp3 + ws(k1,ipntr)*wy(k1,jpntr)
  16        continue
            wn1(iy,jy) = temp1
            wn1(is,js) = temp2
            wn1(is,jy) = temp3
            jpntr = mod(jpntr,m) + 1
  20     continue
 
c          put new column in block (2,1).
         jy = col       
         jpntr = head + col - 1
         if (jpntr .gt. m) jpntr = jpntr - m
         ipntr = head
         do 30 i = 1, col
            is = m + i
            temp3 = zero
c             compute element i of column 'col' of R_z
            do 25 k = pbegin, pend
               k1 = ind(k)
               temp3 = temp3 + ws(k1,ipntr)*wy(k1,jpntr)
  25        continue 
            ipntr = mod(ipntr,m) + 1
            wn1(is,jy) = temp3
  30     continue
         upcl = col - 1
      else
         upcl = col
      endif
 
c       modify the old parts in blocks (1,1) and (2,2) due to changes
c       in the set of free variables.
      ipntr = head      
      do 45 iy = 1, upcl
         is = m + iy
         jpntr = head
         do 40 jy = 1, iy
            js = m + jy
            temp1 = zero
            temp2 = zero
            temp3 = zero
            temp4 = zero
            do 35 k = 1, nenter
               k1 = indx2(k)
               temp1 = temp1 + wy(k1,ipntr)*wy(k1,jpntr)
               temp2 = temp2 + ws(k1,ipntr)*ws(k1,jpntr)
  35        continue
            do 36 k = ileave, n
               k1 = indx2(k)
               temp3 = temp3 + wy(k1,ipntr)*wy(k1,jpntr)
               temp4 = temp4 + ws(k1,ipntr)*ws(k1,jpntr)
  36        continue
            wn1(iy,jy) = wn1(iy,jy) + temp1 - temp3 
            wn1(is,js) = wn1(is,js) - temp2 + temp4 
            jpntr = mod(jpntr,m) + 1
  40     continue
         ipntr = mod(ipntr,m) + 1
  45  continue
 
c       modify the old parts in block (2,1).
      ipntr = head      
      do 60 is = m + 1, m + upcl
         jpntr = head 
         do 55 jy = 1, upcl
            temp1 = zero
            temp3 = zero
            do 50 k = 1, nenter
               k1 = indx2(k)
               temp1 = temp1 + ws(k1,ipntr)*wy(k1,jpntr)
  50        continue
            do 51 k = ileave, n
               k1 = indx2(k)
               temp3 = temp3 + ws(k1,ipntr)*wy(k1,jpntr)
  51        continue
         if (is .le. jy + m) then
               wn1(is,jy) = wn1(is,jy) + temp1 - temp3  
            else
               wn1(is,jy) = wn1(is,jy) - temp1 + temp3  
            endif
            jpntr = mod(jpntr,m) + 1
  55     continue
         ipntr = mod(ipntr,m) + 1
  60  continue
 
c     Form the upper triangle of WN = [D+Y' ZZ'Y/theta   -L_a'+R_z' ] 
c                                     [-L_a +R_z        S'AA'S*theta]

      m2 = 2*m
      do 70 iy = 1, col
         is = col + iy
         is1 = m + iy
         do 65 jy = 1, iy
            js = col + jy
            js1 = m + jy
            wn(jy,iy) = wn1(iy,jy)/theta
            wn(js,is) = wn1(is1,js1)*theta
  65     continue
         do 66 jy = 1, iy - 1
            wn(jy,is) = -wn1(is1,jy)
  66     continue
         do 67 jy = iy, col
            wn(jy,is) = wn1(is1,jy)
  67     continue
         wn(iy,iy) = wn(iy,iy) + sy(iy,iy)
  70  continue

c     Form the upper triangle of WN= [  LL'            L^-1(-L_a'+R_z')] 
c                                    [(-L_a +R_z)L'^-1   S'AA'S*theta  ]

c        first Cholesky factor (1,1) block of wn to get LL'
c                          with L' stored in the upper triangle of wn.
      call dpofa(wn,m2,col,info)
      if (info .ne. 0) then
         info = -1
         return
      endif
c        then form L^-1(-L_a'+R_z') in the (1,2) block.
      col2 = 2*col
      do 71 js = col+1 ,col2
         call dtrsl(wn,m2,col,wn(1,js),11,info)
  71  continue

c     Form S'AA'S*theta + (L^-1(-L_a'+R_z'))'L^-1(-L_a'+R_z') in the
c        upper triangle of (2,2) block of wn.
                      

      do 72 is = col+1, col2
         do 74 js = is, col2
               wn(is,js) = wn(is,js) + ddot(col,wn(1,is),1,wn(1,js),1)
  74        continue
  72     continue

c     Cholesky factorization of (2,2) block of wn.

      call dpofa(wn(col+1,col+1),m2,col,info)
      if (info .ne. 0) then
         info = -2
         return
      endif

      return

formt()

subroutine formt	(	integer	m,
		double precision, dimension(m, m)	wt,
		double precision, dimension(m, m)	sy,
		double precision, dimension(m, m)	ss,
		integer	col,
		double precision	theta,
		integer	info
	)

 
      integer          m, col, info
      double precision theta, wt(m, m), sy(m, m), ss(m, m)

c     ************
c
c     Subroutine formt
c
c       This subroutine forms the upper half of the pos. def. and symm.
c         T = theta*SS + L*D^(-1)*L', stores T in the upper triangle
c         of the array wt, and performs the Cholesky factorization of T
c         to produce J*J', with J' stored in the upper triangle of wt.
c
c     Subprograms called:
c
c       Linpack ... dpofa.
c
c
c                           *  *  *
c
c     NEOS, November 1994. (Latest revision June 1996.)
c     Optimization Technology Center.
c     Argonne National Laboratory and Northwestern University.
c     Written by
c                        Ciyou Zhu
c     in collaboration with R.H. Byrd, P. Lu-Chen and J. Nocedal.
c
c
c     ************

      integer          i,j,k,k1
      double precision ddum
      double precision zero
      parameter(zero=0.0d0)


c     Form the upper half of  T = theta*SS + L*D^(-1)*L',
c        store T in the upper triangle of the array wt.
 
      do 52 j = 1, col
         wt(1,j) = theta*ss(1,j)
  52  continue
      do 55 i = 2, col
         do 54 j = i, col
            k1 = min(i,j) - 1
            ddum  = zero
            do 53 k = 1, k1
               ddum  = ddum + sy(i,k)*sy(j,k)/sy(k,k)
  53        continue
            wt(i,j) = ddum + theta*ss(i,j)
  54     continue
  55  continue
 
c     Cholesky factorize T to J*J' with 
c        J' stored in the upper triangle of wt.
 
      call dpofa(wt,m,col,info)
      if (info .ne. 0) then
         info = -3
      endif

      return

freev()

subroutine freev	(	integer	n,
		integer	nfree,
		integer, dimension(n)	index,
		integer	nenter,
		integer	ileave,
		integer, dimension(n)	indx2,
		integer, dimension(n)	iwhere,
		logical	wrk,
		logical	updatd,
		logical	cnstnd,
		integer	iprint,
		integer	iter
	)

      integer n, nfree, nenter, ileave, iprint, iter, 
     +        index(n), indx2(n), iwhere(n)
      logical wrk, updatd, cnstnd

c     ************
c
c     Subroutine freev 
c
c     This subroutine counts the entering and leaving variables when
c       iter > 0, and finds the index set of free and active variables
c       at the GCP.
c
c     cnstnd is a logical variable indicating whether bounds are present
c
c     index is an integer array of dimension n
c       for i=1,...,nfree, index(i) are the indices of free variables
c       for i=nfree+1,...,n, index(i) are the indices of bound variables
c       On entry after the first iteration, index gives 
c         the free variables at the previous iteration.
c       On exit it gives the free variables based on the determination
c         in cauchy using the array iwhere.
c
c     indx2 is an integer array of dimension n
c       On entry indx2 is unspecified.
c       On exit with iter>0, indx2 indicates which variables
c          have changed status since the previous iteration.
c       For i= 1,...,nenter, indx2(i) have changed from bound to free.
c       For i= ileave+1,...,n, indx2(i) have changed from free to bound.
c 
c
c                           *  *  *
c
c     NEOS, November 1994. (Latest revision June 1996.)
c     Optimization Technology Center.
c     Argonne National Laboratory and Northwestern University.
c     Written by
c                        Ciyou Zhu
c     in collaboration with R.H. Byrd, P. Lu-Chen and J. Nocedal.
c
c
c     ************
 
      integer iact,i,k

      nenter = 0
      ileave = n + 1
      if (iter .gt. 0 .and. cnstnd) then
c                           count the entering and leaving variables.
         do 20 i = 1, nfree
            k = index(i)

c            write(6,*) ' k  = index(i) ', k
c            write(6,*) ' index = ', i

            if (iwhere(k) .gt. 0) then
               ileave = ileave - 1
               indx2(ileave) = k
               if (iprint .ge. 100) write (6,*)
     +             'Variable ',k,' leaves the set of free variables'
            endif
  20     continue
         do 22 i = 1 + nfree, n
            k = index(i)
            if (iwhere(k) .le. 0) then
               nenter = nenter + 1
               indx2(nenter) = k
               if (iprint .ge. 100) write (6,*)
     +             'Variable ',k,' enters the set of free variables'
            endif
  22     continue
         if (iprint .ge. 99) write (6,*)
     +       n+1-ileave,' variables leave; ',nenter,' variables enter'
      endif
      wrk = (ileave .lt. n+1) .or. (nenter .gt. 0) .or. updatd
 
c     Find the index set of free and active variables at the GCP.
 
