dtgevc man page on Scientific

Man page or keyword search:  
man Server   26626 pages
apropos Keyword Search (all sections)
Output format
Scientific logo
[printable version]

DTGEVC(1)		 LAPACK routine (version 3.2)		     DTGEVC(1)

NAME
       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

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) correspond‐
       ing 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	eigen‐
       vectors 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 fac‐
       torization 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,  backtrans‐
	       formed 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 com‐
	       puted.  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  eigenval‐
	       ue, 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 refer‐
	       enced 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 factor‐
	       ization, as computed by DHGEQZ.	2-by-2 diagonal	 blocks	 of  P
	       corresponding  to 2-by-2 blocks of S must be in positive diago‐
	       nal 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  con‐
	       tain  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 correspond‐
	       ing to a complex eigenvalue is stored in two  consecutive  col‐
	       umns, the first holding the real part, and the second the imag‐
	       inary 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 con‐
	       tain 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)	speci‐
	       fied  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 consecu‐
	       tive 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 eigenvec‐
       tors, 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 3.2)	 November 2008			     DTGEVC(1)
[top]

List of man pages available for Scientific

Copyright (c) for man pages and the logo by the respective OS vendor.

For those who want to learn more, the polarhome community provides shell access and support.

[legal] [privacy] [GNU] [policy] [cookies] [netiquette] [sponsors] [FAQ]
Tweet
Polarhome, production since 1999.
Member of Polarhome portal.
Based on Fawad Halim's script.
....................................................................
Vote for polarhome
Free Shell Accounts :: the biggest list on the net