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CHEEVX(3F)							    CHEEVX(3F)


NAME    [Toc]    [Back]

     CHEEVX - compute selected eigenvalues and,	optionally, eigenvectors of a
     complex Hermitian matrix A

SYNOPSIS    [Toc]    [Back]

     SUBROUTINE	CHEEVX(	JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU, ABSTOL,
			M, W, Z, LDZ, WORK, LWORK, RWORK, IWORK, IFAIL,	INFO )

	 CHARACTER	JOBZ, RANGE, UPLO

	 INTEGER	IL, INFO, IU, LDA, LDZ,	LWORK, M, N

	 REAL		ABSTOL,	VL, VU

	 INTEGER	IFAIL( * ), IWORK( * )

	 REAL		RWORK( * ), W( * )

	 COMPLEX	A( LDA,	* ), WORK( * ),	Z( LDZ,	* )

PURPOSE    [Toc]    [Back]

     CHEEVX computes selected eigenvalues and, optionally, eigenvectors	of a
     complex Hermitian matrix A.  Eigenvalues and eigenvectors can be selected
     by	specifying either a range of values or a range of indices for the
     desired eigenvalues.

ARGUMENTS    [Toc]    [Back]

     JOBZ    (input) CHARACTER*1
	     = 'N':  Compute eigenvalues only;
	     = 'V':  Compute eigenvalues and eigenvectors.

     RANGE   (input) CHARACTER*1
	     = 'A': all	eigenvalues will be found.
	     = 'V': all	eigenvalues in the half-open interval (VL,VU] will be
	     found.  = 'I': the	IL-th through IU-th eigenvalues	will be	found.

     UPLO    (input) CHARACTER*1
	     = 'U':  Upper triangle of A is stored;
	     = 'L':  Lower triangle of A is stored.

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

     A	     (input/output) COMPLEX array, dimension (LDA, N)
	     On	entry, the Hermitian matrix A.	If UPLO	= 'U', the leading Nby-N
 upper	triangular part	of A contains the upper	triangular
	     part of the matrix	A.  If UPLO = 'L', the leading N-by-N lower
	     triangular	part of	A contains the lower triangular	part of	the
	     matrix A.	On exit, the lower triangle (if	UPLO='L') or the upper
	     triangle (if UPLO='U') of A, including the	diagonal, is
	     destroyed.



									Page 1






CHEEVX(3F)							    CHEEVX(3F)



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

     VL	     (input) REAL
	     VU	     (input) REAL If RANGE='V',	the lower and upper bounds of
	     the interval to be	searched for eigenvalues. VL < VU.  Not
	     referenced	if RANGE = 'A' or 'I'.

     IL	     (input) INTEGER
	     IU	     (input) INTEGER If	RANGE='I', the indices (in ascending
	     order) of the smallest and	largest	eigenvalues to be returned.  1
	     <=	IL <= IU <= N, if N > 0; IL = 1	and IU = 0 if N	= 0.  Not
	     referenced	if RANGE = 'A' or 'V'.

     ABSTOL  (input) REAL
	     The absolute error	tolerance for the eigenvalues.	An approximate
	     eigenvalue	is accepted as converged when it is determined to lie
	     in	an interval [a,b] of width less	than or	equal to

	     ABSTOL + EPS *   max( |a|,|b| ) ,

	     where EPS is the machine precision.  If ABSTOL is less than or
	     equal to zero, then  EPS*|T|  will	be used	in its place, where
	     |T| is the	1-norm of the tridiagonal matrix obtained by reducing
	     A to tridiagonal form.

	     Eigenvalues will be computed most accurately when ABSTOL is set
	     to	twice the underflow threshold 2*SLAMCH('S'), not zero.	If
	     this routine returns with INFO>0, indicating that some
	     eigenvectors did not converge, try	setting	ABSTOL to
	     2*SLAMCH('S').

	     See "Computing Small Singular Values of Bidiagonal	Matrices with
	     Guaranteed	High Relative Accuracy," by Demmel and Kahan, LAPACK
	     Working Note #3.

     M	     (output) INTEGER
	     The total number of eigenvalues found.  0 <= M <= N.  If RANGE =
	     'A', M = N, and if	RANGE =	'I', M = IU-IL+1.

     W	     (output) REAL array, dimension (N)
	     On	normal exit, the first M elements contain the selected
	     eigenvalues in ascending order.

