*> \brief <b> ZHBEVD computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices</b>
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zhbevd.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE ZHBEVD( JOBZ, UPLO, N, KD, AB, LDAB, W, Z, LDZ, WORK,
* LWORK, RWORK, LRWORK, IWORK, LIWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER JOBZ, UPLO
* INTEGER INFO, KD, LDAB, LDZ, LIWORK, LRWORK, LWORK, N
* ..
* .. Array Arguments ..
* INTEGER IWORK( * )
* DOUBLE PRECISION RWORK( * ), W( * )
* COMPLEX*16 AB( LDAB, * ), WORK( * ), Z( LDZ, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> ZHBEVD computes all the eigenvalues and, optionally, eigenvectors of
*> a complex Hermitian band matrix A. If eigenvectors are desired, it
*> uses a divide and conquer algorithm.
*>
*> The divide and conquer algorithm makes very mild assumptions about
*> floating point arithmetic. It will work on machines with a guard
*> digit in add/subtract, or on those binary machines without guard
*> digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or
*> Cray-2. It could conceivably fail on hexadecimal or decimal machines
*> without guard digits, but we know of none.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] JOBZ
*> \verbatim
*> JOBZ is CHARACTER*1
*> = 'N': Compute eigenvalues only;
*> = 'V': Compute eigenvalues and eigenvectors.
*> \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] KD
*> \verbatim
*> KD is INTEGER
*> The number of superdiagonals of the matrix A if UPLO = 'U',
*> or the number of subdiagonals if UPLO = 'L'. KD >= 0.
*> \endverbatim
*>
*> \param[in,out] AB
*> \verbatim
*> AB is COMPLEX*16 array, dimension (LDAB, N)
*> On entry, the upper or lower triangle of the Hermitian band
*> matrix A, stored in the first KD+1 rows of the array. The
*> j-th column of A is stored in the j-th column of the array AB
*> as follows:
*> if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
*> if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
*>
*> On exit, AB is overwritten by values generated during the
*> reduction to tridiagonal form. If UPLO = 'U', the first
*> superdiagonal and the diagonal of the tridiagonal matrix T
*> are returned in rows KD and KD+1 of AB, and if UPLO = 'L',
*> the diagonal and first subdiagonal of T are returned in the
*> first two rows of AB.
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*> LDAB is INTEGER
*> The leading dimension of the array AB. LDAB >= KD + 1.
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*> W is DOUBLE PRECISION array, dimension (N)
*> If INFO = 0, the eigenvalues in ascending order.
*> \endverbatim
*>
*> \param[out] Z
*> \verbatim
*> Z is COMPLEX*16 array, dimension (LDZ, N)
*> If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal
*> eigenvectors of the matrix A, with the i-th column of Z
*> holding the eigenvector associated with W(i).
*> If JOBZ = 'N', then Z is not referenced.
*> \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*16 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 dimension of the array WORK.
*> If N <= 1, LWORK must be at least 1.
*> If JOBZ = 'N' and N > 1, LWORK must be at least N.
*> If JOBZ = 'V' and N > 1, LWORK must be at least 2*N**2.
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal sizes of the WORK, RWORK and
*> IWORK arrays, returns these values as the first entries of
*> the WORK, RWORK and IWORK arrays, and no error message
*> related to LWORK or LRWORK or LIWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] RWORK
*> \verbatim
*> RWORK is DOUBLE PRECISION array,
*> dimension (LRWORK)
*> On exit, if INFO = 0, RWORK(1) returns the optimal LRWORK.
*> \endverbatim
*>
*> \param[in] LRWORK
*> \verbatim
*> LRWORK is INTEGER
*> The dimension of array RWORK.
*> If N <= 1, LRWORK must be at least 1.
*> If JOBZ = 'N' and N > 1, LRWORK must be at least N.
*> If JOBZ = 'V' and N > 1, LRWORK must be at least
*> 1 + 5*N + 2*N**2.
*>
*> If LRWORK = -1, then a workspace query is assumed; the
*> routine only calculates the optimal sizes of the WORK, RWORK
*> and IWORK arrays, returns these values as the first entries
*> of the WORK, RWORK and IWORK arrays, and no error message
*> related to LWORK or LRWORK or LIWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (MAX(1,LIWORK))
*> On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
*> \endverbatim
*>
*> \param[in] LIWORK
*> \verbatim
*> LIWORK is INTEGER
*> The dimension of array IWORK.
*> If JOBZ = 'N' or N <= 1, LIWORK must be at least 1.
*> If JOBZ = 'V' and N > 1, LIWORK must be at least 3 + 5*N .
*>
*> If LIWORK = -1, then a workspace query is assumed; the
*> routine only calculates the optimal sizes of the WORK, RWORK
*> and IWORK arrays, returns these values as the first entries
*> of the WORK, RWORK and IWORK arrays, and no error message
*> related to LWORK or LRWORK or LIWORK is issued by XERBLA.
*> \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, the algorithm failed to converge; i
*> off-diagonal elements of an intermediate tridiagonal
*> form did not converge to zero.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date December 2016
*
*> \ingroup complex16OTHEReigen
*
* =====================================================================
SUBROUTINE ZHBEVD( JOBZ, UPLO, N, KD, AB, LDAB, W, Z, LDZ, WORK,
$ LWORK, RWORK, LRWORK, IWORK, LIWORK, INFO )
*
* -- LAPACK driver routine (version 3.7.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* December 2016
*
* .. Scalar Arguments ..
