*> \brief CHEEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for HE matrices
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download CHEEVX + dependencies
*>
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*>
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*>
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*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE CHEEVX( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU,
* ABSTOL, M, W, Z, LDZ, WORK, LWORK, RWORK,
* IWORK, IFAIL, INFO )
*
* .. Scalar Arguments ..
* CHARACTER JOBZ, RANGE, UPLO
* INTEGER IL, INFO, IU, LDA, LDZ, LWORK, M, N
* REAL ABSTOL, VL, VU
* ..
* .. Array Arguments ..
* INTEGER IFAIL( * ), IWORK( * )
* REAL RWORK( * ), W( * )
* COMPLEX A( LDA, * ), WORK( * ), Z( LDZ, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> 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.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] JOBZ
*> \verbatim
*> JOBZ is CHARACTER*1
*> = 'N': Compute eigenvalues only;
*> = 'V': Compute eigenvalues and eigenvectors.
*> \endverbatim
*>
*> \param[in] RANGE
*> \verbatim
*> RANGE is 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.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': Upper triangle of A is stored;
*> = 'L': Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is COMPLEX array, dimension (LDA, N)
*> On entry, the Hermitian matrix A. If UPLO = 'U', the
*> leading N-by-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.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in] VL
*> \verbatim
*> VL is REAL
*> \endverbatim
*>
*> \param[in] VU
*> \verbatim
*> VU is 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'.
*> \endverbatim
*>
*> \param[in] IL
*> \verbatim
*> IL is INTEGER
*> \endverbatim
*>
*> \param[in] IU
*> \verbatim
*> IU is 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'.
*> \endverbatim
*>
*> \param[in] ABSTOL
*> \verbatim
*> ABSTOL is 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.
*> \endverbatim
*>
*> \param[out] M
*> \verbatim
*> M is INTEGER
*> The total number of eigenvalues found. 0 <= M <= N.
*> If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*> W is REAL array, dimension (N)
*> On normal exit, the first M elements contain the selected
*> eigenvalues in ascending order.
*> \endverbatim
*>
*> \param[out] Z
*> \verbatim
*> Z is 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 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.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*> LDZ is INTEGER
*> The leading dimension of the array Z. LDZ >= 1, and if
*> JOBZ = 'V', LDZ >= max(1,N).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is COMPLEX array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The length of the array WORK. LWORK >= 1, when N <= 1;
*> otherwise 2*N.
*> For optimal efficiency, LWORK >= (NB+1)*N,
*> where NB is the max of the blocksize for CHETRD and for
*> CUNMTR as returned by ILAENV.
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] RWORK
*> \verbatim
*> RWORK is REAL array, dimension (7*N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (5*N)
*> \endverbatim
*>
*> \param[out] IFAIL
*> \verbatim
*> IFAIL is 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.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is 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.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2011
*
*> \ingroup complexHEeigen
*
* =====================================================================
SUBROUTINE CHEEVX( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU,
$ ABSTOL, M, W, Z, LDZ, WORK, LWORK, RWORK,
$ IWORK, IFAIL, INFO )
*
* -- LAPACK driver routine (version 3.4.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2011
*
* .. Scalar Arguments ..
CHARACTER JOBZ, RANGE, UPLO
INTEGER IL, INFO, IU, LDA, LDZ, LWORK, M, N
REAL ABSTOL, VL, VU
* ..
* .. Array Arguments ..
INTEGER IFAIL( * ), IWORK( * )
REAL RWORK( * ), W( * )
COMPLEX A( LDA, * ), WORK( * ), Z( LDZ, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
REAL ZERO, ONE
PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 )
COMPLEX CONE
PARAMETER ( CONE = ( 1.0E+0, 0.0E+0 ) )
* ..
* .. Local Scalars ..
LOGICAL ALLEIG, INDEIG, LOWER, LQUERY, TEST, VALEIG,
$ WANTZ
CHARACTER ORDER
INTEGER I, IINFO, IMAX, INDD, INDE, INDEE, INDIBL,
$ INDISP, INDIWK, INDRWK, INDTAU, INDWRK, ISCALE,
$ ITMP1, J, JJ, LLWORK, LWKMIN, LWKOPT, NB,
$ NSPLIT
REAL ABSTLL, ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN,
$ SIGMA, SMLNUM, TMP1, VLL, VUU
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
REAL SLAMCH, CLANHE
EXTERNAL LSAME, ILAENV, SLAMCH, CLANHE
* ..
