NAG Library Manual, Mark 27.2
```!   E04GB_P0W_F Example Program Text
!   Mark 27.2 Release. NAG Copyright 2021.
Module e04gb_p0w_fe_mod

!     E04GB_P0W_F Example Program Module:
!            Parameters and User-defined Routines

!     .. Use Statements ..
Use iso_c_binding, Only: c_ptr
Use nag_library, Only: ddot, dgemv, nag_wp
!     .. Implicit None Statement ..
Implicit None
!     .. Accessibility Statements ..
Private
Public                           :: lsqfun, lsqgrd, lsqmon
!     .. Parameters ..
Integer, Parameter               :: inc1 = 1
Integer, Parameter, Public       :: m = 15, n = 3, nin = 5, nout = 6,    &
nt = 3
Integer, Parameter, Public       :: ldfjac = m
Integer, Parameter, Public       :: ldv = n
!     .. Local Arrays ..
Real (Kind=nag_wp), Public, Save :: t(m,nt), y(m)
Contains
!       Routine to evaluate gradient of the sum of squares

!       .. Scalar Arguments ..
Type (c_ptr), Intent (Inout)   :: ad_handle
Integer, Intent (In)           :: ldfjac, m, n
!       .. Array Arguments ..
Real (Kind=nag_wp), Intent (In) :: fjac(ldfjac,n), fvec(m)
Real (Kind=nag_wp), Intent (Out) :: g(n)
!       .. Local Scalars ..
Real (Kind=nag_wp)             :: alpha, beta
Character (1), Save            :: trans = 'T'
!       .. Executable Statements ..
alpha = 1.0_nag_wp
beta = 0.0_nag_wp
Call dgemv(trans,m,n,alpha,fjac,ldfjac,fvec,inc1,beta,g,inc1)

g(1:n) = 2.0_nag_wp*g(1:n)

Return

End Subroutine lsqgrd

!       Routine to evaluate the residuals
!       Always use algorithmic differentiation here for simplicity

!       .. Scalar Arguments ..
Type (c_ptr), Intent (Inout)   :: ad_handle
Integer, Intent (Inout)        :: iflag
Integer, Intent (In)           :: ldfjac, m, n
!       .. Array Arguments ..
Real (Kind=nag_wp), Intent (Inout) :: fjac(ldfjac,n), ruser(*)
Real (Kind=nag_wp), Intent (Out) :: fvec(m)
Real (Kind=nag_wp), Intent (In) :: xc(n)
Integer, Intent (Inout)        :: iuser(*)
!       .. Local Scalars ..
Real (Kind=nag_wp)             :: denom, denom2
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)
If (iflag/=0) Then
fjac(i,1) = 1.0_nag_wp
denom2 = -1.0_nag_wp/(denom*denom)
fjac(i,2) = t(i,1)*t(i,2)*denom2
fjac(i,3) = t(i,1)*t(i,3)*denom2
End If
End Do
Return

End Subroutine lsqfun
iuser,ruser)
!       Monitoring routine

!       .. Parameters ..
Integer, Parameter             :: ndec = 3
!       .. Scalar Arguments ..
Type (c_ptr), Intent (Inout)   :: ad_handle
Integer, Intent (In)           :: igrade, ldfjac, m, n, nf, niter
!       .. Array Arguments ..
Real (Kind=nag_wp), Intent (In) :: fjac(ldfjac,n), fvec(m), s(n),      &
xc(n)
Real (Kind=nag_wp), Intent (Inout) :: ruser(*)
Integer, Intent (Inout)        :: iuser(*)
!       .. Local Scalars ..
Real (Kind=nag_wp)             :: fsumsq, gtg
Integer                        :: j
!       .. Local Arrays ..
Real (Kind=nag_wp)             :: g(ndec)
!       .. Executable Statements ..
fsumsq = ddot(m,fvec,inc1,fvec,inc1)

gtg = ddot(n,g,inc1,g,inc1)

Write (nout,*)
Write (nout,*)                                                         &
'  Itn      F evals        SUMSQ             GTG        Grade'
Write (nout,99999) niter, nf, fsumsq, gtg, igrade
Write (nout,*)
Write (nout,*)                                                         &
'       x                    g           Singular values'

Write (nout,99998)(xc(j),g(j),s(j),j=1,n)

Return

99999   Format (1X,I4,6X,I5,6X,1P,E13.5,6X,1P,E9.1,6X,I3)
99998   Format (1X,1P,E13.5,10X,1P,E9.1,10X,1P,E9.1)
End Subroutine lsqmon
End Module e04gb_p0w_fe_mod
Program e04gb_p0w_fe

!     E04GB_P0W_F Example Main Program

!     .. Use Statements ..
Use e04gb_p0w_fe_mod, Only: ldfjac, ldv, lsqfun, lsqgrd, lsqmon, m, n,   &
nin, nout, nt, t, y
Use iso_c_binding, Only: c_ptr
Use nag_library, Only: nag_wp, x02ajf
!     .. Implicit None Statement ..
Implicit None
!     .. Local Scalars ..
Real (Kind=nag_wp)               :: eta, fsumsq, stepmx, xtol
Integer                          :: i, ifail, iprint, maxcal, nf, niter, &
selct
!     .. Local Arrays ..
Real (Kind=nag_wp)               :: fjac(m,n), fvec(m), g(n), ruser(1),  &
s(n), v(ldv,n), x(n)
Integer                          :: iuser(1)
!     .. Intrinsic Procedures ..
Intrinsic                        :: real, sqrt
!     .. Executable Statements ..
Write (nout,*) 'E04GB_P0W_F Example Program Results'

!     Skip heading in data file

!     Observations of T_j (j = 1:nt) are held in T(1:m, j)

Do i = 1, m
End Do

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

maxcal = 50*n
eta = 0.9_nag_wp
xtol = 10.0_nag_wp*sqrt(x02ajf())

!     We estimate that the minimum will be within 10 units of the
!     starting point
stepmx = 10.0_nag_wp

!     Set up the starting point
Do i = 1, n
x(i) = 0.5_nag_wp*real(i,kind=nag_wp)
End Do

!     Solve the problem
selct = 2
ifail = -1
xtol,stepmx,x,fsumsq,fvec,fjac,ldfjac,s,v,ldv,niter,nf,iuser,ruser,    &
ifail)

Select Case (ifail)
Case (0,2:)
Write (nout,*)
Write (nout,99999) 'Sum of squares = ', fsumsq
Write (nout,99999) 'Solution point = ', x(1:n)

Write (nout,99998) 'Estim gradien  = ', g(1:n)
Write (nout,*) '                    (machine dependent)'
Write (nout,*) 'Residuals:'
Write (nout,99997) fvec(1:m)
Case (:-1)
Write (nout,*)
Write (nout,99996) 'Routine e04gb_p0w_f failed with ifail = ', ifail
End Select

99999 Format (1X,A,3F12.4)
99998 Format (1X,A,1P,3E12.3)
99997 Format (3(1X,1P,E11.1))
99996 Format (1X,A,I5)

End Program e04gb_p0w_fe
```