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DTGEVC(3F)							    DTGEVC(3F)

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 conjugate 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.

									Page 1

DTGEVC(3F)							    DTGEVC(3F)

     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 corresponding 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.

									Page 2

DTGEVC(3F)							    DTGEVC(3F)

     VR	     (input/output) COMPLEX*16 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

									Page 3

DTGEVC(3F)							    DTGEVC(3F)

      (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.

									Page 4

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