ctgsna.f(3)

NAME
ctgsna.f -

SYNOPSIS

  Functions/Subroutines

subroutine ctgsna (JOB, HOWMNY, SELECT, N, A, LDA, B, LDB, VL, LDVL, VR, LDVR, S, DIF, MM, M, WORK, LWORK, IWORK, INFO)
CTGSNA

Function/Subroutine Documentation

  subroutine ctgsna (characterJOB, characterHOWMNY, logical, dimension( * )SELECT, integerN, complex, dimension( lda, * )A, integerLDA, complex, dimension( ldb, * )B, integerLDB, complex, dimension( ldvl, * )VL, integerLDVL, complex, dimension( ldvr, * )VR, integerLDVR, real, dimension( * )S, real, dimension( * )DIF, integerMM, integerM, complex, dimension( * )WORK, integerLWORK, integer, dimension( * )IWORK, integerINFO)

CTGSNA

Purpose:

     CTGSNA estimates reciprocal condition numbers for specified
     eigenvalues and/or eigenvectors of a matrix pair (A, B).

     (A, B) must be in generalized Schur canonical form, that is, A and
     B are both upper triangular.

Parameters:
    JOB

              JOB is CHARACTER*1
              Specifies whether condition numbers are required for
              eigenvalues (S) or eigenvectors (DIF):
              = 'E': for eigenvalues only (S);
              = 'V': for eigenvectors only (DIF);
              = 'B': for both eigenvalues and eigenvectors (S and DIF).

    HOWMNY

              HOWMNY is CHARACTER*1
              = 'A': compute condition numbers for all eigenpairs;
              = 'S': compute condition numbers for selected eigenpairs
                     specified by the array SELECT.

    SELECT

              SELECT is LOGICAL array, dimension (N)
              If HOWMNY = 'S', SELECT specifies the eigenpairs for which
              condition numbers are required. To select condition numbers
              for the corresponding j-th eigenvalue and/or eigenvector,
              SELECT(j) must be set to .TRUE..
              If HOWMNY = 'A', SELECT is not referenced.

    N

              N is INTEGER
              The order of the square matrix pair (A, B). N >= 0.

    A

              A is COMPLEX array, dimension (LDA,N)
              The upper triangular matrix A in the pair (A,B).

    LDA

              LDA is INTEGER
              The leading dimension of the array A. LDA >= max(1,N).

    B

              B is COMPLEX array, dimension (LDB,N)
              The upper triangular matrix B in the pair (A, B).

    LDB

              LDB is INTEGER
              The leading dimension of the array B. LDB >= max(1,N).

    VL

              VL is COMPLEX array, dimension (LDVL,M)
              IF JOB = 'E' or 'B', VL must contain left eigenvectors of
              (A, B), corresponding to the eigenpairs specified by HOWMNY
              and SELECT.  The eigenvectors must be stored in consecutive
              columns of VL, as returned by CTGEVC.
              If JOB = 'V', VL is not referenced.

    LDVL

              LDVL is INTEGER
              The leading dimension of the array VL. LDVL >= 1; and
              If JOB = 'E' or 'B', LDVL >= N.

    VR

              VR is COMPLEX array, dimension (LDVR,M)
              IF JOB = 'E' or 'B', VR must contain right eigenvectors of
              (A, B), corresponding to the eigenpairs specified by HOWMNY
              and SELECT.  The eigenvectors must be stored in consecutive
              columns of VR, as returned by CTGEVC.
              If JOB = 'V', VR is not referenced.

    LDVR

              LDVR is INTEGER
              The leading dimension of the array VR. LDVR >= 1;
              If JOB = 'E' or 'B', LDVR >= N.

    S

              S is REAL array, dimension (MM)
              If JOB = 'E' or 'B', the reciprocal condition numbers of the
              selected eigenvalues, stored in consecutive elements of the
              array.
              If JOB = 'V', S is not referenced.

