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dtgevc.f(3) LAPACK dtgevc.f(3)

NAME

dtgevc.f -

SYNOPSIS

Functions/Subroutines


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

Function/Subroutine Documentation

subroutine dtgevc (characterSIDE, characterHOWMNY, logical, dimension( * )SELECT, integerN, double precision, dimension( lds, * )S, integerLDS, double precision, dimension( ldp, * )P, integerLDP, double precision, dimension( ldvl, * )VL, integerLDVL, double precision, dimension( ldvr, * )VR, integerLDVR, integerMM, integerM, double precision, dimension( * )WORK, integerINFO)

DTGEVC
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).
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 DOUBLE PRECISION array, dimension (LDS,N)
          The upper quasi-triangular matrix S from a generalized Schur
          factorization, as computed by DHGEQZ.
LDS
          LDS is INTEGER
          The leading dimension of array S.  LDS >= max(1,N).
P
          P is 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
          LDP is INTEGER
          The leading dimension of array P.  LDP >= max(1,N).
VL
          VL is 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
          LDVL is INTEGER
          The leading dimension of array VL.  LDVL >= 1, and if
          SIDE = 'L' or 'B', LDVL >= N.
VR
          VR is 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
          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 DOUBLE PRECISION 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.
Date:
November 2011
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.
Definition at line 295 of file dtgevc.f.

Author

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