```!   D02BGF Example Program Text
!   Mark 26 Release. NAG Copyright 2016.

Module d02bgfe_mod

!     D02BGF 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                           :: fcn
!     .. Parameters ..
Integer, Parameter, Public       :: n = 3, nin = 5, nout = 6
!     n: number of differential equations
Contains
Subroutine fcn(x,y,f)

!       .. Scalar Arguments ..
Real (Kind=nag_wp), Intent (In) :: x
!       .. Array Arguments ..
Real (Kind=nag_wp), Intent (Out) :: f(*)
Real (Kind=nag_wp), Intent (In) :: y(*)
!       .. Intrinsic Procedures ..
Intrinsic                      :: cos, tan
!       .. Executable Statements ..
f(1) = tan(y(3))
f(2) = -0.032E0_nag_wp*tan(y(3))/y(2) - 0.02E0_nag_wp*y(2)/cos(y(3))
f(3) = -0.032E0_nag_wp/y(2)**2
Return
End Subroutine fcn
End Module d02bgfe_mod

Program d02bgfe

!     D02BGF Example Main Program

!     .. Use Statements ..
Use nag_library, Only: d02bgf, nag_wp
Use d02bgfe_mod, Only: fcn, n, nin, nout
!     .. Implicit None Statement ..
Implicit None
!     .. Local Scalars ..
Real (Kind=nag_wp)               :: alpha, hmax, tol, val, x, xend,      &
xinit
Integer                          :: i, ifail, m
!     .. Local Arrays ..
Real (Kind=nag_wp), Allocatable  :: w(:,:), y(:), yinit(:)
!     .. Executable Statements ..
Write (nout,*) 'D02BGF Example Program Results'
!     Skip heading in data file
!     m: index of mode of solution to attain value alpha
Allocate (w(n,10),y(n),yinit(n))
!     xinit: initial x value,     xend : final x value.
!     alpha: attain y(m) = alpha, yinit: initial solution values.
hmax = 0.0E0_nag_wp
val = alpha
Do i = 4, 5
tol = 10.0E0_nag_wp**(-i)
x = xinit
y(1:n) = yinit(1:n)

!       ifail: behaviour on error exit
!              =0 for hard exit, =1 for quiet-soft, =-1 for noisy-soft
ifail = 0
Call d02bgf(x,xend,n,y,tol,hmax,m,val,fcn,w,ifail)

Write (nout,*)
Write (nout,99999) 'Calculation with TOL =', tol
Write (nout,99998) ' Y(M) changes sign at X = ', x
If (tol<0.0E0_nag_wp) Then
Write (nout,*) ' Over one-third steps controlled by HMAX'
End If
End Do

99999 Format (1X,A,E8.1)
99998 Format (1X,A,F7.4)
End Program d02bgfe
```