      nfree = 0 
      iact = n + 1
      do 24 i = 1, n
         if (iwhere(i) .le. 0) then
            nfree = nfree + 1
            index(nfree) = i
         else
            iact = iact - 1
            index(iact) = i
         endif
  24  continue
      if (iprint .ge. 99) write (6,*)
     +      nfree,' variables are free at GCP ',iter + 1  

      return

hpsolb()

subroutine hpsolb	(	integer	n,
		double precision, dimension(n)	t,
		integer, dimension(n)	iorder,
		integer	iheap
	)

      integer          iheap, n, iorder(n)
      double precision t(n)
  
c     ************
c
c     Subroutine hpsolb 
c
c     This subroutine sorts out the least element of t, and puts the
c       remaining elements of t in a heap.
c 
c     n is an integer variable.
c       On entry n is the dimension of the arrays t and iorder.
c       On exit n is unchanged.
c
c     t is a double precision array of dimension n.
c       On entry t stores the elements to be sorted,
c       On exit t(n) stores the least elements of t, and t(1) to t(n-1)
c         stores the remaining elements in the form of a heap.
c
c     iorder is an integer array of dimension n.
c       On entry iorder(i) is the index of t(i).
c       On exit iorder(i) is still the index of t(i), but iorder may be
c         permuted in accordance with t.
c
c     iheap is an integer variable specifying the task.
c       On entry iheap should be set as follows:
c         iheap .eq. 0 if t(1) to t(n) is not in the form of a heap,
c         iheap .ne. 0 if otherwise.
c       On exit iheap is unchanged.
c
c
c     References:
c       Algorithm 232 of CACM (J. W. J. Williams): HEAPSORT.
c
c                           *  *  *
c
c     NEOS, November 1994. (Latest revision June 1996.)
c     Optimization Technology Center.
c     Argonne National Laboratory and Northwestern University.
c     Written by
c                        Ciyou Zhu
c     in collaboration with R.H. Byrd, P. Lu-Chen and J. Nocedal.
c
c     ************
  
      integer          i,j,k,indxin,indxou
      double precision ddum,out

      if (iheap .eq. 0) then

c        Rearrange the elements t(1) to t(n) to form a heap.

         do 20 k = 2, n
            ddum  = t(k)
            indxin = iorder(k)

c           Add ddum to the heap.
            i = k
   10       continue
            if (i.gt.1) then
               j = i/2
               if (ddum .lt. t(j)) then
                  t(i) = t(j)
                  iorder(i) = iorder(j)
                  i = j
                  goto 10 
               endif  
            endif  
            t(i) = ddum
            iorder(i) = indxin
   20    continue
      endif
 
c     Assign to 'out' the value of t(1), the least member of the heap,
c        and rearrange the remaining members to form a heap as
c        elements 1 to n-1 of t.
 
      if (n .gt. 1) then
         i = 1
         out = t(1)
         indxou = iorder(1)
         ddum  = t(n)
         indxin  = iorder(n)

c        Restore the heap 
   30    continue
         j = i+i
         if (j .le. n-1) then
            if (t(j+1) .lt. t(j)) j = j+1
            if (t(j) .lt. ddum ) then
               t(i) = t(j)
               iorder(i) = iorder(j)
               i = j
               goto 30
            endif 
         endif 
         t(i) = ddum
         iorder(i) = indxin
 
c     Put the least member in t(n). 

         t(n) = out
         iorder(n) = indxou
      endif 

      return

lnsrlb()

subroutine lnsrlb	(	integer	n,
		double precision, dimension(n)	l,
		double precision, dimension(n)	u,
		integer, dimension(n)	nbd,
		double precision, dimension(n)	x,
		double precision	f,
		double precision	fold,
		double precision	gd,
		double precision	gdold,
		double precision, dimension(n)	g,
		double precision, dimension(n)	d,
		double precision, dimension(n)	r,
		double precision, dimension(n)	t,
		double precision, dimension(n)	z,
		double precision	stp,
		double precision	dnorm,
		double precision	dtd,
		double precision	xstep,
		double precision	stpmx,
		integer	iter,
		integer	ifun,
		integer	iback,
		integer	nfgv,
		integer	info,
		character*60	task,
		logical	boxed,
		logical	cnstnd,
		character*60	csave,
		integer, dimension(2)	isave,
		double precision, dimension(13)	dsave
	)

      character*60     task, csave
      logical          boxed, cnstnd
      integer          n, iter, ifun, iback, nfgv, info,
     +                 nbd(n), isave(2)
      double precision f, fold, gd, gdold, stp, dnorm, dtd, xstep,
     +                 stpmx, x(n), l(n), u(n), g(n), d(n), r(n), t(n),
     +                 z(n), dsave(13)
c     **********
c
c     Subroutine lnsrlb
c
c     this subroutine calls subroutine dcsrch from the minpack2 library
c       to perform the line search.  Subroutine dscrch is safeguarded so
c       that all trial points lie within the feasible region.
c
c     subprograms called:
c
c       minpack2 library ... dcsrch.
c
c       linpack ... dtrsl, ddot.
c
c
c                           *  *  *
c
c     neos, november 1994. (latest revision june 1996.)
c     optimization technology center.
c     argonne national laboratory and northwestern university.
c     written by
c                        ciyou zhu
c     in collaboration with r.h. byrd, p. lu-chen and j. nocedal.
c
c
c     **********

      integer          i
      double           precision ddot,a1,a2
      double precision one,zero,big
      parameter(one=1.0d0,zero=0.0d0,big=1.0d+10)
      double precision ftol,gtol,xtol
      parameter(ftol=1.0d-3,gtol=0.9d0,xtol=0.1d0)

      if (task(1:5) .eq. 'FG_LN') goto 556

      dtd = ddot(n,d,1,d,1)
      dnorm = sqrt(dtd)

c     determine the maximum step length.

      stpmx = big
      if (cnstnd) then
         if (iter .eq. 0) then
            stpmx = one
         else
            do 43 i = 1, n
               a1 = d(i)
               if (nbd(i) .ne. 0) then
                  if (a1 .lt. zero .and. nbd(i) .le. 2) then
                     a2 = l(i) - x(i)
                     if (a2 .ge. zero) then
                        stpmx = zero
                     else if (a1*stpmx .lt. a2) then
                        stpmx = a2/a1
                     endif
                  else if (a1 .gt. zero .and. nbd(i) .ge. 2) then
                     a2 = u(i) - x(i)
                     if (a2 .le. zero) then
                        stpmx = zero
                     else if (a1*stpmx .gt. a2) then
                        stpmx = a2/a1
                     endif
                  endif
               endif
  43        continue
         endif
      endif
 
      if (iter .eq. 0 .and. .not. boxed) then
         stp = min(one/dnorm, stpmx)
      else
         stp = one
      endif 

      call dcopy(n,x,1,t,1)
      call dcopy(n,g,1,r,1)
      fold = f
      ifun = 0
      iback = 0
      csave = 'START'
 556  continue
      gd = ddot(n,g,1,d,1)
      if (ifun .eq. 0) then
         gdold=gd
         if (gd .ge. zero) then
c                               the directional derivative >=0.
c                               line search is impossible.
            write(6,*)' ascent direction in projection gd = ', gd
            info = -4
            return
         endif
      endif

      call dcsrch(f,gd,stp,ftol,gtol,xtol,zero,stpmx,csave,isave,dsave)

      xstep = stp*dnorm
      if (csave(1:4) .ne. 'CONV' .and. csave(1:4) .ne. 'WARN') then
         task = 'FG_LNSRCH'
         ifun = ifun + 1
         nfgv = nfgv + 1
         iback = ifun - 1 
         if (stp .eq. one) then
            call dcopy(n,z,1,x,1)
         else
            do 41 i = 1, n
               x(i) = stp*d(i) + t(i)
  41        continue
         endif
      else
         task = 'NEW_X'
      endif

      return

mainlb()

subroutine mainlb	(	integer	n,
		integer	m,
		double precision, dimension(n)	x,
		double precision, dimension(n)	l,
		double precision, dimension(n)	u,
		integer, dimension(n)	nbd,
		double precision	f,
		double precision, dimension(n)	g,
		double precision	factr,
		double precision	pgtol,
		double precision, dimension(n, m)	ws,
		double precision, dimension(n, m)	wy,
		double precision, dimension(m, m)	sy,
		double precision, dimension(m, m)	ss,
		double precision, dimension(m, m)	wt,
		double precision, dimension(2*m, 2*m)	wn,
		double precision, dimension(2*m, 2*m)	snd,
		double precision, dimension(n)	z,
		double precision, dimension(n)	r,
		double precision, dimension(n)	d,
		double precision, dimension(n)	t,
		double precision, dimension(n)	xp,
		double precision, dimension(8*m)	wa,
		integer, dimension(n)	index,
		integer, dimension(n)	iwhere,
		integer, dimension(n)	indx2,
		character*60	task,
		integer	iprint,
		character*60	csave,
		logical, dimension(4)	lsave,
		integer, dimension(23)	isave,
		double precision, dimension(29)	dsave
	)

      implicit none
      character*60     task, csave
      logical          lsave(4)
      integer          n, m, iprint, nbd(n), index(n),
     +                 iwhere(n), indx2(n), isave(23)
      double precision f, factr, pgtol,
     +                 x(n), l(n), u(n), g(n), z(n), r(n), d(n), t(n), 
c-jlm-jn
     +                 xp(n), 
     +                 wa(8*m), 
     +                 ws(n, m), wy(n, m), sy(m, m), ss(m, m), 
     +                 wt(m, m), wn(2*m, 2*m), snd(2*m, 2*m), dsave(29)

