```!   E04LBF Example Program Text
!   Mark 26.2 Release. NAG Copyright 2017.
Module e04lbfe_mod

!     E04LBF Example Program Module:
!            Parameters and User-defined Routines

!     .. Use Statements ..
Use nag_library, Only: nag_wp
!     .. Implicit None Statement ..
Implicit None
!     .. Accessibility Statements ..
Private
Public                           :: funct, h, monit
!     .. Parameters ..
Integer, Parameter, Public       :: liw = 2, n = 4, nout = 6
Integer, Parameter, Public       :: lh = n*(n-1)/2
Integer, Parameter, Public       :: lw = 7*n + n*(n-1)/2
Contains
Subroutine funct(iflag,n,xc,fc,gc,iw,liw,w,lw)
!       Routine to evaluate objective function and its 1st derivatives.

!       .. Scalar Arguments ..
Real (Kind=nag_wp), Intent (Out) :: fc
Integer, Intent (Inout)        :: iflag
Integer, Intent (In)           :: liw, lw, n
!       .. Array Arguments ..
Real (Kind=nag_wp), Intent (Out) :: gc(n)
Real (Kind=nag_wp), Intent (Inout) :: w(lw)
Real (Kind=nag_wp), Intent (In) :: xc(n)
Integer, Intent (Inout)        :: iw(liw)
!       .. Executable Statements ..
fc = (xc(1)+10.0_nag_wp*xc(2))**2 + 5.0_nag_wp*(xc(3)-xc(4))**2 +      &
(xc(2)-2.0_nag_wp*xc(3))**4 + 10.0_nag_wp*(xc(1)-xc(4))**4
gc(1) = 2.0_nag_wp*(xc(1)+10.0_nag_wp*xc(2)) +                         &
40.0_nag_wp*(xc(1)-xc(4))**3
gc(2) = 20.0_nag_wp*(xc(1)+10.0_nag_wp*xc(2)) +                        &
4.0_nag_wp*(xc(2)-2.0_nag_wp*xc(3))**3
gc(3) = 10.0_nag_wp*(xc(3)-xc(4)) - 8.0_nag_wp*(xc(2)-2.0_nag_wp*xc(3) &
)**3
gc(4) = 10.0_nag_wp*(xc(4)-xc(3)) - 40.0_nag_wp*(xc(1)-xc(4))**3

Return

End Subroutine funct
Subroutine h(iflag,n,xc,fhesl,lh,fhesd,iw,liw,w,lw)
!       Routine to evaluate 2nd derivatives

!       .. Scalar Arguments ..
Integer, Intent (Inout)        :: iflag
Integer, Intent (In)           :: lh, liw, lw, n
!       .. Array Arguments ..
Real (Kind=nag_wp), Intent (Inout) :: fhesd(n), w(lw)
Real (Kind=nag_wp), Intent (Out) :: fhesl(lh)
Real (Kind=nag_wp), Intent (In) :: xc(n)
Integer, Intent (Inout)        :: iw(liw)
!       .. Executable Statements ..
fhesd(1) = 2.0_nag_wp + 120.0_nag_wp*(xc(1)-xc(4))**2
fhesd(2) = 200.0_nag_wp + 12.0_nag_wp*(xc(2)-2.0_nag_wp*xc(3))**2
fhesd(3) = 10.0_nag_wp + 48.0_nag_wp*(xc(2)-2.0_nag_wp*xc(3))**2
fhesd(4) = 10.0_nag_wp + 120.0_nag_wp*(xc(1)-xc(4))**2
fhesl(1) = 20.0_nag_wp
fhesl(2) = 0.0_nag_wp
fhesl(3) = -24.0_nag_wp*(xc(2)-2.0_nag_wp*xc(3))**2
fhesl(4) = -120.0_nag_wp*(xc(1)-xc(4))**2
fhesl(5) = 0.0_nag_wp
fhesl(6) = -10.0_nag_wp

Return

End Subroutine h
Subroutine monit(n,xc,fc,gc,istate,gpjnrm,cond,posdef,niter,nf,iw,liw,w, &
lw)
!       Monitoring routine

!       .. Scalar Arguments ..
Real (Kind=nag_wp), Intent (In) :: cond, fc, gpjnrm
Integer, Intent (In)           :: liw, lw, n, nf, niter
Logical, Intent (In)           :: posdef
!       .. Array Arguments ..
Real (Kind=nag_wp), Intent (In) :: gc(n), xc(n)
Real (Kind=nag_wp), Intent (Inout) :: w(lw)
Integer, Intent (In)           :: istate(n)
Integer, Intent (Inout)        :: iw(liw)
!       .. Local Scalars ..
Integer                        :: isj, j
!       .. Executable Statements ..