* .. External Subroutines ..
EXTERNAL SCOPY, SSCAL, SSTEBZ, SSTERF, XERBLA, CSSCAL,
$ CHETRD, CLACPY, CSTEIN, CSTEQR, CSWAP, CUNGTR,
$ CUNMTR
* ..
* .. Intrinsic Functions ..
INTRINSIC REAL, MAX, MIN, SQRT
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
LOWER = LSAME( UPLO, 'L' )
WANTZ = LSAME( JOBZ, 'V' )
ALLEIG = LSAME( RANGE, 'A' )
VALEIG = LSAME( RANGE, 'V' )
INDEIG = LSAME( RANGE, 'I' )
LQUERY = ( LWORK.EQ.-1 )
*
INFO = 0
IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
INFO = -1
ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
INFO = -2
ELSE IF( .NOT.( LOWER .OR. LSAME( UPLO, 'U' ) ) ) THEN
INFO = -3
ELSE IF( N.LT.0 ) THEN
INFO = -4
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -6
ELSE
IF( VALEIG ) THEN
IF( N.GT.0 .AND. VU.LE.VL )
$ INFO = -8
ELSE IF( INDEIG ) THEN
IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN
INFO = -9
ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN
INFO = -10
END IF
END IF
END IF
IF( INFO.EQ.0 ) THEN
IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
INFO = -15
END IF
END IF
*
IF( INFO.EQ.0 ) THEN
IF( N.LE.1 ) THEN
LWKMIN = 1
WORK( 1 ) = LWKMIN
ELSE
LWKMIN = 2*N
NB = ILAENV( 1, 'CHETRD', UPLO, N, -1, -1, -1 )
NB = MAX( NB, ILAENV( 1, 'CUNMTR', UPLO, N, -1, -1, -1 ) )
LWKOPT = MAX( 1, ( NB + 1 )*N )
WORK( 1 ) = LWKOPT
END IF
*
IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY )
$ INFO = -17
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'CHEEVX', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
M = 0
IF( N.EQ.0 ) THEN
RETURN
END IF
*
IF( N.EQ.1 ) THEN
IF( ALLEIG .OR. INDEIG ) THEN
M = 1
W( 1 ) = A( 1, 1 )
ELSE IF( VALEIG ) THEN
IF( VL.LT.REAL( A( 1, 1 ) ) .AND. VU.GE.REAL( A( 1, 1 ) ) )
$ THEN
M = 1
W( 1 ) = A( 1, 1 )
END IF
END IF
IF( WANTZ )
$ Z( 1, 1 ) = CONE
RETURN
END IF
*
* Get machine constants.
*
SAFMIN = SLAMCH( 'Safe minimum' )
EPS = SLAMCH( 'Precision' )
SMLNUM = SAFMIN / EPS
BIGNUM = ONE / SMLNUM
RMIN = SQRT( SMLNUM )
RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) )
*
* Scale matrix to allowable range, if necessary.
*
ISCALE = 0
ABSTLL = ABSTOL
IF( VALEIG ) THEN
VLL = VL
VUU = VU
END IF
ANRM = CLANHE( 'M', UPLO, N, A, LDA, RWORK )
IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN
ISCALE = 1
SIGMA = RMIN / ANRM
ELSE IF( ANRM.GT.RMAX ) THEN
ISCALE = 1
SIGMA = RMAX / ANRM
END IF
IF( ISCALE.EQ.1 ) THEN
IF( LOWER ) THEN
DO 10 J = 1, N
CALL CSSCAL( N-J+1, SIGMA, A( J, J ), 1 )
10 CONTINUE
ELSE
DO 20 J = 1, N
CALL CSSCAL( J, SIGMA, A( 1, J ), 1 )
20 CONTINUE
END IF
IF( ABSTOL.GT.0 )
$ ABSTLL = ABSTOL*SIGMA
IF( VALEIG ) THEN
VLL = VL*SIGMA
VUU = VU*SIGMA
END IF
END IF
*
* Call CHETRD to reduce Hermitian matrix to tridiagonal form.
