概要
本サンプルはFortran言語によりLAPACKルーチンZGGESXを利用するサンプルプログラムです。
行列対
の一般化Schur分解を行います。
及び
ここで一般化Schur形式
の左上対角要素に の固有値の 
が対応するようにします。選択された固有値群の条件数の推定値と対応する不変部分空間もあわせて戻されます。
入力データ
（本ルーチンの詳細はZGGESX のマニュアルページを参照）| このデータをダウンロード |
ZGGESX Example Program Data 4 : Value of N (-21.10,-22.50) ( 53.50,-50.50) (-34.50,127.50) ( 7.50, 0.50) ( -0.46, -7.78) ( -3.50,-37.50) (-15.50, 58.50) (-10.50, -1.50) ( 4.30, -5.50) ( 39.70,-17.10) (-68.50, 12.50) ( -7.50, -3.50) ( 5.50, 4.40) ( 14.40, 43.30) (-32.50,-46.00) (-19.00,-32.50) : End of A ( 1.00, -5.00) ( 1.60, 1.20) ( -3.00, 0.00) ( 0.00, -1.00) ( 0.80, -0.60) ( 3.00, -5.00) ( -4.00, 3.00) ( -2.40, -3.20) ( 1.00, 0.00) ( 2.40, 1.80) ( -4.00, -5.00) ( 0.00, -3.00) ( 0.00, 1.00) ( -1.80, 2.40) ( 0.00, -4.00) ( 4.00, -5.00) : End of B
出力結果
（本ルーチンの詳細はZGGESX のマニュアルページを参照）| この出力例をダウンロード |
ZGGESX Example Program Results Number of eigenvalues for which SELCTG is true = 2 (dimension of deflating subspaces) Selected generalized eigenvalues 1 ( 2.00, -5.00) 2 ( 3.00, -1.00)
ソースコード
（本ルーチンの詳細はZGGESX のマニュアルページを参照）※本サンプルソースコードのご利用手順は「サンプルのコンパイル及び実行方法」をご参照下さい。
| このソースコードをダウンロード |
! ZGGESX Example Program Text
! Copyright 2017, Numerical Algorithms Group Ltd. http://www.nag.com
Module zggesx_mod
! ZGGESX Example Program Module:
! Parameters and User-defined Routines
! .. Use Statements ..
Use lapack_precision, Only: dp
! .. Implicit None Statement ..
Implicit None
! .. Accessibility Statements ..
Private
Public :: selctg
! .. Parameters ..
Integer, Parameter, Public :: nb = 64, nin = 5, nout = 6
Logical, Parameter, Public :: chkfac = .False., prcond = .False., &
prmat = .False.
Contains
Function selctg(a, b)
! Logical function selctg for use with ZGGESX (ZGGESX)
! Returns the value .TRUE. if the absolute value of the eigenvalue
! a/b < 6.0
! .. Function Return Value ..
Logical :: selctg
! .. Scalar Arguments ..
Complex (Kind=dp), Intent (In) :: a, b
! .. Intrinsic Procedures ..
Intrinsic :: abs
! .. Executable Statements ..
selctg = (abs(a)<6.0_dp*abs(b))
Return
End Function
End Module
Program zggesx_example
! ZGGESX Example Main Program
! .. Use Statements ..
Use blas_interfaces, Only: zgemm
Use zggesx_mod, Only: chkfac, nb, nin, nout, prcond, prmat, selctg
Use lapack_example_aux, Only: nagf_sort_realvec_rank, nagf_blas_dpyth, &
nagf_file_print_matrix_complex_gen_comp, &
nagf_sort_cmplxvec_rank_rearrange
Use lapack_interfaces, Only: zggesx, zlange
Use lapack_precision, Only: dp
! .. Implicit None Statement ..
Implicit None
! .. Local Scalars ..