    DIF

              DIF is REAL array, dimension (MM)
              If JOB = 'V' or 'B', the estimated reciprocal condition
              numbers of the selected eigenvectors, stored in consecutive
              elements of the array.
              If the eigenvalues cannot be reordered to compute DIF(j),
              DIF(j) is set to 0; this can only occur when the true value
              would be very small anyway.
              For each eigenvalue/vector specified by SELECT, DIF stores
              a Frobenius norm-based estimate of Difl.
              If JOB = 'E', DIF is not referenced.

    MM

              MM is INTEGER
              The number of elements in the arrays S and DIF. MM >= M.

    M

              M is INTEGER
              The number of elements of the arrays S and DIF used to store
              the specified condition numbers; for each selected eigenvalue
              one element is used. If HOWMNY = 'A', M is set to N.

    WORK

              WORK is COMPLEX array, dimension (MAX(1,LWORK))
              On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK. LWORK >= max(1,N).
              If JOB = 'V' or 'B', LWORK >= max(1,2*N*N).

    IWORK

              IWORK is INTEGER array, dimension (N+2)
              If JOB = 'E', IWORK is not referenced.

    INFO

              INFO is INTEGER
              = 0: Successful exit
              < 0: If INFO = -i, the i-th argument had an illegal value

Author:
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date:
    November 2011

Further Details:

      The reciprocal of the condition number of the i-th generalized
      eigenvalue w = (a, b) is defined as

              S(I) = (|v**HAu|**2 + |v**HBu|**2)**(1/2) / (norm(u)*norm(v))

      where u and v are the right and left eigenvectors of (A, B)
      corresponding to w; |z| denotes the absolute value of the complex
      number, and norm(u) denotes the 2-norm of the vector u. The pair
      (a, b) corresponds to an eigenvalue w = a/b (= v**HAu/v**HBu) of the
      matrix pair (A, B). If both a and b equal zero, then (A,B) is
      singular and S(I) = -1 is returned.

      An approximate error bound on the chordal distance between the i-th
      computed generalized eigenvalue w and the corresponding exact
      eigenvalue lambda is

              chord(w, lambda) <=   EPS * norm(A, B) / S(I),

      where EPS is the machine precision.

      The reciprocal of the condition number of the right eigenvector u
      and left eigenvector v corresponding to the generalized eigenvalue w
      is defined as follows. Suppose

                       (A, B) = ( a   *  ) ( b  *  )  1
                                ( 0  A22 ),( 0 B22 )  n-1
                                  1  n-1     1 n-1

      Then the reciprocal condition number DIF(I) is

              Difl[(a, b), (A22, B22)]  = sigma-min( Zl )

      where sigma-min(Zl) denotes the smallest singular value of

             Zl = [ kron(a, In-1) -kron(1, A22) ]
                  [ kron(b, In-1) -kron(1, B22) ].

      Here In-1 is the identity matrix of size n-1 and X**H is the conjugate
      transpose of X. kron(X, Y) is the Kronecker product between the
      matrices X and Y.

      We approximate the smallest singular value of Zl with an upper
      bound. This is done by CLATDF.

      An approximate error bound for a computed eigenvector VL(i) or
      VR(i) is given by

                          EPS * norm(A, B) / DIF(i).

      See ref. [2-3] for more details and further references.

Contributors:
    Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden.

References:

      [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the
          Generalized Real Schur Form of a Regular Matrix Pair (A, B), in
          M.S. Moonen et al (eds), Linear Algebra for Large Scale and
          Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218.

      [2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified
          Eigenvalues of a Regular Matrix Pair (A, B) and Condition
          Estimation: Theory, Algorithms and Software, Report
          UMINF - 94.04, Department of Computing Science, Umea University,
          S-901 87 Umea, Sweden, 1994. Also as LAPACK Working Note 87.
          To appear in Numerical Algorithms, 1996.

      [3] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software
          for Solving the Generalized Sylvester Equation and Estimating the
          Separation between Regular Matrix Pairs, Report UMINF - 93.23,
          Department of Computing Science, Umea University, S-901 87 Umea,
          Sweden, December 1993, Revised April 1994, Also as LAPACK Working
          Note 75.
          To appear in ACM Trans. on Math. Software, Vol 22, No 1, 1996.

Definition at line 310 of file ctgsna.f.

Author

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