Provided by: liblapack-doc_3.3.1-1_all bug

NAME

       LAPACK-3  -  estimates  reciprocal  condition  numbers  for  specified  eigenvalues and/or
       eigenvectors of a matrix pair (A, B)

SYNOPSIS

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

           CHARACTER      HOWMNY, JOB

           INTEGER        INFO, LDA, LDB, LDVL, LDVR, LWORK, M, MM, N

           LOGICAL        SELECT( * )

           INTEGER        IWORK( * )

           DOUBLE         PRECISION DIF( * ), S( * )

           COMPLEX*16     A( LDA, * ), B( LDB, * ), VL( LDVL, * ), VR( LDVR, * ), WORK( * )

PURPOSE

       ZTGSNA   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.

ARGUMENTS

        JOB     (input) 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  (input) CHARACTER*1
                = 'A': compute condition numbers for all eigenpairs;
                = 'S': compute condition numbers for selected eigenpairs
                specified by the array SELECT.

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

        A       (input) COMPLEX*16 array, dimension (LDA,N)
                The upper triangular matrix A in the pair (A,B).

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

        B       (input) COMPLEX*16 array, dimension (LDB,N)
                The upper triangular matrix B in the pair (A, B).

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

        VL      (input) COMPLEX*16 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 ZTGEVC.
                If JOB = 'V', VL is not referenced.

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

        VR      (input) COMPLEX*16 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 ZTGEVC.
                If JOB = 'V', VR is not referenced.

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

        S       (output) DOUBLE PRECISION 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     (output) DOUBLE PRECISION 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      (input) INTEGER
                The number of elements in the arrays S and DIF. MM >= M.

        M       (output) 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    (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK))
                On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

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

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

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

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 ZLATDF.
        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.
        Based on contributions by
           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.

 LAPACK routine (version 3.3.1)             April 2011                            ZTGSNA(3lapack)