Complex (Kind=dp) :: alph, bet
Real (Kind=dp) :: abnorm, anorm, bnorm, eps, normd, norme, tol
Integer :: i, ifail, info, lda, ldb, ldc, ldd, lde, ldvsl, ldvsr, &
liwork, lwork, n, sdim
Logical :: factor
! .. Local Arrays ..
Complex (Kind=dp), Allocatable :: a(:, :), alpha(:), b(:, :), beta(:), &
c(:, :), d(:, :), e(:, :), vsl(:, :), vsr(:, :), work(:)
Complex (Kind=dp) :: dummy(1)
Real (Kind=dp) :: rconde(2), rcondv(2)
Real (Kind=dp), Allocatable :: rwork(:)
Integer :: idum(1)
Integer, Allocatable :: iwork(:)
Logical, Allocatable :: bwork(:)
Character (1) :: clabs(1), rlabs(1)
! .. Intrinsic Procedures ..
Intrinsic :: abs, cmplx, epsilon, max, nint, real
! .. Executable Statements ..
Write (nout, *) 'ZGGESX Example Program Results'
Write (nout, *)
Flush (nout)
! Skip heading in data file
Read (nin, *)
Read (nin, *) n
lda = n
ldb = n
ldc = n
ldd = n
lde = n
ldvsl = n
ldvsr = n
Allocate (a(lda,n), alpha(n), b(ldb,n), beta(n), c(ldc,n), d(ldd,n), &
e(lde,n), vsl(ldvsl,n), vsr(ldvsr,n), rwork(8*n), bwork(n))
! Use routine workspace query to get optimal workspace.
lwork = -1
liwork = -1
Call zggesx('Vectors (left)', 'Vectors (right)', 'Sort', selctg, &
'Both reciprocal condition numbers', n, a, lda, b, ldb, sdim, alpha, &
beta, vsl, ldvsl, vsr, ldvsr, rconde, rcondv, dummy, lwork, rwork, &
idum, liwork, bwork, info)
! Make sure that there is enough workspace for block size nb.
lwork = max(n*nb+n*n/2, nint(real(dummy(1))))
liwork = max(n+2, idum(1))
Allocate (work(lwork), iwork(liwork))
! Read in the matrices A and B
Read (nin, *)(a(i,1:n), i=1, n)
Read (nin, *)(b(i,1:n), i=1, n)
If (chkfac) Then
! Copy A and B into D and E respectively
d(1:n, 1:n) = a(1:n, 1:n)
e(1:n, 1:n) = b(1:n, 1:n)
End If
! Find the Frobenius norms of A and B
anorm = zlange('Frobenius', n, n, a, lda, rwork)
bnorm = zlange('Frobenius', n, n, b, ldb, rwork)
If (prmat) Then
! Print matrices A and B
! ifail: behaviour on error exit
! =0 for hard exit, =1 for quiet-soft, =-1 for noisy-soft
ifail = 0
Call nagf_file_print_matrix_complex_gen_comp('General', ' ', n, n, a, &
lda, 'Bracketed', 'F8.4', 'Matrix A', 'Integer', rlabs, 'Integer', &
clabs, 80, 0, ifail)
Write (nout, *)
Flush (nout)
ifail = 0
Call nagf_file_print_matrix_complex_gen_comp('General', ' ', n, n, b, &
ldb, 'Bracketed', 'F8.4', 'Matrix B', 'Integer', rlabs, 'Integer', &
clabs, 80, 0, ifail)
Write (nout, *)
Flush (nout)
End If
factor = .True.
! Find the generalized Schur form
Call zggesx('Vectors (left)', 'Vectors (right)', 'Sort', selctg, &
'Both reciprocal condition numbers', n, a, lda, b, ldb, sdim, alpha, &
beta, vsl, ldvsl, vsr, ldvsr, rconde, rcondv, work, lwork, rwork, &
iwork, liwork, bwork, info)
If (info/=0 .And. info/=(n+2)) Then
Write (nout, 100) 'Failure in ZGGESX. INFO =', info
factor = .False.
