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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Version 3.4.2							  Tue Sep 25 2012						       dtgevc.f(3)