Example description
    Program f08safe

!     F08SAF Example Program Text

!     Mark 27.0 Release. NAG Copyright 2019.

!     .. Use Statements ..
      Use nag_library, Only: blas_damax_val, ddisna, dsygv, dtrcon, f06rcf,    &
                             nag_wp, x02ajf, x04caf
!     .. Implicit None Statement ..
      Implicit None
!     .. Parameters ..
      Real (Kind=nag_wp), Parameter    :: zero = 0.0_nag_wp
      Integer, Parameter               :: nb = 64, nin = 5, nout = 6
!     .. Local Scalars ..
      Real (Kind=nag_wp)               :: anorm, bnorm, eps, r, rcond, rcondb, &
                                          t1, t2, t3
      Integer                          :: i, ifail, info, k, lda, ldb, lwork,  &
!     .. Local Arrays ..
      Real (Kind=nag_wp), Allocatable  :: a(:,:), b(:,:), eerbnd(:),           &
                                          rcondz(:), w(:), work(:), zerbnd(:)
      Real (Kind=nag_wp)               :: dummy(1)
      Integer, Allocatable             :: iwork(:)
!     .. Intrinsic Procedures ..
      Intrinsic                        :: abs, max, nint
!     .. Executable Statements ..
      Write (nout,*) 'F08SAF Example Program Results'
      Write (nout,*)
!     Skip heading in data file
      Read (nin,*)
      Read (nin,*) n
      lda = n
      ldb = n
      Allocate (a(lda,n),b(ldb,n),eerbnd(n),rcondz(n),w(n),zerbnd(n),iwork(n))

!     Use routine workspace query to get optimal workspace.
      lwork = -1
!     The NAG name equivalent of dsygv is f08saf
      Call dsygv(1,'Vectors','Upper',n,a,lda,b,ldb,w,dummy,lwork,info)

!     Make sure that there is enough workspace for block size nb.
      lwork = max((nb+2)*n,nint(dummy(1)))
      Allocate (work(lwork))

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

      Read (nin,*)(a(i,i:n),i=1,n)
      Read (nin,*)(b(i,i:n),i=1,n)

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

      anorm = f06rcf('One norm','Upper',n,a,lda,work)
      bnorm = f06rcf('One norm','Upper',n,b,ldb,work)

!     Solve the generalized symmetric eigenvalue problem
!     A*x = lambda*B*x (ITYPE = 1)

!     The NAG name equivalent of dsygv is f08saf
      Call dsygv(1,'Vectors','Upper',n,a,lda,b,ldb,w,work,lwork,info)

      If (info==0) Then

!       Print solution

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

        Write (nout,*)
        Flush (nout)

!       Normalize the eigenvectors, largest positive
        Do i = 1, n
          Call blas_damax_val(n,a(1,i),1,k,r)
          If (a(k,i)<zero) Then
            a(1:n,i) = -a(1:n,i)
          End If
        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,n,a,lda,'Eigenvectors',ifail)

!       Call DTRCON (F07TGF) to estimate the reciprocal condition
!       number of the Cholesky factor of B.  Note that:
!       cond(B) = 1/rcond**2

!       The NAG name equivalent of dtrcon is f07tgf
        Call dtrcon('One norm','Upper','Non-unit',n,b,ldb,rcond,work,iwork,    &

!       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 - lambda*B)

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

!         Compute the error estimates for the eigenvalues and
!         eigenvectors

          t1 = eps/rcondb
          t2 = anorm/bnorm
          t3 = t2/rcond
          Do i = 1, n
            eerbnd(i) = t1*(t2+abs(w(i)))
            zerbnd(i) = t1*(t3+abs(w(i)))/rcondz(i)
          End Do

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

          Write (nout,*)
          Write (nout,*) 'Error estimates for the eigenvalues'
          Write (nout,99998) eerbnd(1:n)
          Write (nout,*)
          Write (nout,*) 'Error estimates for the eigenvectors'
          Write (nout,99998) zerbnd(1:n)
          Write (nout,*)
          Write (nout,*) 'B is very ill-conditioned, error ',                  &
            'estimates have not been computed'
        End If
      Else If (info>n .And. info<=2*n) Then
        i = info - n
        Write (nout,99997) 'The leading minor of order ', i,                   &
          ' of B is not positive definite'
        Write (nout,99996) 'Failure in DSYGV. 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 f08safe