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

NAME

       LAPACK-3  -  computes  some or all of the right and/or left eigenvectors of a pair of real
       matrices (S,P), where S is a quasi-triangular matrix and P is upper triangular

SYNOPSIS

       SUBROUTINE DTGEVC( SIDE, HOWMNY, SELECT, N, S, LDS, P, LDP, VL, LDVL,  VR,  LDVR,  MM,  M,
                          WORK, INFO )

           CHARACTER      HOWMNY, SIDE

           INTEGER        INFO, LDP, LDS, LDVL, LDVR, M, MM, N

           LOGICAL        SELECT( * )

           DOUBLE         PRECISION P( LDP, * ), S( LDS, * ), VL( LDVL, * ), VR( LDVR, * ), WORK(
                          * )

PURPOSE

       DTGEVC computes some or all of the right and/or  left  eigenvectors  of  a  pair  of  real
       matrices  (S,P),  where  S is a quasi-triangular matrix and P is upper triangular.  Matrix
       pairs of this type are produced by
        the generalized Schur factorization of a matrix pair (A,B):
           A = Q*S*Z**T,  B = Q*P*Z**T
        as computed by DGGHRD + DHGEQZ.
        The right eigenvector x and the left eigenvector y of (S,P)
        corresponding to an eigenvalue w are defined by:

           S*x = w*P*x,  (y**H)*S = w*(y**H)*P,

        where y**H denotes the conjugate tranpose of y.
        The eigenvalues are not input to this routine, but are computed
        directly from the diagonal blocks of S and P.

        This routine returns the matrices X and/or Y of right and left
        eigenvectors of (S,P), or the products Z*X and/or Q*Y,
        where Z and Q are input matrices.
        If Q and Z are the orthogonal factors from the generalized Schur
        factorization of a matrix pair (A,B), then Z*X and Q*Y
        are the matrices of right and left eigenvectors of (A,B).

ARGUMENTS

        SIDE    (input) CHARACTER*1
                = 'R': compute right eigenvectors only;
                = 'L': compute left eigenvectors only;
                = 'B': compute both right and left eigenvectors.

        HOWMNY  (input) CHARACTER*1
                = 'A': compute all right and/or left eigenvectors;
                = 'B': compute all right and/or left eigenvectors,
                backtransformed by the matrices in VR and/or VL;
                = 'S': compute selected right and/or left eigenvectors,
                specified by the logical array SELECT.

        SELECT  (input) LOGICAL array, dimension (N)
                If HOWMNY='S', SELECT specifies the eigenvectors to be
                computed.  If w(j) is a real eigenvalue, the corresponding
                real eigenvector is computed if SELECT(j) is .TRUE..
                If w(j) and w(j+1) are the real and imaginary parts of a
                complex eigenvalue, the corresponding complex eigenvector
                is computed if either SELECT(j) or SELECT(j+1) is .TRUE.,
                and on exit SELECT(j) is set to .TRUE. and SELECT(j+1) is
                set to .FALSE..
                Not referenced if HOWMNY = 'A' or 'B'.

        N       (input) INTEGER
                The order of the matrices S and P.  N >= 0.

        S       (input) DOUBLE PRECISION array, dimension (LDS,N)
                The upper quasi-triangular matrix S from a generalized Schur
                factorization, as computed by DHGEQZ.

        LDS     (input) INTEGER
                The leading dimension of array S.  LDS >= max(1,N).

        P       (input) DOUBLE PRECISION array, dimension (LDP,N)
                The upper triangular matrix P from a generalized Schur
                factorization, as computed by DHGEQZ.
                2-by-2 diagonal blocks of P corresponding to 2-by-2 blocks
                of S must be in positive diagonal form.

        LDP     (input) INTEGER
                The leading dimension of array P.  LDP >= max(1,N).

        VL      (input/output) DOUBLE PRECISION array, dimension (LDVL,MM)
                On entry, if SIDE = 'L' or 'B' and HOWMNY = 'B', VL must
                contain an N-by-N matrix Q (usually the orthogonal matrix Q
                of left Schur vectors returned by DHGEQZ).
                On exit, if SIDE = 'L' or 'B', VL contains:
                if HOWMNY = 'A', the matrix Y of left eigenvectors of (S,P);
                if HOWMNY = 'B', the matrix Q*Y;
                if HOWMNY = 'S', the left eigenvectors of (S,P) specified by
                SELECT, stored consecutively in the columns of
                VL, in the same order as their eigenvalues.
                A complex eigenvector corresponding to a complex eigenvalue
                is stored in two consecutive columns, the first holding the
                real part, and the second the imaginary part.
                Not referenced if SIDE = 'R'.

