SRC/ctgsna.f(3)

NAME
SRC/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 (character job, character howmny, logical, dimension( * ) select, integer n, complex, dimension( lda, * ) a, integer lda, complex, dimension( ldb, * ) b, integer ldb, complex, dimension( ldvl, * ) vl, integer ldvl, complex, dimension( ldvr, * ) vr, integer ldvr, real, dimension( * ) s, real, dimension( * ) dif, integer mm, integer m, complex, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer info)

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.

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 308 of file ctgsna.f.

Author

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