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DTREVC(1)		 LAPACK routine (version 3.2)		     DTREVC(1)

NAME
       DTREVC  - computes 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.	Matrices of this type are pro‐
       duced by the Schur  factorization  of  a	 real  general	matrix:	  A  =
       Q*T*Q**T, as computed by DHSEQR.
       The  right  eigenvector x and the left eigenvector y of T corresponding
       to an eigenvalue w are defined by:
	  T*x = w*x,	 (y**H)*T = w*(y**H)
       where y**H denotes the conjugate transpose of y.
       The eigenvalues are not input to this routine, but  are	read  directly
       from the diagonal blocks of T.
       This  routine  returns the matrices X and/or Y of right and left eigen‐
       vectors of T, or the products Q*X and/or	 Q*Y,  where  Q	 is  an	 input
       matrix.	If Q is the orthogonal factor that reduces a matrix A to Schur
       form T, then Q*X and Q*Y are the matrices of right and  left  eigenvec‐
       tors of A.

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, as indicated by the
	       logical array SELECT.

       SELECT  (input/output) 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 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 con‐
	       tain an N-by-N matrix Q (usually the  orthogonal	 matrix	 Q  of
	       Schur  vectors  returned by DHSEQR).  On exit, if SIDE = 'L' or
	       'B', VL contains: if HOWMNY = 'A', the matrix Y of left	eigen‐
	       vectors of T; if HOWMNY = 'B', the matrix Q*Y; if HOWMNY = 'S',
	       the left eigenvectors of T specified by SELECT, stored consecu‐
	       tively  in the columns of VL, in the same order as their eigen‐
	       values.	A complex eigenvector corresponding to a  complex  ei‐
	       genvalue	 is stored in two consecutive columns, the first hold‐
	       ing the real part, and the second the imaginary part.  Not ref‐
	       erenced if SIDE = 'R'.

       LDVL    (input) INTEGER
	       The  leading dimension of the 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	 Q (usually the orthogonal matrix Q of
	       Schur vectors returned by DHSEQR).  On exit, if SIDE =  'R'  or
	       'B', VR contains: if HOWMNY = 'A', the matrix X of right eigen‐
	       vectors of T; if HOWMNY = 'B', the matrix Q*X; if HOWMNY = 'S',
	       the right eigenvectors of T specified by SELECT, stored consec‐
	       utively in the columns of VR, in the same order as their eigen‐
	       values.	 A  complex eigenvector corresponding to a complex ei‐
	       genvalue is stored in two consecutive columns, the first	 hold‐
	       ing  the real part and the second the imaginary part.  Not ref‐
	       erenced 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 (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) sub‐
       stitution,  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 routine (version 3.2)	 November 2008			     DTREVC(1)
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