*
INDD = 1
INDE = INDD + N
INDRWK = INDE + N
INDTAU = 1
INDWRK = INDTAU + N
LLWORK = LWORK - INDWRK + 1
CALL CHETRD( UPLO, N, A, LDA, RWORK( INDD ), RWORK( INDE ),
$ WORK( INDTAU ), WORK( INDWRK ), LLWORK, IINFO )
*
* If all eigenvalues are desired and ABSTOL is less than or equal to
* zero, then call SSTERF or CUNGTR and CSTEQR. If this fails for
* some eigenvalue, then try SSTEBZ.
*
TEST = .FALSE.
IF( INDEIG ) THEN
IF( IL.EQ.1 .AND. IU.EQ.N ) THEN
TEST = .TRUE.
END IF
END IF
IF( ( ALLEIG .OR. TEST ) .AND. ( ABSTOL.LE.ZERO ) ) THEN
CALL SCOPY( N, RWORK( INDD ), 1, W, 1 )
INDEE = INDRWK + 2*N
IF( .NOT.WANTZ ) THEN
CALL SCOPY( N-1, RWORK( INDE ), 1, RWORK( INDEE ), 1 )
CALL SSTERF( N, W, RWORK( INDEE ), INFO )
ELSE
CALL CLACPY( 'A', N, N, A, LDA, Z, LDZ )
CALL CUNGTR( UPLO, N, Z, LDZ, WORK( INDTAU ),
$ WORK( INDWRK ), LLWORK, IINFO )
CALL SCOPY( N-1, RWORK( INDE ), 1, RWORK( INDEE ), 1 )
CALL CSTEQR( JOBZ, N, W, RWORK( INDEE ), Z, LDZ,
$ RWORK( INDRWK ), INFO )
IF( INFO.EQ.0 ) THEN
DO 30 I = 1, N
IFAIL( I ) = 0
30 CONTINUE
END IF
END IF
IF( INFO.EQ.0 ) THEN
M = N
GO TO 40
END IF
INFO = 0
END IF
*
* Otherwise, call SSTEBZ and, if eigenvectors are desired, CSTEIN.
*
IF( WANTZ ) THEN
ORDER = 'B'
ELSE
ORDER = 'E'
END IF
INDIBL = 1
INDISP = INDIBL + N
INDIWK = INDISP + N
CALL SSTEBZ( RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTLL,
$ RWORK( INDD ), RWORK( INDE ), M, NSPLIT, W,
$ IWORK( INDIBL ), IWORK( INDISP ), RWORK( INDRWK ),
$ IWORK( INDIWK ), INFO )
*
IF( WANTZ ) THEN
CALL CSTEIN( N, RWORK( INDD ), RWORK( INDE ), M, W,
$ IWORK( INDIBL ), IWORK( INDISP ), Z, LDZ,
$ RWORK( INDRWK ), IWORK( INDIWK ), IFAIL, INFO )
*
* Apply unitary matrix used in reduction to tridiagonal
* form to eigenvectors returned by CSTEIN.
*
CALL CUNMTR( 'L', UPLO, 'N', N, M, A, LDA, WORK( INDTAU ), Z,
$ LDZ, WORK( INDWRK ), LLWORK, IINFO )
END IF
*
* If matrix was scaled, then rescale eigenvalues appropriately.
*
40 CONTINUE
IF( ISCALE.EQ.1 ) THEN
IF( INFO.EQ.0 ) THEN
IMAX = M
ELSE
IMAX = INFO - 1
END IF
CALL SSCAL( IMAX, ONE / SIGMA, W, 1 )
END IF
*
* If eigenvalues are not in order, then sort them, along with
* eigenvectors.
*
IF( WANTZ ) THEN
DO 60 J = 1, M - 1
I = 0
TMP1 = W( J )
DO 50 JJ = J + 1, M
IF( W( JJ ).LT.TMP1 ) THEN
I = JJ
TMP1 = W( JJ )
END IF
50 CONTINUE
*
IF( I.NE.0 ) THEN
ITMP1 = IWORK( INDIBL+I-1 )
W( I ) = W( J )
IWORK( INDIBL+I-1 ) = IWORK( INDIBL+J-1 )
W( J ) = TMP1
IWORK( INDIBL+J-1 ) = ITMP1
CALL CSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 )
IF( INFO.NE.0 ) THEN
ITMP1 = IFAIL( I )
IFAIL( I ) = IFAIL( J )
IFAIL( J ) = ITMP1
END IF
END IF
60 CONTINUE
END IF
*
* Set WORK(1) to optimal complex workspace size.
*
WORK( 1 ) = LWKOPT
*
RETURN
*
* End of CHEEVX
*
END