```!   F08YKF Example Program Text
!   Mark 26 Release. NAG Copyright 2016.

Module f08ykfe_mod

!     F08YKF 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                           :: normalize
!     .. Parameters ..
Real (Kind=nag_wp), Parameter, Public :: one = 1.0_nag_wp
Real (Kind=nag_wp), Parameter, Public :: zero = 0.0_nag_wp
Integer, Parameter, Public       :: nin = 5, nout = 6
Contains
Subroutine normalize(n,alphai,v,ldv)

!       .. Use Statements ..
Use nag_library, Only: dnrm2
!       .. Implicit None Statement ..
Implicit None
!       .. Scalar Arguments ..
Integer, Intent (In)           :: ldv, n
!       .. Array Arguments ..
Real (Kind=nag_wp), Intent (In) :: alphai(n)
Real (Kind=nag_wp), Intent (Inout) :: v(ldv,*)
!       .. Local Scalars ..
Real (Kind=nag_wp)             :: a, b, r, r1, r2, v1, v2
Integer                        :: i, j, k
!       .. Intrinsic Procedures ..
Intrinsic                      :: sqrt
!       .. Executable Statements ..

Do j = 1, n

If (alphai(j)>=0.0_nag_wp) Then
If (alphai(j)==0.0_nag_wp) Then
!             Real eigenvalue
!             The 2-norm of Q is calculated using dnrm2 (f06ejf).
r = dnrm2(n,v(1,j),1)
v(1:n,j) = v(1:n,j)/r
Else
!             Complex eigenvalue (positive imaginary part)
!             Make largest element real and positive
r1 = dnrm2(n,v(1,j),1)
r2 = dnrm2(n,v(1,j+1),1)
r1 = sqrt(r1**2+r2**2)
r2 = -1.0_nag_wp
Do i = 1, n
r = v(i,j)**2 + v(i,j+1)**2
If (r>r2) Then
r2 = r
k = i
End If
End Do
r = r1*sqrt(r2)
a = v(k,j)/r
b = v(k,j+1)/r
Do i = 1, n
v1 = v(i,j)
v2 = v(i,j+1)
v(i,j) = v1*a + v2*b
v(i,j+1) = v2*a - v1*b
End Do
End If
End If
End Do
End Subroutine normalize
End Module f08ykfe_mod
Program f08ykfe

!     F08YKF Example Main Program

!     .. Use Statements ..
Use nag_library, Only: dgeqrf, dggbak, dggbal, dgghrd, dhgeqz, dorgqr,   &
dormqr, dtgevc, f06qff, f06qhf, nag_wp, x04cbf
Use f08ykfe_mod, Only: nin, normalize, nout, one, zero
!     .. Implicit None Statement ..
Implicit None
!     .. Local Scalars ..
Integer                          :: i, icols, ifail, ihi, ilo, info,     &
irows, jwork, lda, ldb, ldvl, ldvr,  &
lwork, m, n
Logical                          :: ileft, iright
Character (1)                    :: compq, compz, howmny, job, side
!     .. Local Arrays ..
Real (Kind=nag_wp), Allocatable  :: a(:,:), alphai(:), alphar(:),        &
b(:,:), beta(:), lscale(:),          &
rscale(:), tau(:), vl(:,:), vr(:,:), &
work(:)
Logical, Allocatable             :: select(:)
Character (0)                    :: clabs(1), rlabs(1)
!     .. Intrinsic Procedures ..
Intrinsic                        :: nint
!     .. Executable Statements ..
Write (nout,*) 'F08YKF Example Program Results'
Flush (nout)

!     ileft  is TRUE if left  eigenvectors are required
!     iright is TRUE if right eigenvectors are required

ileft = .True.
iright = .True.

