DGGESX(実一般化非対称固有値問題) LAPACKサンプルソースコード

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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。

$\begin{displaymath} A = \left( \begin{array}{rrrr} 3.9 & 12.5 & -34.5 & -0.5 ... ...& -4.0 & 3.0 \\ 1.0 & 3.0 & -4.0 & 4.0 \end{array} \right), \end{displaymath}$

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

ご案内

実一般化非対称固有値問題: 対称行列ペア : (一般化固有値と一般化実シュール形)

LAPACKサンプルソースコード : 使用ルーチン名：DGGESX

概要

入力データ

出力結果

ソースコード