c     ************
c
c     Subroutine mainlb
c
c     This subroutine solves bound constrained optimization problems by
c       using the compact formula of the limited memory BFGS updates.
c       
c     n is an integer variable.
c       On entry n is the number of variables.
c       On exit n is unchanged.
c
c     m is an integer variable.
c       On entry m is the maximum number of variable metric
c          corrections allowed in the limited memory matrix.
c       On exit m is unchanged.
c
c     x is a double precision array of dimension n.
c       On entry x is an approximation to the solution.
c       On exit x is the current approximation.
c
c     l is a double precision array of dimension n.
c       On entry l is the lower bound of x.
c       On exit l is unchanged.
c
c     u is a double precision array of dimension n.
c       On entry u is the upper bound of x.
c       On exit u is unchanged.
c
c     nbd is an integer array of dimension n.
c       On entry nbd represents the type of bounds imposed on the
c         variables, and must be specified as follows:
c         nbd(i)=0 if x(i) is unbounded,
c                1 if x(i) has only a lower bound,
c                2 if x(i) has both lower and upper bounds,
c                3 if x(i) has only an upper bound.
c       On exit nbd is unchanged.
c
c     f is a double precision variable.
c       On first entry f is unspecified.
c       On final exit f is the value of the function at x.
c
c     g is a double precision array of dimension n.
c       On first entry g is unspecified.
c       On final exit g is the value of the gradient at x.
c
c     factr is a double precision variable.
c       On entry factr >= 0 is specified by the user.  The iteration
c         will stop when
c
c         (f^k - f^{k+1})/max{|f^k|,|f^{k+1}|,1} <= factr*epsmch
c
c         where epsmch is the machine precision, which is automatically
c         generated by the code.
c       On exit factr is unchanged.
c
c     pgtol is a double precision variable.
c       On entry pgtol >= 0 is specified by the user.  The iteration
c         will stop when
c
c                 max{|proj g_i | i = 1, ..., n} <= pgtol
c
c         where pg_i is the ith component of the projected gradient.
c       On exit pgtol is unchanged.
c
c     ws, wy, sy, and wt are double precision working arrays used to
c       store the following information defining the limited memory
c          BFGS matrix:
c          ws, of dimension n x m, stores S, the matrix of s-vectors;
c          wy, of dimension n x m, stores Y, the matrix of y-vectors;
c          sy, of dimension m x m, stores S'Y;
c          ss, of dimension m x m, stores S'S;
c          yy, of dimension m x m, stores Y'Y;
c          wt, of dimension m x m, stores the Cholesky factorization
c                                  of (theta*S'S+LD^(-1)L'); see eq.
c                                  (2.26) in [3].
c
c     wn is a double precision working array of dimension 2m x 2m
c       used to store the LEL^T factorization of the indefinite matrix
c                 K = [-D -Y'ZZ'Y/theta     L_a'-R_z'  ]
c                     [L_a -R_z           theta*S'AA'S ]
c
c       where     E = [-I  0]
c                     [ 0  I]
c
c     snd is a double precision working array of dimension 2m x 2m
c       used to store the lower triangular part of
c                 N = [Y' ZZ'Y   L_a'+R_z']
c                     [L_a +R_z  S'AA'S   ]
c            
c     z(n),r(n),d(n),t(n), xp(n),wa(8*m) are double precision working arrays.
c       z  is used at different times to store the Cauchy point and
c          the Newton point.
c       xp is used to safeguard the projected Newton direction
c
c     sg(m),sgo(m),yg(m),ygo(m) are double precision working arrays. 
c
c     index is an integer working array of dimension n.
c       In subroutine freev, index is used to store the free and fixed
c          variables at the Generalized Cauchy Point (GCP).
c
c     iwhere is an integer working array of dimension n used to record
c       the status of the vector x for GCP computation.
c       iwhere(i)=0 or -3 if x(i) is free and has bounds,
c                 1       if x(i) is fixed at l(i), and l(i) .ne. u(i)
c                 2       if x(i) is fixed at u(i), and u(i) .ne. l(i)
c                 3       if x(i) is always fixed, i.e.,  u(i)=x(i)=l(i)
c                -1       if x(i) is always free, i.e., no bounds on it.
c
c     indx2 is an integer working array of dimension n.
c       Within subroutine cauchy, indx2 corresponds to the array iorder.
c       In subroutine freev, a list of variables entering and leaving
c       the free set is stored in indx2, and it is passed on to
c       subroutine formk with this information.
c
c     task is a working string of characters of length 60 indicating
c       the current job when entering and leaving this subroutine.
c
c     iprint is an INTEGER variable that must be set by the user.
c       It controls the frequency and type of output generated:
c        iprint<0    no output is generated;
c        iprint=0    print only one line at the last iteration;
c        0<iprint<99 print also f and |proj g| every iprint iterations;
c        iprint=99   print details of every iteration except n-vectors;
c        iprint=100  print also the changes of active set and final x;
c        iprint>100  print details of every iteration including x and g;
c       When iprint > 0, the file iterate.dat will be created to
c                        summarize the iteration.
c
c     csave is a working string of characters of length 60.
c
c     lsave is a logical working array of dimension 4.
c
c     isave is an integer working array of dimension 23.
c
c     dsave is a double precision working array of dimension 29.
c
c
c     Subprograms called
c
c       L-BFGS-B Library ... cauchy, subsm, lnsrlb, formk, 
c
c        errclb, prn1lb, prn2lb, prn3lb, active, projgr,
c
c        freev, cmprlb, matupd, formt.
c
c       Minpack2 Library ... timer
c
c       Linpack Library ... dcopy, ddot.
c
c
c     References:
c
c       [1] R. H. Byrd, P. Lu, J. Nocedal and C. Zhu, ``A limited
c       memory algorithm for bound constrained optimization'',
c       SIAM J. Scientific Computing 16 (1995), no. 5, pp. 1190--1208.
c
c       [2] C. Zhu, R.H. Byrd, P. Lu, J. Nocedal, ``L-BFGS-B: FORTRAN
c       Subroutines for Large Scale Bound Constrained Optimization''
c       Tech. Report, NAM-11, EECS Department, Northwestern University,
c       1994.
c 
c       [3] R. Byrd, J. Nocedal and R. Schnabel "Representations of
c       Quasi-Newton Matrices and their use in Limited Memory Methods'',
c       Mathematical Programming 63 (1994), no. 4, pp. 129-156.
c
c       (Postscript files of these papers are available via anonymous
c        ftp to eecs.nwu.edu in the directory pub/lbfgs/lbfgs_bcm.)
c
c                           *  *  *
c
c     NEOS, November 1994. (Latest revision June 1996.)
c     Optimization Technology Center.
c     Argonne National Laboratory and Northwestern University.
c     Written by
c                        Ciyou Zhu
c     in collaboration with R.H. Byrd, P. Lu-Chen and J. Nocedal.
c
c
c     ************
 
      logical          prjctd,cnstnd,boxed,updatd,wrk
      character*3      word
      integer          i,k,nintol,itfile,iback,nskip,
     +                 head,col,iter,itail,iupdat,
     +                 nseg,nfgv,info,ifun,
     +                 iword,nfree,nact,ileave,nenter
      double precision theta,fold,ddot,dr,rr,tol,
     +                 xstep,sbgnrm,ddum,dnorm,dtd,epsmch,
     +                 cpu1,cpu2,cachyt,sbtime,lnscht,time1,time2,
     +                 gd,gdold,stp,stpmx,time
      double precision one,zero
      parameter(one=1.0d0,zero=0.0d0)
      
      if (task .eq. 'START') then

         epsmch = epsilon(one)

         call timer(time1)

c        Initialize counters and scalars when task='START'.

c           for the limited memory BFGS matrices:
         col    = 0
         head   = 1
         theta  = one
         iupdat = 0
         updatd = .false.
         iback  = 0
         itail  = 0
         iword  = 0
         nact   = 0
         ileave = 0
         nenter = 0
         fold   = zero
         dnorm  = zero
         cpu1   = zero
         gd     = zero
         stpmx  = zero
         sbgnrm = zero
         stp    = zero
         gdold  = zero
         dtd    = zero

c           for operation counts:
         iter   = 0
         nfgv   = 0
         nseg   = 0
         nintol = 0
         nskip  = 0
         nfree  = n
         ifun   = 0
c           for stopping tolerance:
         tol = factr*epsmch

c           for measuring running time:
         cachyt = 0
         sbtime = 0
         lnscht = 0
 
c           'word' records the status of subspace solutions.
         word = '---'

c           'info' records the termination information.
         info = 0

         itfile = 8
         if (iprint .ge. 1) then
c                                open a summary file 'iterate.dat'
            open (8, file = 'iterate.dat', status = 'unknown')
         endif            

c        Check the input arguments for errors.

         call errclb(n,m,factr,l,u,nbd,task,info,k)
         if (task(1:5) .eq. 'ERROR') then
            call prn3lb(n,x,f,task,iprint,info,itfile,
     +                  iter,nfgv,nintol,nskip,nact,sbgnrm,
     +                  zero,nseg,word,iback,stp,xstep,k,
     +                  cachyt,sbtime,lnscht)
            return
         endif

         call prn1lb(n,m,l,u,x,iprint,itfile,epsmch)
 
c        Initialize iwhere & project x onto the feasible set.
 
         call active(n,l,u,nbd,x,iwhere,iprint,prjctd,cnstnd,boxed) 

c        The end of the initialization.

      else
c          restore local variables.

         prjctd = lsave(1)
         cnstnd = lsave(2)
         boxed  = lsave(3)
         updatd = lsave(4)