     Z	     (output) COMPLEX array, dimension (LDZ, max(1,M))
	     If	JOBZ = 'V', then if INFO = 0, the first	M columns of Z contain
	     the orthonormal eigenvectors of the matrix	A corresponding	to the
	     selected eigenvalues, with	the i-th column	of Z holding the
	     eigenvector associated with W(i).	If an eigenvector fails	to
	     converge, then that column	of Z contains the latest approximation
	     to	the eigenvector, and the index of the eigenvector is returned
	     in	IFAIL.	If JOBZ	= 'N', then Z is not referenced.  Note:	the



									Page 2






CHEEVX(3F)							    CHEEVX(3F)



	     user must ensure that at least max(1,M) columns are supplied in
	     the array Z; if RANGE = 'V', the exact value of M is not known in
	     advance and an upper bound	must be	used.

     LDZ     (input) INTEGER
	     The leading dimension of the array	Z.  LDZ	>= 1, and if JOBZ =
	     'V', LDZ >= max(1,N).

     WORK    (workspace/output)	COMPLEX	array, dimension (LWORK)
	     On	exit, if INFO =	0, WORK(1) returns the optimal LWORK.

     LWORK   (input) INTEGER
	     The length	of the array WORK.  LWORK >= max(1,2*N-1).  For
	     optimal efficiency, LWORK >= (NB+1)*N, where NB is	the blocksize
	     for CHETRD	returned by ILAENV.

     RWORK   (workspace) REAL array, dimension (7*N)

     IWORK   (workspace) INTEGER array,	dimension (5*N)

     IFAIL   (output) INTEGER array, dimension (N)
	     If	JOBZ = 'V', then if INFO = 0, the first	M elements of IFAIL
	     are zero.	If INFO	> 0, then IFAIL	contains the indices of	the
	     eigenvectors that failed to converge.  If JOBZ = 'N', then	IFAIL
	     is	not referenced.

     INFO    (output) INTEGER
	     = 0:  successful exit
	     < 0:  if INFO = -i, the i-th argument had an illegal value
	     > 0:  if INFO = i,	then i eigenvectors failed to converge.	 Their
	     indices are stored	in array IFAIL.
CHEEVX(3F)							    CHEEVX(3F)


NAME    [Toc]    [Back]

     CHEEVX - compute selected eigenvalues and,	optionally, eigenvectors of a
     complex Hermitian matrix A

SYNOPSIS    [Toc]    [Back]

     SUBROUTINE	CHEEVX(	JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU, ABSTOL,
			M, W, Z, LDZ, WORK, LWORK, RWORK, IWORK, IFAIL,	INFO )

	 CHARACTER	JOBZ, RANGE, UPLO

	 INTEGER	IL, INFO, IU, LDA, LDZ,	LWORK, M, N

	 REAL		ABSTOL,	VL, VU

	 INTEGER	IFAIL( * ), IWORK( * )

	 REAL		RWORK( * ), W( * )

	 COMPLEX	A( LDA,	* ), WORK( * ),	Z( LDZ,	* )

PURPOSE    [Toc]    [Back]

     CHEEVX computes selected eigenvalues and, optionally, eigenvectors	of a
     complex Hermitian matrix A.  Eigenvalues and eigenvectors can be selected
     by	specifying either a range of values or a range of indices for the
     desired eigenvalues.

ARGUMENTS    [Toc]    [Back]

     JOBZ    (input) CHARACTER*1
	     = 'N':  Compute eigenvalues only;
	     = 'V':  Compute eigenvalues and eigenvectors.

     RANGE   (input) CHARACTER*1
	     = 'A': all	eigenvalues will be found.
	     = 'V': all	eigenvalues in the half-open interval (VL,VU] will be
	     found.  = 'I': the	IL-th through IU-th eigenvalues	will be	found.

     UPLO    (input) CHARACTER*1
	     = 'U':  Upper triangle of A is stored;
	     = 'L':  Lower triangle of A is stored.

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

     A	     (input/output) COMPLEX array, dimension (LDA, N)
	     On	entry, the Hermitian matrix A.	If UPLO	= 'U', the leading Nby-N
 upper	triangular part	of A contains the upper	triangular
	     part of the matrix	A.  If UPLO = 'L', the leading N-by-N lower
	     triangular	part of	A contains the lower triangular	part of	the
	     matrix A.	On exit, the lower triangle (if	UPLO='L') or the upper
	     triangle (if UPLO='U') of A, including the	diagonal, is
	     destroyed.



