```!   E04HEF Example Program Text
!   Mark 26.1 Release. NAG Copyright 2016.
Module e04hefe_mod

!     E04HEF 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, lsqgrd, lsqhes, lsqmon
!     .. Parameters ..
Integer, Parameter, Public       :: liw = 1, m = 15, n = 3, nin = 5,     &
nout = 6, nt = 3
Integer, Parameter, Public       :: lb = n*(n+1)/2
Integer, Parameter, Public       :: ldfjac = m
Integer, Parameter, Public       :: ldv = n
Integer, Parameter, Public       :: lw = 7*n + m*n + 2*m + n*n
!     .. Local Arrays ..
Real (Kind=nag_wp), Public, Save :: t(m,nt), y(m)
Contains
Subroutine lsqgrd(m,n,fvec,fjac,ldfjac,g)
!       Routine to evaluate gradient of the sum of squares

!       .. Use Statements ..
Use nag_library, Only: dgemv
!       .. Scalar Arguments ..
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)
!       .. Executable Statements ..
!       The NAG name equivalent of dgemv is f06paf
Call dgemv('T',m,n,1.0_nag_wp,fjac,ldfjac,fvec,1,0.0_nag_wp,g,1)

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

Return

End Subroutine lsqgrd
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) = 1.0_nag_wp
dummy = -1.0_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) = 0.0_nag_wp
b(2) = 0.0_nag_wp
sum22 = 0.0_nag_wp
sum32 = 0.0_nag_wp
sum33 = 0.0_nag_wp

Do i = 1, m
dummy = 2.0_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) = 0.0_nag_wp
b(5) = sum32
b(6) = sum33

Return

End Subroutine lsqhes
!       Monitoring routine

!       .. Use Statements ..
Use nag_library, Only: f06eaf
!       .. Parameters ..
Integer, Parameter             :: ndec = 3
!       .. Scalar Arguments ..
Integer, Intent (In)           :: igrade, ldfjac, liw, lw, 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) :: w(lw)
Integer, Intent (Inout)        :: iw(liw)
!       .. Local Scalars ..
Real (Kind=nag_wp)             :: fsumsq, gtg
Integer                        :: j
!       .. Local Arrays ..
Real (Kind=nag_wp)             :: g(ndec)
!       .. Executable Statements ..
fsumsq = f06eaf(m,fvec,1,fvec,1)

Call lsqgrd(m,n,fvec,fjac,ldfjac,g)

gtg = f06eaf(n,g,1,g,1)

Write (nout,*)
Write (nout,*)                                                         &
' Itns    F evals          SUMSQ             GTG        grade'
Write (nout,99999) niter, nf, fsumsq, gtg, igrade
Write (nout,*)
Write (nout,*)                                                         &
'       X                    G           Singular values'

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

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 e04hefe_mod
Program e04hefe

!     E04HEF Example Main Program

!     .. Use Statements ..
Use e04hefe_mod, Only: lb, ldfjac, ldv, liw, lsqfun, lsqgrd, lsqhes,     &
lsqmon, lw, m, n, nin, nout, nt, t, y
Use nag_library, Only: e04hef, e04yaf, e04ybf, 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
!     .. Local Arrays ..
Real (Kind=nag_wp)               :: b(lb), fjac(ldfjac,n), fvec(m),      &
g(n), s(n), v(ldv,n), w(lw), x(n)
Integer                          :: iw(liw)
!     .. Intrinsic Procedures ..
Intrinsic                        :: sqrt
!     .. Executable Statements ..
Write (nout,*) 'E04HEF Example Program Results'

!     Skip heading in data file

!     Observations of TJ (J = 1, 2, ..., nt) 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:nt) = (/0.19_nag_wp,-1.34_nag_wp,0.88_nag_wp/)

!     Check the 1st derivatives

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

!     Check the evaluation of B

ifail = 0
Call e04ybf(m,n,lsqfun,lsqhes,x,fvec,fjac,ldfjac,b,lb,iw,liw,w,lw,ifail)

!     Continue setting parameters for E04HEF

!     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

x(1:nt) = (/0.5_nag_wp,1.0_nag_wp,1.5_nag_wp/)

ifail = -1
Call e04hef(m,n,lsqfun,lsqhes,lsqmon,iprint,maxcal,eta,xtol,stepmx,x,    &
fsumsq,fvec,fjac,ldfjac,s,v,ldv,niter,nf,iw,liw,w,lw,ifail)

Select Case (ifail)
Case (0,2:)
Write (nout,*)
Write (nout,99999) 'On exit, the sum of squares is', fsumsq
Write (nout,99999) 'at the point', x(1:n)

Call lsqgrd(m,n,fvec,fjac,ldfjac,g)

Write (nout,99998) 'The corresponding gradient is', g(1:n)
Write (nout,*) '                           (machine dependent)'
Write (nout,*) 'and the residuals are'
Write (nout,99997) fvec(1:m)
End Select

99999 Format (1X,A,3F12.4)
99998 Format (1X,A,1P,3E12.3)
99997 Format (1X,1P,E9.1)
End Program e04hefe
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