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

         theta  = dsave(1)
         fold   = dsave(2)
         tol    = dsave(3)
         dnorm  = dsave(4)
         epsmch = dsave(5)
         cpu1   = dsave(6)
         cachyt = dsave(7)
         sbtime = dsave(8)
         lnscht = dsave(9)
         time1  = dsave(10)
         gd     = dsave(11)
         stpmx  = dsave(12)
         sbgnrm = dsave(13)
         stp    = dsave(14)
         gdold  = dsave(15)
         dtd    = dsave(16)
   
c        After returning from the driver go to the point where execution
c        is to resume.

         if (task(1:5) .eq. 'FG_LN') goto 666
         if (task(1:5) .eq. 'NEW_X') goto 777
         if (task(1:5) .eq. 'FG_ST') goto 111
         if (task(1:4) .eq. 'STOP') then
            if (task(7:9) .eq. 'CPU') then
c                                          restore the previous iterate.
               call dcopy(n,t,1,x,1)
               call dcopy(n,r,1,g,1)
               f = fold
            endif
            goto 999
         endif
      endif 

c     Compute f0 and g0.

      task = 'FG_START' 
c          return to the driver to calculate f and g; reenter at 111.
      goto 1000
 111  continue
      nfgv = 1
 
c     Compute the infinity norm of the (-) projected gradient.
 
      call projgr(n,l,u,nbd,x,g,sbgnrm)
  
      if (iprint .ge. 1) then
         write (6,1002) iter,f,sbgnrm
         write (itfile,1003) iter,nfgv,sbgnrm,f
      endif
      if (sbgnrm .le. pgtol) then
c                                terminate the algorithm.
         task = 'CONVERGENCE: NORM_OF_PROJECTED_GRADIENT_<=_PGTOL'
         goto 999
      endif 
 
c ----------------- the beginning of the loop --------------------------
 
 222  continue
      if (iprint .ge. 99) write (6,1001) iter + 1
      iword = -1
c
      if (.not. cnstnd .and. col .gt. 0) then 
c                                            skip the search for GCP.
         call dcopy(n,x,1,z,1)
         wrk = updatd
         nseg = 0
         goto 333
      endif

cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc
c
c     Compute the Generalized Cauchy Point (GCP).
c
cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc

      call timer(cpu1) 
      call cauchy(n,x,l,u,nbd,g,indx2,iwhere,t,d,z,
     +            m,wy,ws,sy,wt,theta,col,head,
     +            wa(1),wa(2*m+1),wa(4*m+1),wa(6*m+1),nseg,
     +            iprint, sbgnrm, info, epsmch)
      if (info .ne. 0) then 
c         singular triangular system detected; refresh the lbfgs memory.
         if(iprint .ge. 1) write (6, 1005)
         info   = 0
         col    = 0
         head   = 1
         theta  = one
         iupdat = 0
         updatd = .false.
         call timer(cpu2) 
         cachyt = cachyt + cpu2 - cpu1
         goto 222
      endif
      call timer(cpu2) 
      cachyt = cachyt + cpu2 - cpu1
      nintol = nintol + nseg

c     Count the entering and leaving variables for iter > 0; 
c     find the index set of free and active variables at the GCP.

      call freev(n,nfree,index,nenter,ileave,indx2,
     +           iwhere,wrk,updatd,cnstnd,iprint,iter)
      nact = n - nfree

 333  continue
 
c     If there are no free variables or B=theta*I, then
c                                        skip the subspace minimization.
 
      if (nfree .eq. 0 .or. col .eq. 0) goto 555
 
cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc
c
c     Subspace minimization.
c
cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc

      call timer(cpu1) 

c     Form  the LEL^T factorization of the indefinite
c       matrix    K = [-D -Y'ZZ'Y/theta     L_a'-R_z'  ]
c                     [L_a -R_z           theta*S'AA'S ]
c       where     E = [-I  0]
c                     [ 0  I]

      if (wrk) call formk(n,nfree,index,nenter,ileave,indx2,iupdat,
     +                 updatd,wn,snd,m,ws,wy,sy,theta,col,head,info)
      if (info .ne. 0) then
c          nonpositive definiteness in Cholesky factorization;
c          refresh the lbfgs memory and restart the iteration.
         if(iprint .ge. 1) write (6, 1006)
         info   = 0
         col    = 0
         head   = 1
         theta  = one
         iupdat = 0
         updatd = .false.
         call timer(cpu2) 
         sbtime = sbtime + cpu2 - cpu1 
         goto 222
      endif 

c        compute r=-Z'B(xcp-xk)-Z'g (using wa(2m+1)=W'(xcp-x)
c                                                   from 'cauchy').
      call cmprlb(n,m,x,g,ws,wy,sy,wt,z,r,wa,index,
     +           theta,col,head,nfree,cnstnd,info)
      if (info .ne. 0) goto 444

c-jlm-jn   call the direct method. 

      call subsm( n, m, nfree, index, l, u, nbd, z, r, xp, ws, wy,
     +           theta, x, g, col, head, iword, wa, wn, iprint, info)
 444  continue
      if (info .ne. 0) then 
c          singular triangular system detected;
c          refresh the lbfgs memory and restart the iteration.
         if(iprint .ge. 1) write (6, 1005)
         info   = 0
         col    = 0
         head   = 1
         theta  = one
         iupdat = 0
         updatd = .false.
         call timer(cpu2) 
         sbtime = sbtime + cpu2 - cpu1 
         goto 222
      endif
 
      call timer(cpu2) 
      sbtime = sbtime + cpu2 - cpu1 
 555  continue
 
cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc
c
c     Line search and optimality tests.
c
cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc
 
c     Generate the search direction d:=z-x.

      do 40 i = 1, n
         d(i) = z(i) - x(i)
  40  continue
      call timer(cpu1) 
 666  continue
      call lnsrlb(n,l,u,nbd,x,f,fold,gd,gdold,g,d,r,t,z,stp,dnorm,
     +            dtd,xstep,stpmx,iter,ifun,iback,nfgv,info,task,
     +            boxed,cnstnd,csave,isave(22),dsave(17))
      if (info .ne. 0 .or. iback .ge. 20) then
c          restore the previous iterate.
         call dcopy(n,t,1,x,1)
         call dcopy(n,r,1,g,1)
         f = fold
         if (col .eq. 0) then
c             abnormal termination.
            if (info .eq. 0) then
               info = -9
c                restore the actual number of f and g evaluations etc.
               nfgv = nfgv - 1
               ifun = ifun - 1
               iback = iback - 1
            endif
            task = 'ABNORMAL_TERMINATION_IN_LNSRCH'
            iter = iter + 1
            goto 999
         else
c             refresh the lbfgs memory and restart the iteration.
            if(iprint .ge. 1) write (6, 1008)
            if (info .eq. 0) nfgv = nfgv - 1
            info   = 0
            col    = 0
            head   = 1
            theta  = one
            iupdat = 0
            updatd = .false.
            task   = 'RESTART_FROM_LNSRCH'
            call timer(cpu2)
            lnscht = lnscht + cpu2 - cpu1
            goto 222
         endif
      else if (task(1:5) .eq. 'FG_LN') then
c          return to the driver for calculating f and g; reenter at 666.
         goto 1000
      else 
c          calculate and print out the quantities related to the new X.
         call timer(cpu2) 
         lnscht = lnscht + cpu2 - cpu1
         iter = iter + 1
 
c        Compute the infinity norm of the projected (-)gradient.
 
         call projgr(n,l,u,nbd,x,g,sbgnrm)
 
c        Print iteration information.

         call prn2lb(n,x,f,g,iprint,itfile,iter,nfgv,nact,
     +               sbgnrm,nseg,word,iword,iback,stp,xstep)
         goto 1000
      endif
 777  continue

c     Test for termination.

      if (sbgnrm .le. pgtol) then
c                                terminate the algorithm.
         task = 'CONVERGENCE: NORM_OF_PROJECTED_GRADIENT_<=_PGTOL'
         goto 999
      endif 

      ddum = max(abs(fold), abs(f), one)
      if ((fold - f) .le. tol*ddum) then
c                                        terminate the algorithm.
         task = 'CONVERGENCE: REL_REDUCTION_OF_F_<=_FACTR*EPSMCH'
         if (iback .ge. 10) info = -5
c           i.e., to issue a warning if iback>10 in the line search.
         goto 999
      endif 

c     Compute d=newx-oldx, r=newg-oldg, rr=y'y and dr=y's.
 
      do 42 i = 1, n
         r(i) = g(i) - r(i)
  42  continue
      rr = ddot(n,r,1,r,1)
      if (stp .eq. one) then  
         dr = gd - gdold
         ddum = -gdold
      else
         dr = (gd - gdold)*stp
         call dscal(n,stp,d,1)
         ddum = -gdold*stp
      endif
 
      if (dr .le. epsmch*ddum) then
c                            skip the L-BFGS update.
         nskip = nskip + 1
         updatd = .false.
         if (iprint .ge. 1) write (6,1004) dr, ddum
         goto 888
      endif 
 
cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc
c
c     Update the L-BFGS matrix.
c
cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc
 
      updatd = .true.
      iupdat = iupdat + 1

c     Update matrices WS and WY and form the middle matrix in B.

      call matupd(n,m,ws,wy,sy,ss,d,r,itail,
     +            iupdat,col,head,theta,rr,dr,stp,dtd)

c     Form the upper half of the pds T = theta*SS + L*D^(-1)*L';
c        Store T in the upper triangular of the array wt;
c        Cholesky factorize T to J*J' with
c           J' stored in the upper triangular of wt.

      call formt(m,wt,sy,ss,col,theta,info)
 
      if (info .ne. 0) then 
c          nonpositive definiteness in Cholesky factorization;
c          refresh the lbfgs memory and restart the iteration.
         if(iprint .ge. 1) write (6, 1007)
         info = 0
         col = 0
         head = 1
         theta = one
         iupdat = 0
         updatd = .false.
         goto 222
      endif

c     Now the inverse of the middle matrix in B is

c       [  D^(1/2)      O ] [ -D^(1/2)  D^(-1/2)*L' ]
c       [ -L*D^(-1/2)   J ] [  0        J'          ]