									Page 1






CHEEVX(3F)							    CHEEVX(3F)



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

     VL	     (input) REAL
	     VU	     (input) REAL If RANGE='V',	the lower and upper bounds of
	     the interval to be	searched for eigenvalues. VL < VU.  Not
	     referenced	if RANGE = 'A' or 'I'.

     IL	     (input) INTEGER
	     IU	     (input) INTEGER If	RANGE='I', the indices (in ascending
	     order) of the smallest and	largest	eigenvalues to be returned.  1
	     <=	IL <= IU <= N, if N > 0; IL = 1	and IU = 0 if N	= 0.  Not
	     referenced	if RANGE = 'A' or 'V'.

     ABSTOL  (input) REAL
	     The absolute error	tolerance for the eigenvalues.	An approximate
	     eigenvalue	is accepted as converged when it is determined to lie
	     in	an interval [a,b] of width less	than or	equal to

	     ABSTOL + EPS *   max( |a|,|b| ) ,

	     where EPS is the machine precision.  If ABSTOL is less than or
	     equal to zero, then  EPS*|T|  will	be used	in its place, where
	     |T| is the	1-norm of the tridiagonal matrix obtained by reducing
	     A to tridiagonal form.

	     Eigenvalues will be computed most accurately when ABSTOL is set
	     to	twice the underflow threshold 2*SLAMCH('S'), not zero.	If
	     this routine returns with INFO>0, indicating that some
	     eigenvectors did not converge, try	setting	ABSTOL to
	     2*SLAMCH('S').

	     See "Computing Small Singular Values of Bidiagonal	Matrices with
	     Guaranteed	High Relative Accuracy," by Demmel and Kahan, LAPACK
	     Working Note #3.

     M	     (output) INTEGER
	     The total number of eigenvalues found.  0 <= M <= N.  If RANGE =
	     'A', M = N, and if	RANGE =	'I', M = IU-IL+1.

     W	     (output) REAL array, dimension (N)
	     On	normal exit, the first M elements contain the selected
	     eigenvalues in ascending order.

     Z	     (output) COMPLEX array, dimension (LDZ, max(1,M))
	     If	JOBZ = 'V', then if INFO = 0, the first	M columns of Z contain
	     the orthonormal eigenvectors of the matrix	A corresponding	to the
	     selected eigenvalues, with	the i-th column	of Z holding the
	     eigenvector associated with W(i).	If an eigenvector fails	to
	     converge, then that column	of Z contains the latest approximation
	     to	the eigenvector, and the index of the eigenvector is returned
	     in	IFAIL.	If JOBZ	= 'N', then Z is not referenced.  Note:	the



									Page 2






CHEEVX(3F)							    CHEEVX(3F)



	     user must ensure that at least max(1,M) columns are supplied in
	     the array Z; if RANGE = 'V', the exact value of M is not known in
	     advance and an upper bound	must be	used.

     LDZ     (input) INTEGER
	     The leading dimension of the array	Z.  LDZ	>= 1, and if JOBZ =
	     'V', LDZ >= max(1,N).

     WORK    (workspace/output)	COMPLEX	array, dimension (LWORK)
	     On	exit, if INFO =	0, WORK(1) returns the optimal LWORK.

     LWORK   (input) INTEGER
	     The length	of the array WORK.  LWORK >= max(1,2*N-1).  For
	     optimal efficiency, LWORK >= (NB+1)*N, where NB is	the blocksize
	     for CHETRD	returned by ILAENV.

     RWORK   (workspace) REAL array, dimension (7*N)

     IWORK   (workspace) INTEGER array,	dimension (5*N)

     IFAIL   (output) INTEGER array, dimension (N)
	     If	JOBZ = 'V', then if INFO = 0, the first	M elements of IFAIL
	     are zero.	If INFO	> 0, then IFAIL	contains the indices of	the
	     eigenvectors that failed to converge.  If JOBZ = 'N', then	IFAIL
	     is	not referenced.

     INFO    (output) INTEGER
	     = 0:  successful exit
	     < 0:  if INFO = -i, the i-th argument had an illegal value
	     > 0:  if INFO = i,	then i eigenvectors failed to converge.	 Their
	     indices are stored	in array IFAIL.


									PPPPaaaaggggeeee 3333
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