DTGEVC(l)								 )								 DTGEVC(l)

NAME

       DTGEVC - compute some or all of the right and/or left generalized eigenvectors of a pair of real upper triangular matrices (A,B)

SYNOPSIS

       SUBROUTINE DTGEVC( SIDE, HOWMNY, SELECT, N, A, LDA, B, LDB, VL, LDVL, VR, LDVR, MM, M, WORK, INFO )

	   CHARACTER	  HOWMNY, SIDE

	   INTEGER	  INFO, LDA, LDB, LDVL, LDVR, M, MM, N

	   LOGICAL	  SELECT( * )

	   DOUBLE	  PRECISION A( LDA, * ), B( LDB, * ), VL( LDVL, * ), VR( LDVR, * ), WORK( * )

PURPOSE

       DTGEVC computes some or all of the right and/or left generalized eigenvectors of a pair of real upper triangular matrices (A,B).  The right
       generalized eigenvector x and the left generalized eigenvector y of (A,B) corresponding to a generalized eigenvalue w are defined by:

	       (A - wB) * x = 0  and  y**H * (A - wB) = 0

       where y**H denotes the conjugate tranpose of y.

       If an eigenvalue w is determined by zero diagonal elements of both A and B, a unit vector is returned as the corresponding eigenvector.

       If all eigenvectors are requested, the routine may either return the matrices X and/or Y of right or left eigenvectors  of  (A,B),  or  the
       products  Z*X and/or Q*Y, where Z and Q are input orthogonal matrices.  If (A,B) was obtained from the generalized real-Schur factorization
       of an original pair of matrices
	  (A0,B0) = (Q*A*Z**H,Q*B*Z**H),
       then Z*X and Q*Y are the matrices of right or left eigenvectors of A.

       A must be block upper triangular, with 1-by-1 and 2-by-2 diagonal blocks.  Corresponding to each 2-by-2 diagonal block is a complex  conju-
       gate pair of eigenvalues and eigenvectors; only one
       eigenvector of the pair is computed, namely the one corresponding to the eigenvalue with positive imaginary part.

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, and backtransform them using the input matrices supplied 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 computed.  If HOWMNY='A' or 'B', SELECT is not referenced.   To  select  the
	       real eigenvector corresponding to the real eigenvalue w(j), SELECT(j) must be set to .TRUE.  To select the complex eigenvector cor-
	       responding to a complex conjugate pair w(j) and w(j+1), either SELECT(j) or SELECT(j+1) must be set to .TRUE..

       N       (input) INTEGER
	       The order of the matrices A and B.  N >= 0.

       A       (input) DOUBLE PRECISION array, dimension (LDA,N)
	       The upper quasi-triangular matrix A.

       LDA     (input) INTEGER
	       The leading dimension of array A.  LDA >= max(1, N).

       B       (input) DOUBLE PRECISION array, dimension (LDB,N)
	       The upper triangular matrix B.  If A has a 2-by-2 diagonal block, then the corresponding 2-by-2 block of B must	be  diagonal  with
	       positive elements.

       LDB     (input) INTEGER
	       The leading dimension of array B.  LDB >= max(1,N).

       VL      (input/output) 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
	       (A,B);  if  HOWMNY = 'B', the matrix Q*Y; if HOWMNY = 'S', the left eigenvectors of (A,B) specified by SELECT, stored consecutively
	       in the columns of VL, in the same order as their eigenvalues.  If SIDE = 'R', VL is not referenced.

	       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.

       LDVL    (input) INTEGER
	       The leading dimension of array VL.  LDVL >= max(1,N) if SIDE = 'L' or 'B'; LDVL >= 1 otherwise.

       VR      (input/output) DOUBLE PRECISION array, dimension (LDVR,MM)
	       On entry, if SIDE = 'R' or 'B' and HOWMNY = 'B', VR must contain an N-by-N matrix Q (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
	       (A,B);  if HOWMNY = 'B', the matrix Z*X; if HOWMNY = 'S', the right eigenvectors of (A,B) specified by SELECT, stored consecutively
	       in the columns of VR, in the same order as their eigenvalues.  If SIDE = 'L', VR is not referenced.

	       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.

       LDVR    (input) INTEGER
	       The leading dimension of the array VR.  LDVR >= max(1,N) if SIDE = 'R' or 'B'; LDVR >= 1 otherwise.

       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 LDA (and LDB) 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.

LAPACK version 3.0						   15 June 2000 							 DTGEVC(l)