 888  continue
 
c -------------------- the end of the loop -----------------------------
 
      goto 222
 999  continue
      call timer(time2)
      time = time2 - time1
      call prn3lb(n,x,f,task,iprint,info,itfile,
     +            iter,nfgv,nintol,nskip,nact,sbgnrm,
     +            time,nseg,word,iback,stp,xstep,k,
     +            cachyt,sbtime,lnscht)
 1000 continue

c     Save local variables.

      lsave(1)  = prjctd
      lsave(2)  = cnstnd
      lsave(3)  = boxed
      lsave(4)  = updatd

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

      dsave(1)  = theta 
      dsave(2)  = fold 
      dsave(3)  = tol 
      dsave(4)  = dnorm 
      dsave(5)  = epsmch 
      dsave(6)  = cpu1 
      dsave(7)  = cachyt 
      dsave(8)  = sbtime 
      dsave(9)  = lnscht 
      dsave(10) = time1 
      dsave(11) = gd 
      dsave(12) = stpmx 
      dsave(13) = sbgnrm
      dsave(14) = stp
      dsave(15) = gdold
      dsave(16) = dtd  

 1001 format (//,'ITERATION ',i5)
 1002 format
     +  (/,'At iterate',i5,4x,'f= ',1p,d12.5,4x,'|proj g|= ',1p,d12.5)
 1003 format (2(1x,i4),5x,'-',5x,'-',3x,'-',5x,'-',5x,'-',8x,'-',3x,
     +        1p,2(1x,d10.3))
 1004 format ('  ys=',1p,e10.3,'  -gs=',1p,e10.3,' BFGS update SKIPPED')
 1005 format (/, 
     +' Singular triangular system detected;',/,
     +'   refresh the lbfgs memory and restart the iteration.')
 1006 format (/, 
     +' Nonpositive definiteness in Cholesky factorization in formk;',/,
     +'   refresh the lbfgs memory and restart the iteration.')
 1007 format (/, 
     +' Nonpositive definiteness in Cholesky factorization in formt;',/,
     +'   refresh the lbfgs memory and restart the iteration.')
 1008 format (/, 
     +' Bad direction in the line search;',/,
     +'   refresh the lbfgs memory and restart the iteration.')

      return   

matupd()

subroutine matupd	(	integer	n,
		integer	m,
		double precision, dimension(n, m)	ws,
		double precision, dimension(n, m)	wy,
		double precision, dimension(m, m)	sy,
		double precision, dimension(m, m)	ss,
		double precision, dimension(n)	d,
		double precision, dimension(n)	r,
		integer	itail,
		integer	iupdat,
		integer	col,
		integer	head,
		double precision	theta,
		double precision	rr,
		double precision	dr,
		double precision	stp,
		double precision	dtd
	)

 
      integer          n, m, itail, iupdat, col, head
      double precision theta, rr, dr, stp, dtd, d(n), r(n), 
     +                 ws(n, m), wy(n, m), sy(m, m), ss(m, m)

c     ************
c
c     Subroutine matupd
c
c       This subroutine updates matrices WS and WY, and forms the
c         middle matrix in B.
c
c     Subprograms called:
c
c       Linpack ... dcopy, ddot.
c
c
c                           *  *  *
c
c     NEOS, November 1994. (Latest revision June 1996.)
c     Optimization Technology Center.
c     Argonne National Laboratory and Northwestern University.
c     Written by
c                        Ciyou Zhu
c     in collaboration with R.H. Byrd, P. Lu-Chen and J. Nocedal.
c
c
c     ************
 
      integer          j,pointr
      double precision ddot
      double precision one
      parameter(one=1.0d0)

c     Set pointers for matrices WS and WY.
 
      if (iupdat .le. m) then
         col = iupdat
         itail = mod(head+iupdat-2,m) + 1
      else
         itail = mod(itail,m) + 1
         head = mod(head,m) + 1
      endif
 
c     Update matrices WS and WY.

      call dcopy(n,d,1,ws(1,itail),1)
      call dcopy(n,r,1,wy(1,itail),1)
 
c     Set theta=yy/ys.
 
      theta = rr/dr
 
c     Form the middle matrix in B.
 
c        update the upper triangle of SS,
c                                         and the lower triangle of SY:
      if (iupdat .gt. m) then
c                              move old information
         do 50 j = 1, col - 1
            call dcopy(j,ss(2,j+1),1,ss(1,j),1)
            call dcopy(col-j,sy(j+1,j+1),1,sy(j,j),1)
  50     continue
      endif
c        add new information: the last row of SY
c                                             and the last column of SS:
      pointr = head
      do 51 j = 1, col - 1
         sy(col,j) = ddot(n,d,1,wy(1,pointr),1)
         ss(j,col) = ddot(n,ws(1,pointr),1,d,1)
         pointr = mod(pointr,m) + 1
  51  continue
      if (stp .eq. one) then
         ss(col,col) = dtd
      else
         ss(col,col) = stp*stp*dtd
      endif
      sy(col,col) = dr
 
      return

prn1lb()

subroutine prn1lb	(	integer	n,
		integer	m,
		double precision, dimension(n)	l,
		double precision, dimension(n)	u,
		double precision, dimension(n)	x,
		integer	iprint,
		integer	itfile,
		double precision	epsmch
	)

 
      integer n, m, iprint, itfile
      double precision epsmch, x(n), l(n), u(n)

c     ************
c
c     Subroutine prn1lb
c
c     This subroutine prints the input data, initial point, upper and
c       lower bounds of each variable, machine precision, as well as 
c       the headings of the output.
c
c
c                           *  *  *
c
c     NEOS, November 1994. (Latest revision June 1996.)
c     Optimization Technology Center.
c     Argonne National Laboratory and Northwestern University.
c     Written by
c                        Ciyou Zhu
c     in collaboration with R.H. Byrd, P. Lu-Chen and J. Nocedal.
c
c
c     ************

      integer i

      if (iprint .ge. 0) then
         write (6,7001) epsmch
         write (6,*) 'N = ',n,'    M = ',m
         if (iprint .ge. 1) then
            write (itfile,2001) epsmch
            write (itfile,*)'N = ',n,'    M = ',m
            write (itfile,9001)
            if (iprint .gt. 100) then
               write (6,1004) 'L =',(l(i),i = 1,n)
               write (6,1004) 'X0 =',(x(i),i = 1,n)
               write (6,1004) 'U =',(u(i),i = 1,n)
            endif 
         endif
      endif 

 1004 format (/,a4, 1p, 6(1x,d11.4),/,(4x,1p,6(1x,d11.4)))
 2001 format ('RUNNING THE L-BFGS-B CODE',/,/,
     + 'it    = iteration number',/,
     + 'nf    = number of function evaluations',/,
     + 'nseg  = number of segments explored during the Cauchy search',/,
     + 'nact  = number of active bounds at the generalized Cauchy point'
     + ,/,
     + 'sub   = manner in which the subspace minimization terminated:'
     + ,/,'        con = converged, bnd = a bound was reached',/,
     + 'itls  = number of iterations performed in the line search',/,
     + 'stepl = step length used',/,
     + 'tstep = norm of the displacement (total step)',/,
     + 'projg = norm of the projected gradient',/,
     + 'f     = function value',/,/,
     + '           * * *',/,/,
     + 'Machine precision =',1p,d10.3)
 7001 format ('RUNNING THE L-BFGS-B CODE',/,/,
     + '           * * *',/,/,
     + 'Machine precision =',1p,d10.3)
 9001 format (/,3x,'it',3x,'nf',2x,'nseg',2x,'nact',2x,'sub',2x,'itls',
     +        2x,'stepl',4x,'tstep',5x,'projg',8x,'f')

      return

prn2lb()

subroutine prn2lb	(	integer	n,
		double precision, dimension(n)	x,
		double precision	f,
		double precision, dimension(n)	g,
		integer	iprint,
		integer	itfile,
		integer	iter,
		integer	nfgv,
		integer	nact,
		double precision	sbgnrm,
		integer	nseg,
		character*3	word,
		integer	iword,
		integer	iback,
		double precision	stp,
		double precision	xstep
	)

 
      character*3      word
      integer          n, iprint, itfile, iter, nfgv, nact, nseg,
     +                 iword, iback
      double precision f, sbgnrm, stp, xstep, x(n), g(n)

c     ************
c
c     Subroutine prn2lb
c
c     this subroutine prints out new information after a successful
c       line search. 
c
c
c                           *  *  *
c
c     neos, november 1994. (latest revision june 1996.)
c     optimization technology center.
c     argonne national laboratory and northwestern university.
c     written by
c                        ciyou zhu
c     in collaboration with r.h. byrd, p. lu-chen and j. nocedal.
c
c
c     ************

      integer i,imod

c           'word' records the status of subspace solutions.
      if (iword .eq. 0) then
c                            the subspace minimization converged.
         word = 'con'
      else if (iword .eq. 1) then
c                          the subspace minimization stopped at a bound.
         word = 'bnd'
      else if (iword .eq. 5) then
c                             the truncated newton step has been used.
         word = 'TNT'
      else
         word = '---'
      endif
      if (iprint .ge. 99) then
         write (6,*) 'LINE SEARCH',iback,' times; norm of step = ',xstep
         write (6,2001) iter,f,sbgnrm
         if (iprint .gt. 100) then      
            write (6,1004) 'X =',(x(i), i = 1, n)
            write (6,1004) 'G =',(g(i), i = 1, n)
         endif
      else if (iprint .gt. 0) then 
         imod = mod(iter,iprint)
         if (imod .eq. 0) write (6,2001) iter,f,sbgnrm
      endif
      if (iprint .ge. 1) write (itfile,3001)
     +          iter,nfgv,nseg,nact,word,iback,stp,xstep,sbgnrm,f

