```!   E04HDF Example Program Text
!   Mark 26.2 Release. NAG Copyright 2017.
Module e04hdfe_mod

!     E04HDF 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
!     .. Parameters ..
Integer, Parameter, Public       :: liw = 1, n = 4, nout = 6
Integer, Parameter, Public       :: lh = n*(n-1)/2
Integer, Parameter, Public       :: lw = 5*n
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
End Module e04hdfe_mod
Program e04hdfe

!     E04HDF Example Main Program

!     .. Use Statements ..
Use e04hdfe_mod, Only: funct, h, lh, liw, lw, n, nout
Use nag_library, Only: e04hcf, e04hdf, nag_wp
!     .. Implicit None Statement ..
Implicit None
!     .. Local Scalars ..
Real (Kind=nag_wp)               :: f
Integer                          :: i, ifail, k
!     .. Local Arrays ..
Real (Kind=nag_wp)               :: g(n), hesd(n), hesl(lh), w(lw), x(n)
Integer                          :: iw(liw)
!     .. Executable Statements ..
Write (nout,*) 'E04HDF Example Program Results'

!     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/)

Write (nout,*)
Write (nout,*) 'The test point is'
Write (nout,99999) x(1:n)

!     Check the 1st derivatives

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

!     Check the 2nd derivatives

ifail = -1
Call e04hdf(n,funct,h,x,g,hesl,lh,hesd,iw,liw,w,lw,ifail)

If (ifail>=0) Then
Write (nout,*)

If (ifail==0) Then
Write (nout,*) '2nd derivatives are consistent with 1st derivatives'
Else If (ifail==2) Then
Write (nout,*) 'Probable error in calculation of 2nd derivatives'
End If

Write (nout,*)
Write (nout,99998) 'At the test point, FUNCT gives the function value' &
, f
Write (nout,*) 'and the 1st derivatives'
Write (nout,99997) g(1:n)
Write (nout,*)
Write (nout,*) 'H gives the lower triangle of the Hessian matrix'
Write (nout,99996) hesd(1)

k = 1

Do i = 2, n
Write (nout,99996) hesl(k:(k+i-2)), hesd(i)
k = k + i - 1
End Do

End If

99999 Format (1X,4F9.4)
99998 Format (1X,A,1P,E12.4)
99997 Format (1X,1P,4E12.3)
99996 Format (1X,1P,4E12.3)
End Program e04hdfe
```