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複素一般化非対称固有値問題: 非対称行列ペア : (一般化固有値およびオプショナルで左および右固有ベクトルおよび条件数の逆数)

LAPACKサンプルソースコード : 使用ルーチン名：ZGGEVX

ホーム > LAPACKサンプルプログラム目次 > 複素一般化非対称固有値問題 > 非対称行列ペア

概要

本サンプルはFortran言語によりLAPACKルーチンZGGEVXを利用するサンプルプログラムです。

行列対 $ (A,B)$の全ての固有値と右固有ベクトルを求めます。

\begin{displaymath} A = \left( \begin{array}{rrrr} -21.10 - 22.50i & 53.50 - ... ...3.30i & -32.50 - 46.00i & -19.00 - 32.50i \end{array} \right) \end{displaymath}

及び

\begin{displaymath} B = \left( \begin{array}{rrrr} 1.00 - 5.00i & 1.60 + 1.20... ...+ 2.40i & 0.00 - 4.00i & 4.00 - 5.00i \end{array} \right), \; \end{displaymath}

条件数の推定値とそれぞれの固有値と固有ベクトルの前方誤差限界も合わせて求めます。行列対を均衡化するオプションが使用されます。

入力データ

（本ルーチンの詳細はZGGEVX のマニュアルページを参照） 
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このデータをダウンロード 
ZGGEVX 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ZGGEVX のマニュアルページを参照） 
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この出力例をダウンロード 
Warning: Floating underflow occurred
 ZGGEVX Example Program Results

 Eigenvalues

          Eigenvalue
  1 ( 3.0000E+00,-9.0000E+00)
  2 ( 4.0000E+00,-5.0000E+00)
  3 ( 2.0000E+00,-5.0000E+00)
  4 ( 3.0000E+00,-1.0000E+00)

 Eigenvectors

          Eigenvector

  1 ( 1.0000E+00, 0.0000E+00)
    ( 1.6000E-01,-1.2000E-01)
    ( 1.2000E-01, 1.6000E-01)
    (-1.6000E-01, 1.2000E-01)

  2 ( 1.0000E+00, 0.0000E+00)
    ( 8.8889E-03,-6.6667E-03)
    (-3.3333E-02, 1.1796E-16)
    ( 4.1633E-16, 1.5556E-01)

  3 ( 1.0000E+00, 0.0000E+00)
    ( 4.5714E-03,-3.4286E-03)
    ( 6.2857E-02,-1.4572E-16)
    ( 1.4572E-16, 6.2857E-02)

  4 ( 1.0000E+00, 0.0000E+00)
    ( 1.6000E-01,-1.2000E-01)
    ( 1.2000E-01,-1.6000E-01)
    ( 1.6000E-01, 1.2000E-01)

ソースコード

（本ルーチンの詳細はZGGEVX 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。 

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このソースコードをダウンロード 
    Program zggevx_example

!     ZGGEVX Example Program Text

!     Copyright 2017, Numerical Algorithms Group Ltd. http://www.nag.com

!     .. Use Statements ..
      Use lapack_example_aux, Only: nagf_sort_realvec_rank_rearrange, &
        nagf_blas_dpyth, nagf_sort_cmplxvec_rank_rearrange, &
        nagf_sort_realvec_rank
      Use lapack_interfaces, Only: zggevx
      Use lapack_precision, Only: dp
!     .. Implicit None Statement ..
      Implicit None
!     .. Parameters ..
      Integer, Parameter :: nb = 64, nin = 5, nout = 6
      Logical, Parameter :: verbose = .False.
!     .. Local Scalars ..
      Complex (Kind=dp) :: eig, scal
      Real (Kind=dp) :: abnorm, abnrm, bbnrm, eps, small, tol
      Integer :: i, ifail, ihi, ilo, info, j, k, lda, ldb, ldvr, lwork, n
!     .. Local Arrays ..
      Complex (Kind=dp), Allocatable :: a(:, :), alpha(:), b(:, :), beta(:), &
        temp(:), vr(:, :), work(:)
      Complex (Kind=dp) :: dummy(1, 1)
      Real (Kind=dp), Allocatable :: lscale(:), rconde(:), rcondv(:), &
        rscale(:), rwork(:)
      Integer, Allocatable :: irank(:), iwork(:)
      Logical, Allocatable :: bwork(:)
!     .. Intrinsic Procedures ..
      Intrinsic :: abs, epsilon, max, maxloc, nint, real, tiny
!     .. Executable Statements ..
      Write (nout, *) 'ZGGEVX Example Program Results'
!     Skip heading in data file
      Read (nin, *)
      Read (nin, *) n
      lda = n
      ldb = n
      ldvr = n
      Allocate (a(lda,n), alpha(n), b(ldb,n), beta(n), vr(ldvr,n), lscale(n), &
        rconde(n), rcondv(n), rscale(n), rwork(6*n), iwork(n+2), bwork(n), &
        temp(n))

!     Use routine workspace query to get optimal workspace.
      lwork = -1
      Call zggevx('Balance', 'No vectors (left)', 'Vectors (right)', &
        'Both reciprocal condition numbers', n, a, lda, b, ldb, alpha, beta, &
        dummy, 1, vr, ldvr, ilo, ihi, lscale, rscale, abnrm, bbnrm, rconde, &
        rcondv, dummy, lwork, rwork, iwork, bwork, info)