Else If (chkfac) Then
! Compute A - Q*S*Z^H from the factorization of (A,B) and store in
! matrix D
alph = cmplx(1, kind=dp)
bet = cmplx(0, kind=dp)
Call zgemm('N', 'N', n, n, n, alph, vsl, ldvsl, a, lda, bet, c, ldc)
alph = cmplx(-1, kind=dp)
bet = cmplx(1, kind=dp)
Call zgemm('N', 'C', n, n, n, alph, c, ldc, vsr, ldvsr, bet, d, ldd)
! Compute B - Q*T*Z^H from the factorization of (A,B) and store in
! matrix E
alph = cmplx(1, kind=dp)
bet = cmplx(0, kind=dp)
Call zgemm('N', 'N', n, n, n, alph, vsl, ldvsl, b, ldb, bet, c, ldc)
alph = cmplx(-1, kind=dp)
bet = cmplx(1, kind=dp)
Call zgemm('N', 'C', n, n, n, alph, c, ldc, vsr, ldvsr, bet, e, lde)
! Find norms of matrices D and E and warn if either is too large
normd = zlange('O', ldd, n, d, ldd, rwork)
If (normd>epsilon(1.0E0_dp)**0.75_dp) Then
Write (nout, *) 'Norm of A-(Q*S*Z^T) is much greater than 0.'
factor = .False.
Write (nout, *) 'Schur factorization has failed.'
End If
norme = zlange('O', lde, n, e, lde, rwork)
If (norme>epsilon(1.0E0_dp)**0.75_dp) Then
Write (nout, *) 'Norm of B-(Q*T*Z^T) is much greater than 0.'
factor = .False.
End If
End If
If (factor) Then
! Print eigenvalue details
Write (nout, 100) 'Number of eigenvalues for which SELCTG is true = ', &
sdim, '(dimension of deflating subspaces)'
Write (nout, *)
! Print selected (finite) generalized eigenvalues
Write (nout, *) 'Selected generalized eigenvalues'
! Store absolute values of eigenvalues for ranking
work(1:n) = alpha(1:n)/beta(1:n)
rwork(1:n) = abs(work(1:n))
! Rank eigenvalues
ifail = 0
Call nagf_sort_realvec_rank(rwork, 1, sdim, 'Descending', iwork, &
ifail)
! Sort eigenvalues in work(1:n)
Call nagf_sort_cmplxvec_rank_rearrange(work, 1, sdim, iwork, ifail)
Do i = 1, sdim
Write (nout, 110) i, work(i)
End Do
If (info==(n+2)) Then
Write (nout, 120) '*** Note that rounding errors mean ', &
'that leading eigenvalues in the', &
'generalized Schur form no longer satisfy SELCTG = .TRUE.'
Write (nout, *)
End If
Flush (nout)
If (prcond) Then
! Compute the machine precision and sqrt(anorm**2+bnorm**2)
eps = epsilon(1.0E0_dp)
abnorm = nagf_blas_dpyth(anorm, bnorm)
tol = eps*abnorm
! Print out the reciprocal condition numbers and error bound for
! selected eigenvalues
Write (nout, *)
Write (nout, 130) &
'Reciprocal condition numbers for the average of the', &
'selected eigenvalues and their asymptotic error bound', &
'rcond-left = ', rconde(1), ', rcond-right = ', rconde(2), &
', error = ', tol/rconde(1)
Write (nout, *)
Write (nout, 130) &
'Reciprocal condition numbers for the deflating subspaces', &
'and their approximate asymptotic error bound', 'rcond-left = ', &
rcondv(1), ', rcond-right = ', rcondv(2), ', error = ', &
tol/rcondv(2)
End If
Else
Write (nout, *) 'Schur factorization has failed.'
End If
100 Format (1X, A, I4, /, 1X, A)
110 Format (1X, I2, 1X, '(', F6.2, ',', F6.2, ')')
120 Format (1X, 2A, /, 1X, A)
130 Format (1X, A, /, 1X, A, /, 1X, 3(A,1P,E8.1))
End Program