Write (nout,*)
Write (nout,*) ' Itn     Fn evals              Fn value            ',  &
Write (nout,99999) niter, nf, fc, gpjnrm
Write (nout,*)
Write (nout,*)                                                         &
' J           X(J)                G(J)           Status'

Do j = 1, n
isj = istate(j)

Select Case (isj)
Case (1:)
Write (nout,99998) j, xc(j), gc(j), '    Free'
Case (-1)
Write (nout,99998) j, xc(j), gc(j), '    Upper Bound'
Case (-2)
Write (nout,99998) j, xc(j), gc(j), '    Lower Bound'
Case (-3)
Write (nout,99998) j, xc(j), gc(j), '    Constant'
End Select

End Do

If (cond/=0.0_nag_wp) Then

If (cond>1.0E6_nag_wp) Then
Write (nout,*)
Write (nout,*)                                                     &
'Estimated condition number of projected Hessian is more than ', &
'1.0E+6'
Else
Write (nout,*)
Write (nout,99997)                                                 &
'Estimated condition number of projected Hessian = ', cond
End If

If (.Not. posdef) Then
Write (nout,*)
Write (nout,*) 'Projected Hessian matrix is not positive definite'
End If

End If

Return

99999   Format (1X,I3,6X,I5,2(6X,1P,E20.4))
99998   Format (1X,I2,1X,1P,2E20.4,A)
99997   Format (1X,A,1P,E10.2)
End Subroutine monit
End Module e04lbfe_mod
Program e04lbfe

!     E04LBF Example Main Program

!     .. Use Statements ..
Use e04lbfe_mod, Only: funct, h, lh, liw, lw, monit, n, nout
Use nag_library, Only: e04hcf, e04hdf, e04lbf, nag_wp
!     .. Implicit None Statement ..
Implicit None
!     .. Local Scalars ..
Real (Kind=nag_wp)               :: eta, f, stepmx, xtol
Integer                          :: ibound, ifail, iprint, maxcal, nz
!     .. Local Arrays ..
Real (Kind=nag_wp)               :: bl(n), bu(n), g(n), hesd(n),         &
hesl(lh), w(lw), x(n)
Integer                          :: istate(n), iw(liw)
!     .. Intrinsic Procedures ..
Intrinsic                        :: count
!     .. Executable Statements ..
Write (nout,*) 'E04LBF Example Program Results'
Flush (nout)

!     Set up an arbitrary point at which to check the derivatives

x(1:n) = (/1.46_nag_wp,-0.82_nag_wp,0.57_nag_wp,1.21_nag_wp/)

!     Check the 1st derivatives

ifail = 0
Call e04hcf(n,funct,x,f,g,iw,liw,w,lw,ifail)

!     Check the 2nd derivatives

ifail = 0
Call e04hdf(n,funct,h,x,g,hesl,lh,hesd,iw,liw,w,lw,ifail)

!     Continue setting parameters for E04LBF

!     Set IPRINT to 1 to obtain output from MONIT at each iteration
iprint = -1

maxcal = 50*n
eta = 0.9_nag_wp

!     Set XTOL to zero so that E04LBF will use the default tolerance

xtol = 0.0_nag_wp

!     We estimate that the minimum will be within 4 units of the
!     starting point

stepmx = 4.0_nag_wp

ibound = 0

!     X(3) is unconstrained, so we set BL(3) to a large negative
!     number and BU(3) to a large positive number.

bl(1:n) = (/1.0_nag_wp,-2.0_nag_wp,-1.0E6_nag_wp,1.0_nag_wp/)
bu(1:n) = (/3.0_nag_wp,0.0_nag_wp,1.0E6_nag_wp,3.0_nag_wp/)

!     Set up starting point

x(1:n) = (/3.0_nag_wp,-1.0_nag_wp,0.0_nag_wp,1.0_nag_wp/)

ifail = -1
Call e04lbf(n,funct,h,monit,iprint,maxcal,eta,xtol,stepmx,ibound,bl,bu,  &
x,hesl,lh,hesd,istate,f,g,iw,liw,w,lw,ifail)

Select Case (ifail)
Case (0,2:)
Write (nout,*)
Write (nout,99999) 'Function value on exit is ', f
Write (nout,99998) 'at the point', x(1:n)
Write (nout,*) 'The corresponding (machine dependent) gradient is'
Write (nout,99997) g(1:n)
Write (nout,99996) 'ISTATE contains', istate(1:n)

nz = count(istate(1:n)>0)

Write (nout,99995) 'and HESD contains', hesd(1:nz)
End Select

99999 Format (1X,A,F9.4)
99998 Format (1X,A,4F9.4)
99997 Format (23X,1P,4E12.3)
99996 Format (1X,A,4I5)
99995 Format (1X,A,4E12.4)
End Program e04lbfe
```