概要
本サンプルはFortran言語によりLAPACKルーチンDGGESX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。
ここで
の実固有値が一般化Schur形式 
の左上対角要素に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DGGESX のマニュアルページを参照）| このデータをダウンロード |
DGGESX Example Program Data 4 :Value of N 3.9 12.5 -34.5 -0.5 4.3 21.5 -47.5 7.5 4.3 21.5 -43.5 3.5 4.4 26.0 -46.0 6.0 :End of matrix A 1.0 2.0 -3.0 1.0 1.0 3.0 -5.0 4.0 1.0 3.0 -4.0 3.0 1.0 3.0 -4.0 4.0 :End of matrix B
出力結果
（本ルーチンの詳細はDGGESX のマニュアルページを参照）| この出力例をダウンロード |
DGGESX Example Program Results
Matrix A
1 2 3 4
1 3.9000 12.5000 -34.5000 -0.5000
2 4.3000 21.5000 -47.5000 7.5000
3 4.3000 21.5000 -43.5000 3.5000
4 4.4000 26.0000 -46.0000 6.0000
Matrix B
1 2 3 4
1 1.0000 2.0000 -3.0000 1.0000
2 1.0000 3.0000 -5.0000 4.0000
3 1.0000 3.0000 -4.0000 3.0000
4 1.0000 3.0000 -4.0000 4.0000
Number of eigenvalues for which SELCTG is true = 2
(dimension of deflating subspaces)
Selected generalized eigenvalues
1 ( 2.000, 0.000)
2 ( 4.000, 0.000)
Reciprocals of left and right projection norms onto
the deflating subspaces for the selected eigenvalues
RCONDE(1) = 1.9E-01, RCONDE(2) = 1.8E-02
Reciprocal condition numbers for the left and right
deflating subspaces
RCONDV(1) = 5.4E-02, RCONDV(2) = 9.0E-02
Warning: Floating underflow occurred
Approximate asymptotic error bound for selected eigenvalues = 1.1E-13
Approximate asymptotic error bound for the deflating subspaces = 2.4E-13
ソースコード
（本ルーチンの詳細はDGGESX 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 |
! DGGESX Example Program Text
! Copyright 2017, Numerical Algorithms Group Ltd. http://www.nag.com
Module dggesx_mod
! DGGESX 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
Contains
Function selctg(ar, ai, b)
! Logical function selctg for use with DGGESX (DGGESX)
! Returns the value .TRUE. if the eigenvalue is real and positive
! .. Function Return Value ..
Logical :: selctg
! .. Scalar Arguments ..
Real (Kind=dp), Intent (In) :: ai, ar, b
! .. Executable Statements ..
selctg = (ar>0._dp .And. ai==0._dp .And. b/=0._dp)
Return
End Function
End Module
Program dggesx_example
! DGGESX Example Main Program
! .. Use Statements ..
Use blas_interfaces, Only: dgemm
Use dggesx_mod, Only: nb, nin, nout, selctg
Use lapack_example_aux, Only: nagf_blas_dpyth, &
nagf_file_print_matrix_real_gen
Use lapack_interfaces, Only: dggesx, dlange
Use lapack_precision, Only: dp
! .. Implicit None Statement ..
Implicit None
! .. Local Scalars ..
Real (Kind=dp) :: abnorm, alph, anorm, bet, bnorm, eps, normd, norme, &
tol
Integer :: i, ifail, info, lda, ldb, ldc, ldd, lde, ldvsl, ldvsr, &
liwork, lwork, n, sdim
! .. Local Arrays ..
Real (Kind=dp), Allocatable :: a(:, :), alphai(:), alphar(:), b(:, :), &
beta(:), c(:, :), d(:, :), e(:, :), vsl(:, :), vsr(:, :), work(:)
Real (Kind=dp) :: rconde(2), rcondv(2), rdum(1)
Integer :: idum(1)
Integer, Allocatable :: iwork(:)
Logical, Allocatable :: bwork(:)
! .. Intrinsic Procedures ..
Intrinsic :: epsilon, max, nint
! .. Executable Statements ..