CHARACTER JOBZ, UPLO
INTEGER INFO, KD, LDAB, LDZ, LIWORK, LRWORK, LWORK, N
* ..
* .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION RWORK( * ), W( * )
COMPLEX*16 AB( LDAB, * ), WORK( * ), Z( LDZ, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
COMPLEX*16 CZERO, CONE
PARAMETER ( CZERO = ( 0.0D0, 0.0D0 ),
$ CONE = ( 1.0D0, 0.0D0 ) )
* ..
* .. Local Scalars ..
LOGICAL LOWER, LQUERY, WANTZ
INTEGER IINFO, IMAX, INDE, INDWK2, INDWRK, ISCALE,
$ LIWMIN, LLRWK, LLWK2, LRWMIN, LWMIN
DOUBLE PRECISION ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA,
$ SMLNUM
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DLAMCH, ZLANHB
EXTERNAL LSAME, DLAMCH, ZLANHB
* ..
* .. External Subroutines ..
EXTERNAL DSCAL, DSTERF, XERBLA, ZGEMM, ZHBTRD, ZLACPY,
$ ZLASCL, ZSTEDC
* ..
* .. Intrinsic Functions ..
INTRINSIC SQRT
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
WANTZ = LSAME( JOBZ, 'V' )
LOWER = LSAME( UPLO, 'L' )
LQUERY = ( LWORK.EQ.-1 .OR. LIWORK.EQ.-1 .OR. LRWORK.EQ.-1 )
*
INFO = 0
IF( N.LE.1 ) THEN
LWMIN = 1
LRWMIN = 1
LIWMIN = 1
ELSE
IF( WANTZ ) THEN
LWMIN = 2*N**2
LRWMIN = 1 + 5*N + 2*N**2
LIWMIN = 3 + 5*N
ELSE
LWMIN = N
LRWMIN = N
LIWMIN = 1
END IF
END IF
IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
INFO = -1
ELSE IF( .NOT.( LOWER .OR. LSAME( UPLO, 'U' ) ) ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( KD.LT.0 ) THEN
INFO = -4
ELSE IF( LDAB.LT.KD+1 ) THEN
INFO = -6
ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
INFO = -9
END IF
*
IF( INFO.EQ.0 ) THEN
WORK( 1 ) = LWMIN
RWORK( 1 ) = LRWMIN
IWORK( 1 ) = LIWMIN
*
IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
INFO = -11
ELSE IF( LRWORK.LT.LRWMIN .AND. .NOT.LQUERY ) THEN
INFO = -13
ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
INFO = -15
END IF
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'ZHBEVD', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
IF( N.EQ.1 ) THEN
W( 1 ) = AB( 1, 1 )
IF( WANTZ )
$ Z( 1, 1 ) = CONE
RETURN
END IF
*
* Get machine constants.
*
SAFMIN = DLAMCH( 'Safe minimum' )
EPS = DLAMCH( 'Precision' )
SMLNUM = SAFMIN / EPS
BIGNUM = ONE / SMLNUM
RMIN = SQRT( SMLNUM )
RMAX = SQRT( BIGNUM )
*
* Scale matrix to allowable range, if necessary.
*
ANRM = ZLANHB( 'M', UPLO, N, KD, AB, LDAB, RWORK )
ISCALE = 0
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
CALL ZLASCL( 'B', KD, KD, ONE, SIGMA, N, N, AB, LDAB, INFO )
ELSE
CALL ZLASCL( 'Q', KD, KD, ONE, SIGMA, N, N, AB, LDAB, INFO )
END IF
END IF
*
* Call ZHBTRD to reduce Hermitian band matrix to tridiagonal form.
*
INDE = 1
INDWRK = INDE + N
INDWK2 = 1 + N*N
LLWK2 = LWORK - INDWK2 + 1
LLRWK = LRWORK - INDWRK + 1
CALL ZHBTRD( JOBZ, UPLO, N, KD, AB, LDAB, W, RWORK( INDE ), Z,
$ LDZ, WORK, IINFO )
*
* For eigenvalues only, call DSTERF. For eigenvectors, call ZSTEDC.
*
IF( .NOT.WANTZ ) THEN
CALL DSTERF( N, W, RWORK( INDE ), INFO )
ELSE
CALL ZSTEDC( 'I', N, W, RWORK( INDE ), WORK, N, WORK( INDWK2 ),
$ LLWK2, RWORK( INDWRK ), LLRWK, IWORK, LIWORK,
$ INFO )
CALL ZGEMM( 'N', 'N', N, N, N, CONE, Z, LDZ, WORK, N, CZERO,
$ WORK( INDWK2 ), N )
CALL ZLACPY( 'A', N, N, WORK( INDWK2 ), N, Z, LDZ )
END IF
*
* If matrix was scaled, then rescale eigenvalues appropriately.
*
IF( ISCALE.EQ.1 ) THEN
IF( INFO.EQ.0 ) THEN
IMAX = N
ELSE
IMAX = INFO - 1
END IF
CALL DSCAL( IMAX, ONE / SIGMA, W, 1 )
END IF
*
WORK( 1 ) = LWMIN
RWORK( 1 ) = LRWMIN
IWORK( 1 ) = LIWMIN
RETURN
*
* End of ZHBEVD
*
END