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

!     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)

!     Solve the generalized eigenvalue problem

      Call zggevx('Balance', 'No vectors (left)', 'Vectors (right)', &
        'Both reciprocal condition numbers', n, a, lda, b, ldb, alpha, beta, &
        dummy, 1, vr, ldvr, ilo, ihi, lscale, rscale, abnrm, bbnrm, rconde, &
        rcondv, work, lwork, rwork, iwork, bwork, info)

      If (info>0) Then
        Write (nout, *)
        Write (nout, 100) 'Failure in ZGGEVX. INFO =', info
      Else

!       Compute the machine precision, the safe range parameter
!       SMALL and sqrt(ABNRM**2+BBNRM**2)

        eps = epsilon(1.0E0_dp)
        small = tiny(1.0E0_dp)
        abnorm = nagf_blas_dpyth(abnrm, bbnrm)
        tol = eps*abnorm

!       Reorder eigenvalues by descending absolute value
        rwork(1:n) = abs(alpha(1:n)/beta(1:n))
        Allocate (irank(n))
        ifail = 0
        Call nagf_sort_realvec_rank(rwork, 1, n, 'Descending', irank, ifail)
        Call nagf_sort_cmplxvec_rank_rearrange(alpha, 1, n, irank, ifail)
        Call nagf_sort_cmplxvec_rank_rearrange(beta, 1, n, irank, ifail)
        Call nagf_sort_realvec_rank_rearrange(rconde, 1, n, irank, ifail)
!       Reorder eigenvectors accordingly
        Do j = 1, n
          temp(1:n) = vr(j, 1:n)
          Call nagf_sort_cmplxvec_rank_rearrange(temp, 1, n, irank, ifail)
          vr(j, 1:n) = temp(1:n)
        End Do
        Call nagf_sort_realvec_rank_rearrange(rcondv, 1, n, irank, ifail)

!       Print out eigenvalues and vectors and associated condition
!       number and bounds

        Write (nout, *)
        Write (nout, *) 'Eigenvalues'
        Write (nout, *)
        If (verbose) Then
          Write (nout, *) '         Eigenvalue           rcond    error'
        Else
          Write (nout, *) '         Eigenvalue'
        End If

        Do j = 1, n

!         Print out information on the j-th eigenvalue

          If ((abs(alpha(j)))*small>=abs(beta(j))) Then
            If (rconde(j)>0.0_dp) Then
              If (tol/rconde(j)<500.0_dp*eps) Then
                Write (nout, 140) j, rconde(j), '-'
              Else
                Write (nout, 150) j, rconde(j), tol/rconde(j)
              End If
            Else
              Write (nout, 140) j, rconde(j), 'Inf'
            End If
          Else
            eig = alpha(j)/beta(j)
            If (verbose) Then
              If (rconde(j)>0.0_dp) Then
                If (tol/rconde(j)<500.0_dp*eps) Then
                  Write (nout, 110) j, eig, rconde(j), '-'
                Else
                  Write (nout, 120) j, eig, rconde(j), tol/rconde(j)
                End If
              Else
                Write (nout, 110) j, eig, rconde(j), 'Inf'
              End If
            Else
              Write (nout, 110) j, eig
            End If
          End If

        End Do

        Write (nout, *)
        Write (nout, *) 'Eigenvectors'
        Write (nout, *)
        If (verbose) Then
          Write (nout, *) '         Eigenvector          rcond    error'
        Else
          Write (nout, *) '         Eigenvector'
        End If

        Do j = 1, n

!         Print information on j-th eigenvector
          Write (nout, *)

!         Re-normalize eigenvector, largest absolute element real (=1)
          rwork(1:n) = abs(vr(1:n,j))
          k = maxloc(rwork(1:n), 1)
          scal = (1.0_dp, 0.0_dp)/vr(k, j)
          vr(1:n, j) = vr(1:n, j)*scal

          If (verbose) Then
            If (rcondv(j)>0.0_dp) Then
              If (tol/rcondv(j)<500.0_dp*eps) Then
                Write (nout, 110) j, vr(1, j), rcondv(j), '-'
              Else
                Write (nout, 120) j, vr(1, j), rcondv(j), tol/rcondv(j)
              End If
            Else
              Write (nout, 110) j, vr(1, j), rcondv(j), 'Inf'
            End If
          Else
            Write (nout, 110) j, vr(1, j)
          End If
          Write (nout, 130) vr(2:n, j)

        End Do

        If (verbose) Then
          Write (nout, *)
          Write (nout, *) &
            'Errors below 500*machine precision are not displayed'
        End If
      End If

100   Format (1X, A, I4)
110   Format (1X, I2, 1X, '(', 1P, E11.4, ',', E11.4, ')', 1X, 0P, F7.4, 4X, &
        A)
120   Format (1X, I2, 1X, '(', 1P, E11.4, ',', E11.4, ')', 1X, 0P, F7.4, 1X, &
        1P, E8.1)
130   Format (1X, 3X, '(', 1P, E11.4, ',', E11.4, ')')
140   Format (1X, I2, 1X, '  Infinite or undetermined', 1X, 0P, F7.4, 4X, A)
150   Format (1X, I2, 1X, '  Infinite or undetermined', 1X, 0P, F7.4, 1X, 1P, &
        E8.1)

    End Program


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