 1004 format (/,a4, 1p, 6(1x,d11.4),/,(4x,1p,6(1x,d11.4)))
 2001 format
     +  (/,'At iterate',i5,4x,'f= ',1p,d12.5,4x,'|proj g|= ',1p,d12.5)
 3001 format(2(1x,i4),2(1x,i5),2x,a3,1x,i4,1p,2(2x,d7.1),1p,2(1x,d10.3))

      return

prn3lb()

subroutine prn3lb	(	integer	n,
		double precision, dimension(n)	x,
		double precision	f,
		character*60	task,
		integer	iprint,
		integer	info,
		integer	itfile,
		integer	iter,
		integer	nfgv,
		integer	nintol,
		integer	nskip,
		integer	nact,
		double precision	sbgnrm,
		double precision	time,
		integer	nseg,
		character*3	word,
		integer	iback,
		double precision	stp,
		double precision	xstep,
		integer	k,
		double precision	cachyt,
		double precision	sbtime,
		double precision	lnscht
	)

 
      character*60     task
      character*3      word
      integer          n, iprint, info, itfile, iter, nfgv, nintol,
     +                 nskip, nact, nseg, iback, k
      double precision f, sbgnrm, time, stp, xstep, cachyt, sbtime,
     +                 lnscht, x(n)

c     ************
c
c     Subroutine prn3lb
c
c     this subroutine prints out information when either a built-in
c       convergence test is satisfied or when an error message is
c       generated.
c       
c
c                           *  *  *
c
c     neos, november 1994. (latest revision june 1996.)
c     optimization technology center.
c     argonne national laboratory and northwestern university.
c     written by
c                        ciyou zhu
c     in collaboration with r.h. byrd, p. lu-chen and j. nocedal.
c
c
c     ************

      integer i

      if (task(1:5) .eq. 'ERROR') goto 999

      if (iprint .ge. 0) then
         write (6,3003)
         write (6,3004)
         write(6,3005) n,iter,nfgv,nintol,nskip,nact,sbgnrm,f
         if (iprint .ge. 100) then
            write (6,1004) 'X =',(x(i),i = 1,n)
         endif  
         if (iprint .ge. 1) write (6,*) ' F =',f
      endif 
 999  continue
      if (iprint .ge. 0) then
         write (6,3009) task
         if (info .ne. 0) then
            if (info .eq. -1) write (6,9011)
            if (info .eq. -2) write (6,9012)
            if (info .eq. -3) write (6,9013)
            if (info .eq. -4) write (6,9014)
            if (info .eq. -5) write (6,9015)
            if (info .eq. -6) write (6,*)' Input nbd(',k,') is invalid.'
            if (info .eq. -7) 
     +      write (6,*)' l(',k,') > u(',k,').  No feasible solution.'
            if (info .eq. -8) write (6,9018)
            if (info .eq. -9) write (6,9019)
         endif
         if (iprint .ge. 1) write (6,3007) cachyt,sbtime,lnscht
         write (6,3008) time
         if (iprint .ge. 1) then
            if (info .eq. -4 .or. info .eq. -9) then
               write (itfile,3002)
     +             iter,nfgv,nseg,nact,word,iback,stp,xstep
            endif
            write (itfile,3009) task
            if (info .ne. 0) then
               if (info .eq. -1) write (itfile,9011)
               if (info .eq. -2) write (itfile,9012)
               if (info .eq. -3) write (itfile,9013)
               if (info .eq. -4) write (itfile,9014)
               if (info .eq. -5) write (itfile,9015)
               if (info .eq. -8) write (itfile,9018)
               if (info .eq. -9) write (itfile,9019)
            endif
            write (itfile,3008) time
         endif
      endif

 1004 format (/,a4, 1p, 6(1x,d11.4),/,(4x,1p,6(1x,d11.4)))
 3002 format(2(1x,i4),2(1x,i5),2x,a3,1x,i4,1p,2(2x,d7.1),6x,'-',10x,'-')
 3003 format (/,
     + '           * * *',/,/,
     + 'Tit   = total number of iterations',/,
     + 'Tnf   = total number of function evaluations',/,
     + 'Tnint = total number of segments explored during',
     +           ' Cauchy searches',/,
     + 'Skip  = number of BFGS updates skipped',/,
     + 'Nact  = number of active bounds at final generalized',
     +          ' Cauchy point',/,
     + 'Projg = norm of the final projected gradient',/,
     + 'F     = final function value',/,/,
     + '           * * *')
 3004 format (/,3x,'N',4x,'Tit',5x,'Tnf',2x,'Tnint',2x,
     +       'Skip',2x,'Nact',5x,'Projg',8x,'F')
 3005 format (i5,2(1x,i6),(1x,i6),(2x,i4),(1x,i5),1p,2(2x,d10.3))
 3007 format (/,' Cauchy                time',1p,e10.3,' seconds.',/ 
     +        ' Subspace minimization time',1p,e10.3,' seconds.',/
     +        ' Line search           time',1p,e10.3,' seconds.')
 3008 format (/,' Total User time',1p,e10.3,' seconds.',/)
 3009 format (/,a60)
 9011 format (/,
     +' Matrix in 1st Cholesky factorization in formk is not Pos. Def.')
 9012 format (/,
     +' Matrix in 2st Cholesky factorization in formk is not Pos. Def.')
 9013 format (/,
     +' Matrix in the Cholesky factorization in formt is not Pos. Def.')
 9014 format (/,
     +' Derivative >= 0, backtracking line search impossible.',/,
     +'   Previous x, f and g restored.',/,
     +' Possible causes: 1 error in function or gradient evaluation;',/,
     +'                  2 rounding errors dominate computation.')
 9015 format (/,
     +' Warning:  more than 10 function and gradient',/,
     +'   evaluations in the last line search.  Termination',/,
     +'   may possibly be caused by a bad search direction.')
 9018 format (/,' The triangular system is singular.')
 9019 format (/,
     +' Line search cannot locate an adequate point after 20 function',/
     +,'  and gradient evaluations.  Previous x, f and g restored.',/,
     +' Possible causes: 1 error in function or gradient evaluation;',/,
     +'                  2 rounding error dominate computation.')

      return

projgr()

subroutine projgr	(	integer	n,
		double precision, dimension(n)	l,
		double precision, dimension(n)	u,
		integer, dimension(n)	nbd,
		double precision, dimension(n)	x,
		double precision, dimension(n)	g,
		double precision	sbgnrm
	)

      integer          n, nbd(n)
      double precision sbgnrm, x(n), l(n), u(n), g(n)

c     ************
c
c     Subroutine projgr
c
c     This subroutine computes the infinity norm of the projected
c       gradient.
c
c
c                           *  *  *
c
c     NEOS, November 1994. (Latest revision June 1996.)
c     Optimization Technology Center.
c     Argonne National Laboratory and Northwestern University.
c     Written by
c                        Ciyou Zhu
c     in collaboration with R.H. Byrd, P. Lu-Chen and J. Nocedal.
c
c
c     ************

      integer i
      double precision gi
      double precision one,zero
      parameter(one=1.0d0,zero=0.0d0)

      sbgnrm = zero
      do 15 i = 1, n
        gi = g(i)
        if (nbd(i) .ne. 0) then
           if (gi .lt. zero) then
              if (nbd(i) .ge. 2) gi = max((x(i)-u(i)),gi)
           else
              if (nbd(i) .le. 2) gi = min((x(i)-l(i)),gi)
           endif
        endif
        sbgnrm = max(sbgnrm,abs(gi))
  15  continue

      return

setulb()

subroutine setulb	(	integer	n,
		integer	m,
		double precision, dimension(n)	x,
		double precision, dimension(n)	l,
		double precision, dimension(n)	u,
		integer, dimension(n)	nbd,
		double precision	f,
		double precision, dimension(n)	g,
		double precision	factr,
		double precision	pgtol,
		double precision, dimension(2*m*n + 5*n + 11*m*m + 8*m)	wa,
		integer, dimension(3*n)	iwa,
		character*60	task,
		integer	iprint,
		character*60	csave,
		logical, dimension(4)	lsave,
		integer, dimension(44)	isave,
		double precision, dimension(29)	dsave
	)

 
      character*60     task, csave
      logical          lsave(4)
      integer          n, m, iprint, 
     +                 nbd(n), iwa(3*n), isave(44)
      double precision f, factr, pgtol, x(n), l(n), u(n), g(n),
c
c-jlm-jn
     +                 wa(2*m*n + 5*n + 11*m*m + 8*m), dsave(29)
 
