DTREVC(l)								 )								 DTREVC(l)

NAME

       DTREVC - compute some or all of the right and/or left eigenvectors of a real upper quasi-triangular matrix T

SYNOPSIS

       SUBROUTINE DTREVC( SIDE, HOWMNY, SELECT, N, T, LDT, VL, LDVL, VR, LDVR, MM, M, WORK, INFO )

	   CHARACTER	  HOWMNY, SIDE

	   INTEGER	  INFO, LDT, LDVL, LDVR, M, MM, N

	   LOGICAL	  SELECT( * )

	   DOUBLE	  PRECISION T( LDT, * ), VL( LDVL, * ), VR( LDVR, * ), WORK( * )

PURPOSE

       DTREVC  computes  some or all of the right and/or left eigenvectors of a real upper quasi-triangular matrix T.  The right eigenvector x and
       the left eigenvector y of T corresponding to an eigenvalue w are defined by:

		    T*x = w*x,	   y'*T = w*y'

       where y' denotes the conjugate transpose of the vector y.

       If all eigenvectors are requested, the routine may either return the matrices X and/or Y of right or left eigenvectors of T, or	the  prod-
       ucts Q*X and/or Q*Y, where Q is an input orthogonal
       matrix. If T was obtained from the real-Schur factorization of an original matrix A = Q*T*Q', then Q*X and Q*Y are the matrices of right or
       left eigenvectors of A.

       T must be in Schur canonical form (as returned by DHSEQR), that is, block upper triangular with 1-by-1 and  2-by-2  diagonal  blocks;  each
       2-by-2  diagonal block has its diagonal elements equal and its off-diagonal elements of opposite sign.  Corresponding to each 2-by-2 diago-
       nal block is a complex conjugate pair of eigenvalues and eigenvectors; only one eigenvector of the pair is computed, namely the one  corre-
       sponding 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/output) 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 a 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.;  then  on  exit
	       SELECT(j) is .TRUE. and SELECT(j+1) is .FALSE..

       N       (input) INTEGER
	       The order of the matrix T. N >= 0.

       T       (input) DOUBLE PRECISION array, dimension (LDT,N)
	       The upper quasi-triangular matrix T in Schur canonical form.

       LDT     (input) INTEGER
	       The leading dimension of the array T. LDT >= 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 Schur vec-
	       tors returned by DHSEQR).  On exit, if SIDE = 'L' or 'B', VL contains: if HOWMNY = 'A', the matrix Y of left eigenvectors of T;	VL
	       has  the  same  quasi-lower  triangular form as T'. If T(i,i) is a real eigenvalue, then the i-th column VL(i) of VL  is its corre-
	       sponding eigenvector. If T(i:i+1,i:i+1) is  a  2-by-2  block  whose  eigenvalues  are  complex-conjugate  eigenvalues  of  T,  then
	       VL(i)+sqrt(-1)*VL(i+1)  is  the	complex eigenvector corresponding to the eigenvalue with positive real part.  if HOWMNY = 'B', the
	       matrix Q*Y; if HOWMNY = 'S', the left eigenvectors of T 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.	If SIDE = 'R', VL is not referenced.

       LDVL    (input) INTEGER
	       The leading dimension of the 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 Q of Schur  vec-
	       tors returned by DHSEQR).  On exit, if SIDE = 'R' or 'B', VR contains: if HOWMNY = 'A', the matrix X of right eigenvectors of T; VR
	       has the same quasi-upper triangular form as T. If T(i,i) is a real eigenvalue, then the i-th column VR(i) of VR	is its correspond-
	       ing   eigenvector.  If  T(i:i+1,i:i+1)  is  a  2-by-2  block  whose  eigenvalues  are  complex-conjugate  eigenvalues  of  T,  then
	       VR(i)+sqrt(-1)*VR(i+1) is the complex eigenvector corresponding to the eigenvalue with positive real part.  if HOWMNY  =  'B',  the
	       matrix  Q*X;  if  HOWMNY  = 'S', the right eigenvectors of T 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.  If SIDE = 'L', VR is not referenced.

       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 (3*N)

       INFO    (output) INTEGER
	       = 0:  successful exit
	       < 0:  if INFO = -i, the i-th argument had an illegal value

FURTHER DETAILS

       The algorithm used in this program is basically backward (forward) substitution, with scaling to make the the code robust against  possible
       overflow.

       Each  eigenvector  is  normalized so that the element of largest magnitude has magnitude 1; here the magnitude of a complex number (x,y) is
       taken to be |x| + |y|.

LAPACK version 3.0						   15 June 2000 							 DTREVC(l)