!     Skip heading in data file

lda = n
ldb = n
ldvl = n
ldvr = n
lwork = 6*n
Allocate (a(lda,n),alphai(n),alphar(n),b(ldb,n),beta(n),lscale(n),       &
rscale(n),tau(n),vl(ldvl,ldvl),vr(ldvr,ldvr),work(lwork),select(n))

!     READ matrix A from data file

!     READ matrix B from data file

!     Balance matrix pair (A,B)
job = 'B'
!     The NAG name equivalent of dggbal is f08whf
Call dggbal(job,n,a,lda,b,ldb,ilo,ihi,lscale,rscale,work,info)

!     Matrix A after balancing
!     ifail: behaviour on error exit
!             =0 for hard exit, =1 for quiet-soft, =-1 for noisy-soft
ifail = 0
Call x04cbf('General',' ',n,n,a,lda,'F8.4','Matrix A after balancing',   &
'I',rlabs,'I',clabs,80,0,ifail)
Write (nout,*)
Flush (nout)

!     Matrix B after balancing
ifail = 0
Call x04cbf('General',' ',n,n,b,ldb,'F8.4','Matrix B after balancing',   &
'I',rlabs,'I',clabs,80,0,ifail)
Write (nout,*)
Flush (nout)

!     Reduce B to triangular form using QR
irows = ihi + 1 - ilo
icols = n + 1 - ilo
!     The NAG name equivalent of dgeqrf is f08aef
Call dgeqrf(irows,icols,b(ilo,ilo),ldb,tau,work,lwork,info)

!     Apply the orthogonal transformation to matrix A
!     The NAG name equivalent of dormqr is f08agf
Call dormqr('L','T',irows,icols,irows,b(ilo,ilo),ldb,tau,a(ilo,ilo),lda, &
work,lwork,info)

!     Initialize VL (if left eigenvectors are required)
If (ileft) Then

Call f06qhf('General',n,n,zero,one,vl,ldvl)
Call f06qff('Lower',irows-1,irows-1,b(ilo+1,ilo),ldb,vl(ilo+1,ilo),    &
ldvl)

!       The NAG name equivalent of dorgqr is f08aff
Call dorgqr(irows,irows,irows,vl(ilo,ilo),ldvl,tau,work,lwork,info)
End If

!     Initialize VR (if right eigenvectors are required)
If (iright) Then
Call f06qhf('General',n,n,zero,one,vr,ldvr)
End If

!     Compute the generalized Hessenberg form of (A,B)
compq = 'V'
compz = 'V'
!     The NAG name equivalent of dgghrd is f08wef
Call dgghrd(compq,compz,n,ilo,ihi,a,lda,b,ldb,vl,ldvl,vr,ldvr,info)

!     Matrix A in generalized Hessenberg form
ifail = 0
Call x04cbf('General',' ',n,n,a,lda,'F8.4','Matrix A in Hessenberg form' &
,'I',rlabs,'I',clabs,80,0,ifail)
Write (nout,*)
Flush (nout)

!     Matrix B in generalized Hessenberg form
ifail = 0
Call x04cbf('General',' ',n,n,b,ldb,'F8.4','Matrix B in Hessenberg form' &
,'I',rlabs,'I',clabs,80,0,ifail)

!     Routine DHGEQZ
!     Workspace query: jwork = -1

jwork = -1
job = 'S'

!     The NAG name equivalent of dhgeqz is f08xef
Call dhgeqz(job,compq,compz,n,ilo,ihi,a,lda,b,ldb,alphar,alphai,beta,vl, &
ldvl,vr,ldvr,work,jwork,info)

Write (nout,*)
Write (nout,99999) nint(work(1))
Write (nout,99998) lwork
Write (nout,*)
Write (nout,99997)
Write (nout,99996)

!     Compute the generalized Schur form
!     if the workspace lwork is adequate
!     The Schur form also gives parameters
!     required to compute generalized eigenvalues

If (nint(work(1))<=lwork) Then

!       The NAG name equivalent of dhgeqz is f08xef
Call dhgeqz(job,compq,compz,n,ilo,ihi,a,lda,b,ldb,alphar,alphai,beta,  &
vl,ldvl,vr,ldvr,work,lwork,info)

!       Print the generalized eigenvalues

Do i = 1, n
If (beta(i)/=0.0E0_nag_wp) Then
Write (nout,99995) i, '(', alphar(i)/beta(i), ',',                 &
alphai(i)/beta(i), ')'
Else
Write (nout,99996) i
End If
End Do
Write (nout,*)
Flush (nout)

!       Compute left and right generalized eigenvectors
!       of the balanced matrix

howmny = 'B'
If (ileft .And. iright) Then
side = 'B'
Else If (ileft) Then
side = 'L'
Else If (iright) Then
side = 'R'
End If

!       The NAG name equivalent of dtgevc is f08ykf
Call dtgevc(side,howmny,select,n,a,lda,b,ldb,vl,ldvl,vr,ldvr,n,m,work, &
info)

If (iright) Then

!         Compute right eigenvectors of the original matrix

job = 'B'
side = 'R'

!         The NAG name equivalent of dggbak is f08wjf
Call dggbak(job,side,n,ilo,ihi,lscale,rscale,n,vr,ldvr,info)

Call normalize(n,alphai,vr,ldvr)
!         Print the right eigenvectors

ifail = 0
Call x04cbf('General',' ',n,n,vr,ldvr,'F8.4','Right eigenvectors',   &
'I',rlabs,'I',clabs,80,0,ifail)

Write (nout,*)
Flush (nout)
End If

!       Compute left eigenvectors of the original matrix

If (ileft) Then
job = 'B'
side = 'L'

!         The NAG name equivalent of dggbak is f08wjf
Call dggbak(job,side,n,ilo,ihi,lscale,rscale,n,vl,ldvl,info)

Call normalize(n,alphai,vl,ldvl)
!         Print the left eigenvectors

ifail = 0
Call x04cbf('General',' ',n,n,vl,ldvl,'F8.4','Left eigenvectors',    &
'I',rlabs,'I',clabs,80,0,ifail)

End If
Else
Write (nout,99994)
End If

99999 Format (1X,'Minimal required LWORK = ',I6)
99998 Format (1X,'Actual value of  LWORK = ',I6)
99997 Format (1X,'Generalized eigenvalues')
99996 Format (1X,I4,5X,'Infinite eigenvalue')
99995 Format (1X,I4,5X,A,F7.3,A,F7.3,A)
99994 Format (1X,'Insufficient workspace for array WORK',/,' in F08XEF/',      &
'DHGEQZ')
End Program f08ykfe
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