SRC/stgevc.f(3)

NAME
SRC/stgevc.f

SYNOPSIS

  Functions/Subroutines
subroutine stgevc (side, howmny, select, n, s, lds, p, ldp, vl, ldvl, vr, ldvr, mm, m, work, info)
STGEVC

Function/Subroutine Documentation

  subroutine stgevc (character side, character howmny, logical, dimension( * ) select, integer n, real, dimension( lds, * ) s, integer lds, real, dimension( ldp, * ) p, integer ldp, real, dimension( ldvl, * ) vl, integer ldvl, real, dimension( ldvr, * ) vr, integer ldvr, integer mm, integer m, real, dimension( * ) work, integer info)

STGEVC

Purpose:

    !>
    !> STGEVC 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 SGGHRD + SHGEQZ.
    !>
    !> 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 transpose 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).
    !>
    !>

Parameters
    SIDE

    !>          SIDE is CHARACTER*1
    !>          = 'R': compute right eigenvectors only;
    !>          = 'L': compute left eigenvectors only;
    !>          = 'B': compute both right and left eigenvectors.
    !>

    HOWMNY

    !>          HOWMNY is 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

    !>          SELECT is 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

    !>          N is INTEGER
    !>          The order of the matrices S and P.  N >= 0.
    !>

    S

    !>          S is REAL array, dimension (LDS,N)
    !>          The upper quasi-triangular matrix S from a generalized Schur
    !>          factorization, as computed by SHGEQZ.
    !>

    LDS

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

    P

    !>          P is REAL array, dimension (LDP,N)
    !>          The upper triangular matrix P from a generalized Schur
    !>          factorization, as computed by SHGEQZ.
    !>          2-by-2 diagonal blocks of P corresponding to 2-by-2 blocks
    !>          of S must be in positive diagonal form.
    !>

    LDP

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

    VL

    !>          VL is REAL 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 SHGEQZ).
    !>          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

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

    VR

    !>          VR is REAL 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 SHGEQZ).
    !>
    !>          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

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

    MM

    !>          MM is INTEGER
    !>          The number of columns in the arrays VL and/or VR. MM >= M.
    !>

    M

    !>          M is 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

    !>          WORK is REAL array, dimension (6*N)
    !>

    INFO

    !>          INFO is 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.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

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  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  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  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.
    !>

Definition at line 293 of file stgevc.f.

Author

Generated automatically by Doxygen for LAPACK from the source code.