```!   E04YBF Example Program Text
!   Mark 26.2 Release. NAG Copyright 2017.

Module e04ybfe_mod

!     E04YBF 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                           :: lsqfun, lsqhes
!     .. Parameters ..
Integer, Parameter, Public       :: liw = 1, mdec = 15, ndec = 3,        &
nin = 5, nout = 6
Integer, Parameter, Public       :: lb = ndec*(ndec+1)/2
Integer, Parameter, Public       :: ldfjac = mdec
Integer, Parameter, Public       :: lw = 5*ndec + mdec + mdec*ndec +     &
ndec*(ndec-1)/2
!     .. Local Arrays ..
Real (Kind=nag_wp), Public, Save :: t(mdec,ndec), y(mdec)
Contains
Subroutine lsqfun(iflag,m,n,xc,fvec,fjac,ldfjac,iw,liw,w,lw)

!       Routine to evaluate the residuals and their 1st derivatives

!       .. Scalar Arguments ..
Integer, Intent (Inout)        :: iflag
Integer, Intent (In)           :: ldfjac, liw, lw, m, n
!       .. Array Arguments ..
Real (Kind=nag_wp), Intent (Inout) :: fjac(ldfjac,n), w(lw)
Real (Kind=nag_wp), Intent (Out) :: fvec(m)
Real (Kind=nag_wp), Intent (In) :: xc(n)
Integer, Intent (Inout)        :: iw(liw)
!       .. Local Scalars ..
Real (Kind=nag_wp)             :: denom, dummy
Integer                        :: i
!       .. Executable Statements ..
Do i = 1, m
denom = xc(2)*t(i,2) + xc(3)*t(i,3)
fvec(i) = xc(1) + t(i,1)/denom - y(i)
fjac(i,1) = 1.0E0_nag_wp
dummy = -1.0E0_nag_wp/(denom*denom)
fjac(i,2) = t(i,1)*t(i,2)*dummy
fjac(i,3) = t(i,1)*t(i,3)*dummy
End Do

Return

End Subroutine lsqfun
Subroutine lsqhes(iflag,m,n,fvec,xc,b,lb,iw,liw,w,lw)

!       Routine to compute the lower triangle of the matrix B
!       (stored by rows in the array B)

!       .. Scalar Arguments ..
Integer, Intent (Inout)        :: iflag
Integer, Intent (In)           :: lb, liw, lw, m, n
!       .. Array Arguments ..
Real (Kind=nag_wp), Intent (Out) :: b(lb)
Real (Kind=nag_wp), Intent (In) :: fvec(m), xc(n)
Real (Kind=nag_wp), Intent (Inout) :: w(lw)
Integer, Intent (Inout)        :: iw(liw)
!       .. Local Scalars ..
Real (Kind=nag_wp)             :: dummy, sum22, sum32, sum33
Integer                        :: i
!       .. Executable Statements ..
b(1) = 0.0E0_nag_wp
b(2) = 0.0E0_nag_wp
sum22 = 0.0E0_nag_wp
sum32 = 0.0E0_nag_wp
sum33 = 0.0E0_nag_wp

Do i = 1, m
dummy = 2.0E0_nag_wp*t(i,1)/(xc(2)*t(i,2)+xc(3)*t(i,3))**3
sum22 = sum22 + fvec(i)*dummy*t(i,2)**2
sum32 = sum32 + fvec(i)*dummy*t(i,2)*t(i,3)
sum33 = sum33 + fvec(i)*dummy*t(i,3)**2
End Do

b(3) = sum22
b(4) = 0.0E0_nag_wp
b(5) = sum32
b(6) = sum33

Return

End Subroutine lsqhes
End Module e04ybfe_mod
Program e04ybfe

!     E04YBF Example Main Program

!     .. Use Statements ..
Use e04ybfe_mod, Only: lb, ldfjac, liw, lsqfun, lsqhes, lw, mdec, ndec,  &
nin, nout, t, y
Use nag_library, Only: e04yaf, e04ybf, nag_wp
!     .. Implicit None Statement ..
Implicit None
!     .. Local Scalars ..
Integer                          :: i, ifail, k, m, n
!     .. Local Arrays ..
Real (Kind=nag_wp)               :: b(lb), fjac(ldfjac,ndec),            &
fvec(mdec), w(lw), x(ndec)
Integer                          :: iw(liw)
!     .. Executable Statements ..
Write (nout,*) 'E04YBF Example Program Results'

!     Skip heading in data file

m = mdec
n = ndec

!     Observations of TJ (J = 1, 2, ..., n) are held in T(I, J)
!     (I = 1, 2, ..., m)

Do i = 1, m
End Do

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

x(1:n) = (/0.19E0_nag_wp,-1.34E0_nag_wp,0.88E0_nag_wp/)

!     Check the 1st derivatives

ifail = 0
Call e04yaf(m,n,lsqfun,x,fvec,fjac,ldfjac,iw,liw,w,lw,ifail)

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

!     Check the evaluation of B

ifail = -1
Call e04ybf(m,n,lsqfun,lsqhes,x,fvec,fjac,ldfjac,b,lb,iw,liw,w,lw,ifail)

If (ifail>=0 .And. ifail/=1) Then

Select Case (ifail)
Case (0)
Write (nout,*)
Write (nout,*) 'The matrix B is consistent with 1st derivatives'
Case (2)
Write (nout,*)
Write (nout,*) 'Probable error in calculation of the matrix B'
End Select

Write (nout,*)
Write (nout,*) 'At the test point, LSQFUN gives'
Write (nout,*)
Write (nout,*) '      Residuals                   1st derivatives'
Write (nout,99998)(fvec(i),fjac(i,1:n),i=1,m)
Write (nout,*)
Write (nout,*) 'and LSQHES gives the lower triangle of the matrix B'
Write (nout,*)

k = 1

Do i = 1, n
Write (nout,99998) b(k:(k+i-1))
k = k + i
End Do

End If

99999 Format (1X,4F10.5)
99998 Format (1X,1P,4E15.3)
End Program e04ybfe
```