NAG Library Manual, Mark 28.5
```    Program f08nhfe

!     F08NHF Example Program Text

!     Mark 28.5 Release. NAG Copyright 2022.

!     .. Use Statements ..
Use nag_library, Only: blas_damax_val, dgebak, dgebal, dgehrd, dhseqr,   &
dnrm2, dorghr, dtrevc, nag_wp, x04caf
!     .. Implicit None Statement ..
Implicit None
!     .. Parameters ..
Real (Kind=nag_wp), Parameter    :: zero = 0.0_nag_wp
Integer, Parameter               :: nin = 5, nout = 6
!     .. Local Scalars ..
Real (Kind=nag_wp)               :: r
Integer                          :: i, ifail, ihi, ilo, info, k, lda,    &
ldh, ldvl, ldvr, lwork, m, n
!     .. Local Arrays ..
Real (Kind=nag_wp), Allocatable  :: a(:,:), h(:,:), scale(:), tau(:),    &
vl(:,:), vr(:,:), wi(:), work(:),    &
wr(:)
Logical                          :: select(1)
!     .. Executable Statements ..
Write (nout,*) 'F08NHF Example Program Results'
!     Skip heading in data file
ldvl = 1
lda = n
ldh = n
ldvr = n
lwork = 64*n
Allocate (a(lda,n),h(ldh,n),scale(n),tau(n),vl(ldvl,1),vr(ldvr,n),wi(n), &
work(lwork),wr(n))

!     Read A from data file

!     Balance A
!     The NAG name equivalent of dgebal is f08nhf
Call dgebal('Both',n,a,lda,ilo,ihi,scale,info)

!     Reduce A to upper Hessenberg form H = (Q**T)*A*Q
!     The NAG name equivalent of dgehrd is f08nef
Call dgehrd(n,ilo,ihi,a,lda,tau,work,lwork,info)

!     Copy A to H and VR
h(1:n,1:n) = a(1:n,1:n)
vr(1:n,1:n) = a(1:n,1:n)

!     Form Q explicitly, storing the result in VR
!     The NAG name equivalent of dorghr is f08nff
Call dorghr(n,1,n,vr,ldvr,tau,work,lwork,info)

!     Calculate the eigenvalues and Schur factorization of A
!     The NAG name equivalent of dhseqr is f08pef
Call dhseqr('Schur form','Vectors',n,ilo,ihi,h,ldh,wr,wi,vr,ldvr,work,   &
lwork,info)

Write (nout,*)
If (info>0) Then
Write (nout,*) 'Failure to converge.'
Else
Write (nout,*) 'Eigenvalues'
Write (nout,99999)(' (',wr(i),',',wi(i),')',i=1,n)

!       Calculate the eigenvectors of A, storing the result in VR

!       The NAG name equivalent of dtrevc is f08qkf
Call dtrevc('Right','Backtransform',select,n,h,ldh,vl,ldvl,vr,ldvr,n,  &
m,work,info)

!       The NAG name equivalent of dgebak is f08njf
Call dgebak('Both','Right',n,ilo,ihi,scale,m,vr,ldvr,info)

!       Print eigenvectors

Write (nout,*)
Flush (nout)

!       Normalize the eigenvectors, largest positive
Do i = 1, m
Call blas_damax_val(n,vr(1,i),1,k,r)
If (vr(k,i)<zero) Then
vr(1:n,i) = -vr(1:n,i)
End If
r = dnrm2(n,vr(1,i),1)
vr(1:n,i) = vr(1:n,i)/r
End Do

!       ifail: behaviour on error exit
!              =0 for hard exit, =1 for quiet-soft, =-1 for noisy-soft
ifail = 0
Call x04caf('General',' ',n,m,vr,ldvr,'Contents of array VR',ifail)

End If

99999 Format (1X,A,F8.4,A,F8.4,A)
End Program f08nhfe
```