Loading...
Searching...
No Matches

## stgevc()

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

Download STGEVC + dependencies [TGZ] [ZIP] [TXT]

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

  [in]

  SIDE

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

  [in]

  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.
      !>

  [in]

  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'.
      !>

  [in]

  N

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

  [in]

  S

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

  [in]

  LDS

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

  [in]

  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.
      !>

  [in]

  LDP

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

  [in,out]

  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'.
      !>

  [in]

  LDVL

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

  [in,out]

  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'.
      !>

  [in]

  LDVR

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

  [in]

  MM

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

  [out]

  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.
      !>

  [out]

  WORK

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

  [out]

  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.
      !>