c     ************
c
c     Subroutine setulb
c
c     this subroutine partitions the working arrays wa and iwa, and 
c       then uses the limited memory bfgs method to solve the bound
c       constrained optimization problem by calling mainlb.
c(the direct method will be used in the subspace minimization.)
c
c     n is an integer variable.
c       on entry n is the dimension of the problem.
c       on exit n is unchanged.
c
c     m is an integer variable.
c       on entry m is the maximum number of variable metric corrections
c         used to define the limited memory matrix.
c       on exit m is unchanged.
c
c     x is a double precision array of dimension n.
c       on entry x is an approximation to the solution.
c       on exit x is the current approximation.
c
c     l is a double precision array of dimension n.
c       on entry l is the lower bound on x.
c       on exit l is unchanged.
c
c     u is a double precision array of dimension n.
c       on entry u is the upper bound on x.
c       on exit u is unchanged.
c
c     nbd is an integer array of dimension n.
c       on entry nbd represents the type of bounds imposed on the
c         variables, and must be specified as follows:
c         nbd(i)=0 if x(i) is unbounded,
c                1 if x(i) has only a lower bound,
c                2 if x(i) has both lower and upper bounds, and
c                3 if x(i) has only an upper bound.
c       on exit nbd is unchanged.
c
c     f is a double precision variable.
c       on first entry f is unspecified.
c       on final exit f is the value of the function at x.
c
c     g is a double precision array of dimension n.
c       on first entry g is unspecified.
c       on final exit g is the value of the gradient at x.
c
c     factr is a double precision variable.
c       on entry factr >= 0 is specified by the user.  the iteration
c         will stop when
c
c(f^k - f^{k+1})/max{|f^k|,|f^{k+1}|,1} <= factr*epsmch
c
c         where epsmch is the machine precision, which is automatically
c         generated by the code. typical values for factr: 1.d+12 for
c         low accuracy; 1.d+7 for moderate accuracy; 1.d+1 for extremely
c         high accuracy.
c       on exit factr is unchanged.
c
c     pgtol is a double precision variable.
c       on entry pgtol >= 0 is specified by the user.  the iteration
c         will stop when
c
c                 max{|proj g_i | i = 1, ..., n} <= pgtol
c
c         where pg_i is the ith component of the projected gradient.   
c       on exit pgtol is unchanged.
c
c     wa is a double precision working array of length 
c(2mmax + 5)nmax + 12mmax^2 + 12mmax.
c
c     iwa is an integer working array of length 3nmax.
c
c     task is a working string of characters of length 60 indicating
c       the current job when entering and quitting this subroutine.
c
c     iprint is an integer variable that must be set by the user.
c       it controls the frequency and type of output generated:
c        iprint<0    no output is generated;
c        iprint=0    print only one line at the last iteration;
c        0<iprint<99 print also f and |proj g| every iprint iterations;
c        iprint=99   print details of every iteration except n-vectors;
c        iprint=100  print also the changes of active set and final x;
c        iprint>100  print details of every iteration including x and g;
c       when iprint > 0, the file iterate.dat will be created to
c                        summarize the iteration.
c
c     csave is a working string of characters of length 60.
c
c     lsave is a logical working array of dimension 4.
c       on exit with 'task' = new_x, the following information is 
c                                                             available:
c         If lsave(1) = .true.  then  the initial x has been replaced by
c                                     its projection in the feasible set;
c         If lsave(2) = .true.  then  the problem is constrained;
c         If lsave(3) = .true.  then  each variable has upper and lower
c                                     bounds;
c
c     isave is an integer working array of dimension 44.
c       on exit with 'task' = new_x, the following information is 
c                                                             available:
c         isave(22) = the total number of intervals explored in the 
c                         search of cauchy points;
c         isave(26) = the total number of skipped bfgs updates before 
c                         the current iteration;
c         isave(30) = the number of current iteration;
c         isave(31) = the total number of bfgs updates prior the current
c                         iteration;
c         isave(33) = the number of intervals explored in the search of
c                         cauchy point in the current iteration;
c         isave(34) = the total number of function and gradient 
c                         evaluations;
c         isave(36) = the number of function value or gradient
c                                  evaluations in the current iteration;
c         if isave(37) = 0  then the subspace argmin is within the box;
c         if isave(37) = 1  then the subspace argmin is beyond the box;
c         isave(38) = the number of free variables in the current
c                         iteration;
c         isave(39) = the number of active constraints in the current
c                         iteration;
c         n + 1 - isave(40) = the number of variables leaving the set of
c                           active constraints in the current iteration;
c         isave(41) = the number of variables entering the set of active
c                         constraints in the current iteration.
c
c     dsave is a double precision working array of dimension 29.
c       on exit with 'task' = new_x, the following information is
c                                                             available:
c         dsave(1) = current 'theta' in the bfgs matrix;
c         dsave(2) = f(x) in the previous iteration;
c         dsave(3) = factr*epsmch;
c         dsave(4) = 2-norm of the line search direction vector;
c         dsave(5) = the machine precision epsmch generated by the code;
c         dsave(7) = the accumulated time spent on searching for
c                                                         cauchy points;
c         dsave(8) = the accumulated time spent on
c                                                 subspace minimization;
c         dsave(9) = the accumulated time spent on line search;
c         dsave(11) = the slope of the line search function at
c                                  the current point of line search;
c         dsave(12) = the maximum relative step length imposed in
c                                                           line search;
c         dsave(13) = the infinity norm of the projected gradient;
c         dsave(14) = the relative step length in the line search;
c         dsave(15) = the slope of the line search function at
c                                 the starting point of the line search;
c         dsave(16) = the square of the 2-norm of the line search
c                                                      direction vector.
c
c     subprograms called:
c
c       l-bfgs-b library ... mainlb.    
c
c
c     references:
c
c       [1] r. h. byrd, p. lu, j. nocedal and c. zhu, ``a limited
c       memory algorithm for bound constrained optimization'',
c       siam j. scientific computing 16 (1995), no. 5, pp. 1190--1208.
c
c       [2] c. zhu, r.h. byrd, p. lu, j. nocedal, ``l-bfgs-b: a
c       limited memory fortran code for solving bound constrained
c       optimization problems'', tech. report, nam-11, eecs department,
c       northwestern university, 1994.
c
c(postscript files of these papers are available via anonymous
c        ftp to eecs.nwu.edu in the directory pub/lbfgs/lbfgs_bcm.)
c
c                           *  *  *
c
c     neos, november 1994. (latest revision june 1996.)
c     optimization technology center.
c     argonne national laboratory and northwestern university.
c     written by
c                        ciyou zhu
c     in collaboration with r.h. byrd, p. lu-chen and j. nocedal.
c
c
c     ************
c-jlm-jn 
      integer   lws,lr,lz,lt,ld,lxp,lwa,
     +          lwy,lsy,lss,lwt,lwn,lsnd

      if (task .eq. 'START') then
         isave(1)  = m*n
         isave(2)  = m**2
         isave(3)  = 4*m**2
         isave(4)  = 1                      ! ws      m*n
         isave(5)  = isave(4)  + isave(1)   ! wy      m*n
         isave(6)  = isave(5)  + isave(1)   ! wsy     m**2
         isave(7)  = isave(6)  + isave(2)   ! wss     m**2
         isave(8)  = isave(7)  + isave(2)   ! wt      m**2
         isave(9)  = isave(8)  + isave(2)   ! wn      4*m**2
         isave(10) = isave(9)  + isave(3)   ! wsnd    4*m**2
         isave(11) = isave(10) + isave(3)   ! wz      n
         isave(12) = isave(11) + n          ! wr      n
         isave(13) = isave(12) + n          ! wd      n
         isave(14) = isave(13) + n          ! wt      n
         isave(15) = isave(14) + n          ! wxp     n
         isave(16) = isave(15) + n          ! wa      8*m
      endif
      lws  = isave(4)
      lwy  = isave(5)
      lsy  = isave(6)
      lss  = isave(7)
      lwt  = isave(8)
      lwn  = isave(9)
      lsnd = isave(10)
      lz   = isave(11)
      lr   = isave(12)
      ld   = isave(13)
      lt   = isave(14)
      lxp  = isave(15)
      lwa  = isave(16)

      call 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(lxp),
     +  wa(lwa),
     +  iwa(1),iwa(n+1),iwa(2*n+1),task,iprint, 
     +  csave,lsave,isave(22),dsave)

      return

subsm()

subroutine subsm	(	integer	n,
		integer	m,
		integer	nsub,
		integer, dimension(nsub)	ind,
		double precision, dimension(n)	l,
		double precision, dimension(n)	u,
		integer, dimension(n)	nbd,
		double precision, dimension(n)	x,
		double precision, dimension(n)	d,
		double precision, dimension(n)	xp,
		double precision, dimension(n, m)	ws,
		double precision, dimension(n, m)	wy,
		double precision	theta,
		double precision, dimension(n)	xx,
		double precision, dimension(n)	gg,
		integer	col,
		integer	head,
		integer	iword,
		double precision, dimension(2*m)	wv,
		double precision, dimension(2*m, 2*m)	wn,
		integer	iprint,
		integer	info
	)

      implicit none
      integer          n, m, nsub, col, head, iword, iprint, info, 
     +                 ind(nsub), nbd(n)
      double precision theta, 
     +                 l(n), u(n), x(n), d(n), xp(n), xx(n), gg(n),
     +                 ws(n, m), wy(n, m), 
     +                 wv(2*m), wn(2*m, 2*m)

