NAG Library Manual, Mark 27.3
```    Program f08sqfe

!     F08SQF Example Program Text

!     Mark 27.3 Release. NAG Copyright 2021.

!     .. Use Statements ..
Use nag_library, Only: ddisna, nag_wp, x02ajf, x04daf, zhegvd,           &
zlanhe => f06ucf, ztrcon
!     .. Implicit None Statement ..
Implicit None
!     .. Parameters ..
Real (Kind=nag_wp), Parameter    :: one = 1.0E+0_nag_wp
Integer, Parameter               :: nb = 64, nin = 5, nout = 6
!     .. Local Scalars ..
Complex (Kind=nag_wp)            :: scal
Real (Kind=nag_wp)               :: anorm, bnorm, eps, rcond, rcondb,    &
t1, t2
Integer                          :: i, ifail, info, k, lda, ldb, liwork, &
lrwork, lwork, n
!     .. Local Arrays ..
Complex (Kind=nag_wp), Allocatable :: a(:,:), b(:,:), work(:)
Complex (Kind=nag_wp)            :: cdum(1)
Real (Kind=nag_wp), Allocatable  :: eerbnd(:), rcondz(:), rwork(:),      &
w(:), zerbnd(:)
Real (Kind=nag_wp)               :: rdum(1)
Integer                          :: idum(1)
Integer, Allocatable             :: iwork(:)
!     .. Intrinsic Procedures ..
Intrinsic                        :: abs, conjg, max, maxloc, nint, real
!     .. Executable Statements ..
Write (nout,*) 'F08SQF Example Program Results'
Write (nout,*)
!     Skip heading in data file
lda = n
ldb = n
Allocate (a(lda,n),b(ldb,n),eerbnd(n),rcondz(n),w(n),zerbnd(n))

!     Use routine workspace query to get optimal workspace.
lwork = -1
liwork = -1
lrwork = -1
!     The NAG name equivalent of zhegvd is f08sqf
Call zhegvd(2,'Vectors','Upper',n,a,lda,b,ldb,w,cdum,lwork,rdum,lrwork,  &
idum,liwork,info)

!     Make sure that there is enough workspace for block size nb.
lwork = max((nb+2+n)*n,nint(real(cdum(1))))
lrwork = max(1+(5+2*n)*n,nint(rdum(1)))
liwork = max(3+5*n,idum(1))
Allocate (work(lwork),rwork(lrwork),iwork(liwork))

!     Read the upper triangular parts of the matrices A and B

!     Compute the one-norms of the symmetric matrices A and B

!     f06ucf is the NAG name equivalent of the LAPACK auxiliary zlanhe
anorm = zlanhe('One norm','Upper',n,a,lda,rwork)
bnorm = zlanhe('One norm','Upper',n,b,ldb,rwork)

!     Solve the generalized Hermitian eigenvalue problem
!     A*B*x = lambda*x (itype = 2)
!     The NAG name equivalent of zhegvd is f08sqf
Call zhegvd(2,'Vectors','Upper',n,a,lda,b,ldb,w,work,lwork,rwork,lrwork, &
iwork,liwork,info)

If (info==0) Then

!       Print solution

Write (nout,*) 'Eigenvalues'
Write (nout,99999) w(1:n)
Flush (nout)

!       Normalize the eigenvectors, largest element real
Do i = 1, n
rwork(1:n) = abs(a(1:n,i))
k = maxloc(rwork(1:n),1)
scal = conjg(a(k,i))/abs(a(k,i))
a(1:n,i) = a(1:n,i)*scal
End Do

!       ifail: behaviour on error exit
!              =0 for hard exit, =1 for quiet-soft, =-1 for noisy-soft
ifail = 0
Call x04daf('General',' ',n,n,a,lda,'Eigenvectors',ifail)

!       Call ZTRCON (F07TUF) to estimate the reciprocal condition
!       number of the Cholesky factor of B.  Note that:
!       cond(B) = 1/rcond**2
Call ztrcon('One norm','Upper','Non-unit',n,b,ldb,rcond,work,rwork,    &
info)

!       Print the reciprocal condition number of B

rcondb = rcond**2
Write (nout,*)
Write (nout,*) 'Estimate of reciprocal condition number for B'
Write (nout,99998) rcondb
Flush (nout)

!       Get the machine precision, eps, and if rcondb is not less
!       than eps**2, compute error estimates for the eigenvalues and
!       eigenvectors

eps = x02ajf()
If (rcond>=eps) Then

!         Call DDISNA (F08FLF) to estimate reciprocal condition
!         numbers for the eigenvectors of (A*B - lambda*I)

Call ddisna('Eigenvectors',n,n,w,rcondz,info)

!         Compute the error estimates for the eigenvalues and
!         eigenvectors

t1 = one/rcond
t2 = anorm*bnorm
Do i = 1, n
eerbnd(i) = (t2+abs(w(i))/rcondb)
zerbnd(i) = t1*(t2/rcondz(i)+t1)
End Do

!         Print the approximate error bounds for the eigenvalues
!         and vectors

Write (nout,*)
Write (nout,*) 'Error estimates (relative to machine precision)'
Write (nout,*) 'for the eigenvalues:'
Write (nout,99998) eerbnd(1:n)
Write (nout,*)
Write (nout,*) 'for the eigenvectors:'
Write (nout,99998) zerbnd(1:n)
Else
Write (nout,*)
Write (nout,*) 'B is very ill-conditioned, error ',                  &
'estimates have not been computed'
End If
Else If (info>n) Then
i = info - n
Write (nout,99997) 'The leading minor of order ', i,                   &
' of B is not positive definite'
Else
Write (nout,99996) 'Failure in ZHEGVD. INFO =', info
End If

99999 Format (3X,(6F11.4))
99998 Format (4X,1P,6E11.1)
99997 Format (1X,A,I4,A)
99996 Format (1X,A,I4)
End Program f08sqfe
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