        LDVL    (input) INTEGER
                The leading dimension of array VL.  LDVL >= 1, and if
                SIDE = 'L' or 'B', LDVL >= N.

        VR      (input/output) DOUBLE PRECISION array, dimension (LDVR,MM)
                On entry, if SIDE = 'R' or 'B' and HOWMNY = 'B', VR must
                contain an N-by-N matrix Z (usually the orthogonal matrix Z
                of right Schur vectors returned by DHGEQZ).
                On exit, if SIDE = 'R' or 'B', VR contains:
                if HOWMNY = 'A', the matrix X of right eigenvectors of (S,P);
                if HOWMNY = 'B' or 'b', the matrix Z*X;
                if HOWMNY = 'S' or 's', the right eigenvectors of (S,P)
                specified by SELECT, stored consecutively in the
                columns of VR, in the same order as their
                eigenvalues.
                A complex eigenvector corresponding to a complex eigenvalue
                is stored in two consecutive columns, the first holding the
                real part and the second the imaginary part.
                 Not referenced if SIDE = 'L'.

        LDVR    (input) INTEGER
                The leading dimension of the array VR.  LDVR >= 1, and if
                SIDE = 'R' or 'B', LDVR >= N.

        MM      (input) INTEGER
                The number of columns in the arrays VL and/or VR. MM >= M.

        M       (output) INTEGER
                The number of columns in the arrays VL and/or VR actually
                used to store the eigenvectors.  If HOWMNY = 'A' or 'B', M
                is set to N.  Each selected real eigenvector occupies one
                column and each selected complex eigenvector occupies two
                columns.

        WORK    (workspace) DOUBLE PRECISION array, dimension (6*N)

        INFO    (output) INTEGER
                = 0:  successful exit.
                < 0:  if INFO = -i, the i-th argument had an illegal value.
                > 0:  the 2-by-2 block (INFO:INFO+1) does not have a complex
                eigenvalue.

FURTHER DETAILS

        Allocation of workspace:
        ---------- -- ---------
           WORK( j ) = 1-norm of j-th column of A, above the diagonal
           WORK( N+j ) = 1-norm of j-th column of B, above the diagonal
           WORK( 2*N+1:3*N ) = real part of eigenvector
           WORK( 3*N+1:4*N ) = imaginary part of eigenvector
           WORK( 4*N+1:5*N ) = real part of back-transformed eigenvector
           WORK( 5*N+1:6*N ) = imaginary part of back-transformed eigenvector
        Rowwise vs. columnwise solution methods:
        ------- --  ---------- -------- -------
        Finding a generalized eigenvector consists basically of solving the
        singular triangular system
         (A - w B) x = 0     (for right) or:   (A - w B)**H y = 0  (for left)
        Consider finding the i-th right eigenvector (assume all eigenvalues
        are real). The equation to be solved is:
             n                   i
        0 = sum  C(j,k) v(k)  = sum  C(j,k) v(k)     for j = i,. . .,1
            k=j                 k=j
        where  C = (A - w B)  (The components v(i+1:n) are 0.)
        The "rowwise" method is:
        (1)  v(i) := 1
        for j = i-1,. . .,1:
                                i
            (2) compute  s = - sum C(j,k) v(k)   and
                              k=j+1
            (3) v(j) := s / C(j,j)
        Step 2 is sometimes called the "dot product" step, since it is an
        inner product between the j-th row and the portion of the eigenvector
        that has been computed so far.
        The "columnwise" method consists basically in doing the sums
        for all the rows in parallel.  As each v(j) is computed, the
        contribution of v(j) times the j-th column of C is added to the
        partial sums.  Since FORTRAN arrays are stored columnwise, this has
        the advantage that at each step, the elements of C that are accessed
        are adjacent to one another, whereas with the rowwise method, the
        elements accessed at a step are spaced LDS (and LDP) words apart.
        When finding left eigenvectors, the matrix in question is the
        transpose of the one in storage, so the rowwise method then
        actually accesses columns of A and B at each step, and so is the
        preferred method.

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