```!   E04KDF Example Program Text
!   Mark 26.1 Release. NAG Copyright 2016.
Module e04kdfe_mod

!     E04KDF 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, 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.
!       A global variable could be updated here to count the number of
!       calls of FUNCT with IFLAG = 1 (since NF in MONIT only counts
!       calls with IFLAG = 2)

!       .. 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 ..
If (iflag/=1) Then
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
End If

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 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 e04kdfe_mod
Program e04kdfe

!     E04KDF Example Main Program

!     .. Use Statements ..
Use e04kdfe_mod, Only: funct, lh, liw, lw, monit, n, nout
Use nag_library, Only: e04hcf, e04kdf, nag_wp
!     .. Implicit None Statement ..
Implicit None
!     .. Local Scalars ..
Real (Kind=nag_wp)               :: delta, eta, f, stepmx, xtol
Integer                          :: ibound, ifail, iprint, maxcal
!     .. 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)
!     .. Executable Statements ..
Write (nout,*) 'E04KDF Example Program Results'
Flush (nout)

!     Check FUNCT by calling E04HCF at an arbitrary point. Since E04HCF
!     only checks the derivatives calculated when IFLAG = 2, a separate
!     program should be run before using E04HCF or E04KDF to check that
!     FUNCT gives the same values for the GC(J) when IFLAG is set to 1
!     as when IFLAG is set to 2.

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

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

!     Continue setting parameters for E04KDF

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

maxcal = 50*n
eta = 0.5_nag_wp

!     Set XTOL and DELTA to zero so that E04KDF will use the default
!     values

xtol = 0.0_nag_wp
delta = 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 not bounded, 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 e04kdf(n,funct,monit,iprint,maxcal,eta,xtol,delta,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,99999) 'at the point', x(1:n)
Write (nout,*) 'The corresponding (machine dependent) gradient is'
Write (nout,99998) g(1:n)
Write (nout,99997) 'ISTATE contains', istate(1:n)
Write (nout,99996) 'and HESD contains', hesd(1:n)
End Select

99999 Format (1X,A,4F12.4)
99998 Format (24X,1P,4E12.3)
99997 Format (1X,A,4I5)
99996 Format (1X,A,4E12.4)
End Program e04kdfe
```