Write (nout, *) 'DGGESX 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), alphai(n), alphar(n), b(ldb,n), beta(n), &
vsl(ldvsl,n), vsr(ldvsr,n), bwork(n), c(ldc,n), d(ldd,n), e(lde,n))
! Use routine workspace query to get optimal workspace.
lwork = -1
liwork = -1
Call dggesx('Vectors (left)', 'Vectors (right)', 'Sort', selctg, &
'Both reciprocal condition numbers', n, a, lda, b, ldb, sdim, alphar, &
alphai, beta, vsl, ldvsl, vsr, ldvsr, rconde, rcondv, rdum, lwork, &
idum, liwork, bwork, info)
! Make sure that there is enough workspace for block size nb.
lwork = max(8*(n+1)+16+n*nb+n*n/2, nint(rdum(1)))
liwork = max(n+6, 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)
! 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)
! 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_real_gen('General', ' ', n, n, a, lda, &
'Matrix A', ifail)
Write (nout, *)
Flush (nout)
ifail = 0
Call nagf_file_print_matrix_real_gen('General', ' ', n, n, b, ldb, &
'Matrix B', ifail)
Write (nout, *)
Flush (nout)
! Find the Frobenius norms of A and B
anorm = dlange('Frobenius', n, n, a, lda, work)
bnorm = dlange('Frobenius', n, n, b, ldb, work)
! Find the generalized Schur form
Call dggesx('Vectors (left)', 'Vectors (right)', 'Sort', selctg, &
'Both reciprocal condition numbers', n, a, lda, b, ldb, sdim, alphar, &
alphai, beta, vsl, ldvsl, vsr, ldvsr, rconde, rcondv, work, lwork, &
iwork, liwork, bwork, info)
If (info==0 .Or. info==(n+2)) Then
! Compute A - Q*S*Z^T from the factorization of (A,B) and store in
! matrix D
alph = 1.0_dp
bet = 0.0_dp
Call dgemm('N', 'N', n, n, n, alph, vsl, ldvsl, a, lda, bet, c, ldc)
alph = -1.0_dp
bet = 1.0_dp
Call dgemm('N', 'T', n, n, n, alph, c, ldc, vsr, ldvsr, bet, d, ldd)
! Compute B - Q*T*Z^T from the factorization of (A,B) and store in
! matrix E
alph = 1.0_dp
bet = 0.0_dp
Call dgemm('N', 'N', n, n, n, alph, vsl, ldvsl, b, ldb, bet, c, ldc)
alph = -1.0_dp
bet = 1.0_dp
Call dgemm('N', 'T', 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 = dlange('O', ldd, n, d, ldd, work)
norme = dlange('O', lde, n, e, lde, work)
If (normd>epsilon(1.0E0_dp)**0.8_dp .Or. norme>epsilon(1.0E0_dp)** &
0.8_dp) Then
Write (nout, *) 'Norm of A-(Q*S*Z^T) or norm of B-(Q*T*Z^T) &
&is much greater than 0.'
Write (nout, *) 'Schur factorization has failed.'
Else
! Print solution
Write (nout, 100) &
'Number of eigenvalues for which SELCTG is true = ', sdim, &
'(dimension of deflating subspaces)'
Write (nout, *)
! Print generalized eigenvalues
Write (nout, *) 'Selected generalized eigenvalues'
Do i = 1, sdim
If (beta(i)/=0.0_dp) Then
Write (nout, 110) i, '(', alphar(i)/beta(i), ',', &
alphai(i)/beta(i), ')'
Else
Write (nout, 120) i
End If
End Do
If (info==(n+2)) Then
Write (nout, 130) '***Note that rounding errors mean ', &
'that leading eigenvalues in the generalized', &
'Schur form no longer satisfy SELCTG = .TRUE.'
Write (nout, *)
End If
Flush (nout)
! Print out the reciprocal condition numbers
Write (nout, *)
Write (nout, 140) &
'Reciprocals of left and right projection norms onto', &
'the deflating subspaces for the selected eigenvalues', &
'RCONDE(1) = ', rconde(1), ', RCONDE(2) = ', rconde(2)
Write (nout, *)
Write (nout, 140) &
'Reciprocal condition numbers for the left and right', &
'deflating subspaces', 'RCONDV(1) = ', rcondv(1), &
', RCONDV(2) = ', rcondv(2)
Flush (nout)
! 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 approximate asymptotic error bound on the
! average absolute error of the selected eigenvalues given by
! eps*norm((A, B))/PL, where PL = RCONDE(1)
Write (nout, *)
Write (nout, 150) 'Approximate asymptotic error bound for selected ' &
, 'eigenvalues = ', tol/rconde(1)
! Print out an approximate asymptotic bound on the maximum
! angular error in the computed deflating subspaces given by
! eps*norm((A, B))/DIF(2), where DIF(2) = RCONDV(2)
Write (nout, 150) &
'Approximate asymptotic error bound for the deflating ', &
'subspaces = ', tol/rcondv(2)
End If
Else
Write (nout, 100) 'Failure in DGGESX. INFO =', info
End If
100 Format (1X, A, I4, /, 1X, A)
110 Format (1X, I4, 5X, A, F7.3, A, F7.3, A)
120 Format (1X, I4, 'Eigenvalue is infinite')
130 Format (1X, 2A, /, 1X, A)
140 Format (1X, A, /, 1X, A, /, 1X, 2(A,1P,E8.1))
150 Format (1X, 2A, 1P, E8.1)
End Program