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ZTREVC(3F)							    ZTREVC(3F)


NAME    [Toc]    [Back]

     ZTREVC - compute some or all of the right and/or left eigenvectors	of a
     complex upper triangular matrix T

SYNOPSIS    [Toc]    [Back]

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

	 CHARACTER	HOWMNY,	SIDE

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

	 LOGICAL	SELECT(	* )

	 DOUBLE		PRECISION RWORK( * )

	 COMPLEX*16	T( LDT,	* ), VL( LDVL, * ), VR(	LDVR, *	), WORK( * )

PURPOSE    [Toc]    [Back]

     ZTREVC computes some or all of the	right and/or left eigenvectors of a
     complex upper 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 products
     Q*X and/or	Q*Y, where Q is	an input unitary
     matrix. If	T was obtained from the	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.

ARGUMENTS    [Toc]    [Back]

     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.






									Page 1






ZTREVC(3F)							    ZTREVC(3F)



     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	eigenvector corresponding to the j-th eigenvalue,
	     SELECT(j) must be set to .TRUE..

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

     T	     (input/output) COMPLEX*16 array, dimension	(LDT,N)
	     The upper triangular matrix T.  T is modified, but	restored on
	     exit.

     LDT     (input) INTEGER
	     The leading dimension of the array	T. LDT >= max(1,N).

     VL	     (input/output) COMPLEX*16 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 unitary matrix Q of Schur vectors
	     returned by ZHSEQR).  On exit, if SIDE = 'L' or 'B', VL contains:
	     if	HOWMNY = 'A', the matrix Y of left eigenvectors	of T; 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.	 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) 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 unitary matrix Q of Schur vectors
	     returned by ZHSEQR).  On exit, if SIDE = 'R' or 'B', VR contains:
	     if	HOWMNY = 'A', the matrix X of right eigenvectors of T; 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.	 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 eigenvector occupies	one column.




									Page 2






ZTREVC(3F)							    ZTREVC(3F)



     WORK    (workspace) COMPLEX*16 array, dimension (2*N)

     RWORK   (workspace) DOUBLE	PRECISION array, dimension (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|.
ZTREVC(3F)							    ZTREVC(3F)


NAME    [Toc]    [Back]

     ZTREVC - compute some or all of the right and/or left eigenvectors	of a
     complex upper triangular matrix T

SYNOPSIS    [Toc]    [Back]

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

	 CHARACTER	HOWMNY,	SIDE

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

	 LOGICAL	SELECT(	* )

	 DOUBLE		PRECISION RWORK( * )

	 COMPLEX*16	T( LDT,	* ), VL( LDVL, * ), VR(	LDVR, *	), WORK( * )

PURPOSE    [Toc]    [Back]

     ZTREVC computes some or all of the	right and/or left eigenvectors of a
     complex upper 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 products
     Q*X and/or	Q*Y, where Q is	an input unitary
     matrix. If	T was obtained from the	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.

ARGUMENTS    [Toc]    [Back]

     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.






									Page 1






ZTREVC(3F)							    ZTREVC(3F)



     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	eigenvector corresponding to the j-th eigenvalue,
	     SELECT(j) must be set to .TRUE..

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

     T	     (input/output) COMPLEX*16 array, dimension	(LDT,N)
	     The upper triangular matrix T.  T is modified, but	restored on
	     exit.

     LDT     (input) INTEGER
	     The leading dimension of the array	T. LDT >= max(1,N).

     VL	     (input/output) COMPLEX*16 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 unitary matrix Q of Schur vectors
	     returned by ZHSEQR).  On exit, if SIDE = 'L' or 'B', VL contains:
	     if	HOWMNY = 'A', the matrix Y of left eigenvectors	of T; 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.	 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) 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 unitary matrix Q of Schur vectors
	     returned by ZHSEQR).  On exit, if SIDE = 'R' or 'B', VR contains:
	     if	HOWMNY = 'A', the matrix X of right eigenvectors of T; 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.	 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 eigenvector occupies	one column.




									Page 2






ZTREVC(3F)							    ZTREVC(3F)



     WORK    (workspace) COMPLEX*16 array, dimension (2*N)

     RWORK   (workspace) DOUBLE	PRECISION array, dimension (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|.


									PPPPaaaaggggeeee 3333
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