c     **********************************************************************
c
c     This routine contains the major changes in the updated version.
c     The changes are described in the accompanying paper
c
c      Jose Luis Morales, Jorge Nocedal
c      "Remark On Algorithm 788: L-BFGS-B: Fortran Subroutines for Large-Scale
c       Bound Constrained Optimization". Decemmber 27, 2010.
c
c             J.L. Morales  Departamento de Matematicas, 
c                           Instituto Tecnologico Autonomo de Mexico
c                           Mexico D.F.
c
c             J, Nocedal    Department of Electrical Engineering and
c                           Computer Science.
c                           Northwestern University. Evanston, IL. USA
c
c                           January 17, 2011
c
c      **********************************************************************
c                           
c
c     Subroutine subsm
c
c     Given xcp, l, u, r, an index set that specifies
c       the active set at xcp, and an l-BFGS matrix B 
c       (in terms of WY, WS, SY, WT, head, col, and theta), 
c       this subroutine computes an approximate solution
c       of the subspace problem
c
c       (P)   min Q(x) = r'(x-xcp) + 1/2 (x-xcp)' B (x-xcp)
c
c             subject to l<=x<=u
c                       x_i=xcp_i for all i in A(xcp)
c                     
c       along the subspace unconstrained Newton direction 
c       
c          d = -(Z'BZ)^(-1) r.
c
c       The formula for the Newton direction, given the L-BFGS matrix
c       and the Sherman-Morrison formula, is
c
c          d = (1/theta)r + (1/theta*2) Z'WK^(-1)W'Z r.
c 
c       where
c                 K = [-D -Y'ZZ'Y/theta     L_a'-R_z'  ]
c                     [L_a -R_z           theta*S'AA'S ]
c
c     Note that this procedure for computing d differs 
c     from that described in [1]. One can show that the matrix K is
c     equal to the matrix M^[-1]N in that paper.
c
c     n is an integer variable.
c       On entry n is the dimension of the problem.
c       On exit n is unchanged.
c
c     m is an integer variable.
c       On entry m is the maximum number of variable metric corrections
c         used to define the limited memory matrix.
c       On exit m is unchanged.
c
c     nsub is an integer variable.
c       On entry nsub is the number of free variables.
c       On exit nsub is unchanged.
c
c     ind is an integer array of dimension nsub.
c       On entry ind specifies the coordinate indices of free variables.
c       On exit ind is unchanged.
c
c     l is a double precision array of dimension n.
c       On entry l is the lower bound of x.
c       On exit l is unchanged.
c
c     u is a double precision array of dimension n.
c       On entry u is the upper bound of x.
c       On exit u is unchanged.
c
c     nbd is a integer array of dimension n.
c       On entry nbd represents the type of bounds imposed on the
c         variables, and must be specified as follows:
c         nbd(i)=0 if x(i) is unbounded,
c                1 if x(i) has only a lower bound,
c                2 if x(i) has both lower and upper bounds, and
c                3 if x(i) has only an upper bound.
c       On exit nbd is unchanged.
c
c     x is a double precision array of dimension n.
c       On entry x specifies the Cauchy point xcp. 
c       On exit x(i) is the minimizer of Q over the subspace of
c                                                        free variables. 
c
c     d is a double precision array of dimension n.
c       On entry d is the reduced gradient of Q at xcp.
c       On exit d is the Newton direction of Q. 
c
c    xp is a double precision array of dimension n.
c       used to safeguard the projected Newton direction 
c
c    xx is a double precision array of dimension n
c       On entry it holds the current iterate
c       On output it is unchanged

c    gg is a double precision array of dimension n
c       On entry it holds the gradient at the current iterate
c       On output it is unchanged
c
c     ws and wy are double precision arrays;
c     theta is a double precision variable;
c     col is an integer variable;
c     head is an integer variable.
c       On entry they store the information defining the
c                                          limited memory BFGS matrix:
c         ws(n,m) stores S, a set of s-vectors;
c         wy(n,m) stores Y, a set of y-vectors;
c         theta is the scaling factor specifying B_0 = theta I;
c         col is the number of variable metric corrections stored;
c         head is the location of the 1st s- (or y-) vector in S (or Y).
c       On exit they are unchanged.
c
c     iword is an integer variable.
c       On entry iword is unspecified.
c       On exit iword specifies the status of the subspace solution.
c         iword = 0 if the solution is in the box,
c                 1 if some bound is encountered.
c
c     wv is a double precision working array of dimension 2m.
c
c     wn is a double precision array of dimension 2m x 2m.
c       On entry the upper triangle of wn stores the LEL^T factorization
c         of the indefinite matrix
c
c              K = [-D -Y'ZZ'Y/theta     L_a'-R_z'  ]
c                  [L_a -R_z           theta*S'AA'S ]
c                                                    where E = [-I  0]
c                                                              [ 0  I]
c       On exit wn is unchanged.
c
c     iprint is an INTEGER variable that must be set by the user.
c       It controls the frequency and type of output generated:
c        iprint<0    no output is generated;
c        iprint=0    print only one line at the last iteration;
c        0<iprint<99 print also f and |proj g| every iprint iterations;
c        iprint=99   print details of every iteration except n-vectors;
c        iprint=100  print also the changes of active set and final x;
c        iprint>100  print details of every iteration including x and g;
c       When iprint > 0, the file iterate.dat will be created to
c                        summarize the iteration.
c
c     info is an integer variable.
c       On entry info is unspecified.
c       On exit info = 0       for normal return,
c                    = nonzero for abnormal return 
c                                  when the matrix K is ill-conditioned.
c
c     Subprograms called:
c
c       Linpack dtrsl.
c
c
c     References:
c
c       [1] R. H. Byrd, P. Lu, J. Nocedal and C. Zhu, ``A limited
c       memory algorithm for bound constrained optimization'',
c       SIAM J. Scientific Computing 16 (1995), no. 5, pp. 1190--1208.
c
c
c
c                           *  *  *
c
c     NEOS, November 1994. (Latest revision June 1996.)
c     Optimization Technology Center.
c     Argonne National Laboratory and Northwestern University.
c     Written by
c                        Ciyou Zhu
c     in collaboration with R.H. Byrd, P. Lu-Chen and J. Nocedal.
c
c
c     ************

      integer          pointr,m2,col2,ibd,jy,js,i,j,k
      double precision alpha, xk, dk, temp1, temp2 
      double precision one,zero
      parameter(one=1.0d0,zero=0.0d0)
c
      double precision dd_p

      if (nsub .le. 0) return
      if (iprint .ge. 99) write (6,1001)

c     Compute wv = W'Zd.

      pointr = head 
      do 20 i = 1, col
         temp1 = zero
         temp2 = zero
         do 10 j = 1, nsub
            k = ind(j)
            temp1 = temp1 + wy(k,pointr)*d(j)
            temp2 = temp2 + ws(k,pointr)*d(j)
  10     continue
         wv(i) = temp1
         wv(col + i) = theta*temp2
         pointr = mod(pointr,m) + 1
  20  continue
 
c     Compute wv:=K^(-1)wv.

      m2 = 2*m
      col2 = 2*col
      call dtrsl(wn,m2,col2,wv,11,info)
      if (info .ne. 0) return
      do 25 i = 1, col
         wv(i) = -wv(i)
  25     continue
      call dtrsl(wn,m2,col2,wv,01,info)
      if (info .ne. 0) return
 
c     Compute d = (1/theta)d + (1/theta**2)Z'W wv.
 
      pointr = head
      do 40 jy = 1, col
         js = col + jy
         do 30 i = 1, nsub
            k = ind(i)
            d(i) = d(i) + wy(k,pointr)*wv(jy)/theta     
     +                  + ws(k,pointr)*wv(js)
  30     continue
         pointr = mod(pointr,m) + 1
  40  continue

      call dscal( nsub, one/theta, d, 1 )
c 
c-----------------------------------------------------------------
c     Let us try the projection, d is the Newton direction

      iword = 0

      call dcopy ( n, x, 1, xp, 1 )
c
      do 50 i=1, nsub
         k  = ind(i)
         dk = d(i)
         xk = x(k)
         if ( nbd(k) .ne. 0 ) then
c
            if ( nbd(k).eq.1 ) then          ! lower bounds only
               x(k) = max( l(k), xk + dk )
               if ( x(k).eq.l(k) ) iword = 1
            else 
c     
               if ( nbd(k).eq.2 ) then       ! upper and lower bounds
                  xk   = max( l(k), xk + dk ) 
                  x(k) = min( u(k), xk )
                  if ( x(k).eq.l(k) .or. x(k).eq.u(k) ) iword = 1
               else
c
                  if ( nbd(k).eq.3 ) then    ! upper bounds only
                     x(k) = min( u(k), xk + dk )
                     if ( x(k).eq.u(k) ) iword = 1
                  end if 
               end if
            end if
c            
         else                                ! free variables
            x(k) = xk + dk
         end if 
 50   continue
c
      if ( iword.eq.0 ) then
         go to 911
      end if
c
c     check sign of the directional derivative
c
      dd_p = zero
      do 55 i=1, n
         dd_p  = dd_p + (x(i) - xx(i))*gg(i)
 55   continue
      if ( dd_p .gt.zero ) then
         call dcopy( n, xp, 1, x, 1 )
         write(6,*) ' Positive dir derivative in projection '
         write(6,*) ' Using the backtracking step '
      else
         go to 911
      endif
c
c-----------------------------------------------------------------
c
      alpha = one
      temp1 = alpha
      ibd   = 0 
      do 60 i = 1, nsub
         k = ind(i)
         dk = d(i)
         if (nbd(k) .ne. 0) then
            if (dk .lt. zero .and. nbd(k) .le. 2) then
               temp2 = l(k) - x(k)
               if (temp2 .ge. zero) then
                  temp1 = zero
               else if (dk*alpha .lt. temp2) then
                  temp1 = temp2/dk
               endif
            else if (dk .gt. zero .and. nbd(k) .ge. 2) then
               temp2 = u(k) - x(k)
               if (temp2 .le. zero) then
                  temp1 = zero
               else if (dk*alpha .gt. temp2) then
                  temp1 = temp2/dk
               endif
            endif
            if (temp1 .lt. alpha) then
               alpha = temp1
               ibd = i
            endif
         endif
 60   continue
      
      if (alpha .lt. one) then
         dk = d(ibd)
         k = ind(ibd)
         if (dk .gt. zero) then
            x(k) = u(k)
            d(ibd) = zero
         else if (dk .lt. zero) then
            x(k) = l(k)
            d(ibd) = zero
         endif
      endif
      do 70 i = 1, nsub
         k    = ind(i)
         x(k) = x(k) + alpha*d(i)
 70   continue
cccccc
 911  continue

      if (iprint .ge. 99) write (6,1004)

 1001 format (/,'----------------SUBSM entered-----------------',/)
 1004 format (/,'----------------exit SUBSM --------------------',/)

      return