! F08PBF Example Program Text ! Mark 24 Release. NAG Copyright 2012. Module f08pbfe_mod ! F08PBF Example Program Module: ! Parameters and User-defined Routines ! .. Use Statements .. Use nag_library, Only: nag_wp ! .. Implicit None Statement .. Implicit None ! .. Parameters .. Integer, Parameter :: nb = 64, nin = 5, nout = 6 Contains Function select(wr,wi) ! Logical function select for use with DGEESX (F08PBF) ! Returns the value .TRUE. if the eigenvalue is real ! .. Function Return Value .. Logical :: select ! .. Scalar Arguments .. Real (Kind=nag_wp), Intent (In) :: wi, wr ! .. Local Scalars .. Logical :: d ! .. Executable Statements .. If (wi==0.0_nag_wp) Then d = .True. Else d = .False. End If select = d Return End Function select End Module f08pbfe_mod Program f08pbfe ! F08PBF Example Main Program ! .. Use Statements .. Use nag_library, Only: dgeesx, dgemm, dlange => f06raf, nag_wp, x02ajf, & x04caf Use f08pbfe_mod, Only: nb, nin, nout, select ! .. Implicit None Statement .. Implicit None ! .. Local Scalars .. Real (Kind=nag_wp) :: alpha, anorm, beta, eps, norm, & rconde, rcondv, tol Integer :: i, ifail, info, lda, ldc, ldd, & ldvs, liwork, lwork, n, sdim ! .. Local Arrays .. Real (Kind=nag_wp), Allocatable :: a(:,:), c(:,:), d(:,:), vs(:,:), & wi(:), work(:), wr(:) Real (Kind=nag_wp) :: dummy(1) Integer :: idum(1) Integer, Allocatable :: iwork(:) Logical, Allocatable :: bwork(:) ! .. Intrinsic Procedures .. Intrinsic :: max, nint ! .. Executable Statements .. Write (nout,*) 'F08PBF Example Program Results' Write (nout,*) Flush (nout) ! Skip heading in data file Read (nin,*) Read (nin,*) n lda = n ldc = n ldd = n ldvs = n Allocate (a(lda,n),c(ldc,n),d(ldd,n),vs(ldvs,n),wi(n),wr(n),bwork(n)) ! Use routine workspace query to get optimal workspace. lwork = -1 liwork = -1 ! The NAG name equivalent of dgeesx is f08pbf Call dgeesx('Vectors (Schur)','Sort',select, & 'Both reciprocal condition numbers',n,a,lda,sdim,wr,wi,vs,ldvs,rconde, & rcondv,dummy,lwork,idum,liwork,bwork,info) ! Make sure that there is enough workspace for blocksize nb. liwork = max((n*n)/4,idum(1)) lwork = max(n*(nb+2+n/2),nint(dummy(1))) Allocate (work(lwork),iwork(liwork)) ! Read in the matrix A Read (nin,*)(a(i,1:n),i=1,n) ! Copy A into D d(1:n,1:n) = a(1:n,1:n) ! Print Matrix A ! 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,'Matrix A',ifail) Write (nout,*) Flush (nout) ! Find the Frobenius norms of A ! The NAG name equivalent of the LAPACK auxiliary dlange is f06raf anorm = dlange('Frobenius',n,n,a,lda,work) ! Find the Schur factorization of A ! The NAG name equivalent of dgeesx is f08pbf Call dgeesx('Vectors (Schur)','Sort',select, & 'Both reciprocal condition numbers',n,a,lda,sdim,wr,wi,vs,ldvs,rconde, & rcondv,work,lwork,iwork,liwork,bwork,info) If (info==0 .Or. info==(n+2)) Then ! Compute A - Z*T*Z^T from the factorization of A and store in matrix D ! The NAG name equivelent of dgemm is f06yaf alpha = 1.0_nag_wp beta = 0.0_nag_wp Call dgemm('N','N',n,n,n,alpha,vs,ldvs,a,lda,beta,c,ldc) alpha = -1.0_nag_wp beta = 1.0_nag_wp Call dgemm('N','T',n,n,n,alpha,c,ldc,vs,ldvs,beta,d,ldd) ! Find norm of matrix D and print warning if it is too large norm = dlange('O',ldd,n,d,ldd,work) If (norm>x02ajf()**0.8_nag_wp) Then Write (nout,*) 'Norm of A-(Z*T*Z^T) is much greater than 0.' Write (nout,*) 'Schur factorization has failed.' Else ! Print solution Write (nout,99999) & 'Number of eigenvalues for which SELECT is true = ', sdim, & '(dimension of invariant subspace)' Write (nout,*) ! Print eigenvalues. Write (nout,*) 'Selected eigenvalues' Write (nout,99998)(' (',wr(i),',',wi(i),')',i=1,sdim) Write (nout,*) If (info==(n+2)) Then Write (nout,99997) '***Note that rounding errors mean ', & 'that leading eigenvalues in the Schur form', & 'no longer satisfy SELECT = .TRUE.' Write (nout,*) End If Flush (nout) ! Print out the reciprocal condition numbers Write (nout,99996) & 'Reciprocal of projection norm onto the invariant', & 'subspace for the selected eigenvalues', 'RCONDE = ', rconde Write (nout,*) Write (nout,99995) & 'Reciprocal condition number for the invariant subspace', & 'RCONDV = ', rcondv ! Compute the machine precision eps = x02ajf() tol = eps*anorm ! Print out the approximate asymptotic error bound on the ! average absolute error of the selected eigenvalues given by ! eps*norm(A)/RCONDE Write (nout,*) Write (nout,99994) & 'Approximate asymptotic error bound for selected ', & 'eigenvalues = ', tol/rconde ! Print out an approximate asymptotic bound on the maximum ! angular error in the computed invariant subspace given by ! eps*norm(A)/RCONDV Write (nout,99994) & 'Approximate asymptotic error bound for the invariant ', & 'subspace = ', tol/rcondv End If Else Write (nout,99993) 'Failure in DGEESX. INFO =', info End If 99999 Format (1X,A,I4/1X,A) 99998 Format (1X,A,F8.4,A,F8.4,A) 99997 Format (1X,2A/1X,A) 99996 Format (1X,A/1X,A/1X,A,1P,E8.1) 99995 Format (1X,A/1X,A,1P,E8.1) 99994 Format (1X,2A,1P,E8.1) 99993 Format (1X,A,I4) End Program f08pbfe