*> \brief \b ZHBGVD * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download ZHBGVD + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE ZHBGVD( JOBZ, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, W, * Z, LDZ, WORK, LWORK, RWORK, LRWORK, IWORK, * LIWORK, INFO ) * * .. Scalar Arguments .. * CHARACTER JOBZ, UPLO * INTEGER INFO, KA, KB, LDAB, LDBB, LDZ, LIWORK, LRWORK, * $ LWORK, N * .. * .. Array Arguments .. * INTEGER IWORK( * ) * DOUBLE PRECISION RWORK( * ), W( * ) * COMPLEX*16 AB( LDAB, * ), BB( LDBB, * ), WORK( * ), * $ Z( LDZ, * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> ZHBGVD computes all the eigenvalues, and optionally, the eigenvectors *> of a complex generalized Hermitian-definite banded eigenproblem, of *> the form A*x=(lambda)*B*x. Here A and B are assumed to be Hermitian *> and banded, and B is also positive definite. 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 triangles of A and B are stored; *> = 'L': Lower triangles of A and B are stored. *> \endverbatim *> *> \param[in] N *> \verbatim *> N is INTEGER *> The order of the matrices A and B. N >= 0. *> \endverbatim *> *> \param[in] KA *> \verbatim *> KA is INTEGER *> The number of superdiagonals of the matrix A if UPLO = 'U', *> or the number of subdiagonals if UPLO = 'L'. KA >= 0. *> \endverbatim *> *> \param[in] KB *> \verbatim *> KB is INTEGER *> The number of superdiagonals of the matrix B if UPLO = 'U', *> or the number of subdiagonals if UPLO = 'L'. KB >= 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 ka+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(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j; *> if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+ka). *> *> On exit, the contents of AB are destroyed. *> \endverbatim *> *> \param[in] LDAB *> \verbatim *> LDAB is INTEGER *> The leading dimension of the array AB. LDAB >= KA+1. *> \endverbatim *> *> \param[in,out] BB *> \verbatim *> BB is COMPLEX*16 array, dimension (LDBB, N) *> On entry, the upper or lower triangle of the Hermitian band *> matrix B, stored in the first kb+1 rows of the array. The *> j-th column of B is stored in the j-th column of the array BB *> as follows: *> if UPLO = 'U', BB(kb+1+i-j,j) = B(i,j) for max(1,j-kb)<=i<=j; *> if UPLO = 'L', BB(1+i-j,j) = B(i,j) for j<=i<=min(n,j+kb). *> *> On exit, the factor S from the split Cholesky factorization *> B = S**H*S, as returned by ZPBSTF. *> \endverbatim *> *> \param[in] LDBB *> \verbatim *> LDBB is INTEGER *> The leading dimension of the array BB. LDBB >= KB+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 matrix Z of *> eigenvectors, with the i-th column of Z holding the *> eigenvector associated with W(i). The eigenvectors are *> normalized so that Z**H*B*Z = 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 >= 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 >= 1. *> If JOBZ = 'N' and N > 1, LWORK >= N. *> If JOBZ = 'V' and N > 1, LWORK >= 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 (MAX(1,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 >= 1. *> If JOBZ = 'N' and N > 1, LRWORK >= N. *> If JOBZ = 'V' and N > 1, LRWORK >= 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 >= 1. *> If JOBZ = 'V' and N > 1, LIWORK >= 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, and i is: *> <= N: the algorithm failed to converge: *> i off-diagonal elements of an intermediate *> tridiagonal form did not converge to zero; *> > N: if INFO = N + i, for 1 <= i <= N, then ZPBSTF *> returned INFO = i: B is not positive definite. *> The factorization of B could not be completed and *> no eigenvalues or eigenvectors were computed. *> \endverbatim * * Authors: * ======== * *> \author Univ. of Tennessee *> \author Univ. of California Berkeley *> \author Univ. of Colorado Denver *> \author NAG Ltd. * *> \date June 2016 * *> \ingroup complex16OTHEReigen * *> \par Contributors: * ================== *> *> Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA * * ===================================================================== SUBROUTINE ZHBGVD( JOBZ, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, W, $ Z, LDZ, WORK, LWORK, RWORK, LRWORK, IWORK, $ LIWORK, INFO ) * * -- LAPACK driver routine (version 3.6.1) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * June 2016 * * .. Scalar Arguments .. CHARACTER JOBZ, UPLO INTEGER INFO, KA, KB, LDAB, LDBB, LDZ, LIWORK, LRWORK, $ LWORK, N * .. * .. Array Arguments .. INTEGER IWORK( * ) DOUBLE PRECISION RWORK( * ), W( * ) COMPLEX*16 AB( LDAB, * ), BB( LDBB, * ), WORK( * ), $ Z( LDZ, * ) * .. * * ===================================================================== * * .. Parameters .. COMPLEX*16 CONE, CZERO PARAMETER ( CONE = ( 1.0D+0, 0.0D+0 ), $ CZERO = ( 0.0D+0, 0.0D+0 ) ) * .. * .. Local Scalars .. LOGICAL LQUERY, UPPER, WANTZ CHARACTER VECT INTEGER IINFO, INDE, INDWK2, INDWRK, LIWMIN, LLRWK, $ LLWK2, LRWMIN, LWMIN * .. * .. External Functions .. LOGICAL LSAME EXTERNAL LSAME * .. * .. External Subroutines .. EXTERNAL DSTERF, XERBLA, ZGEMM, ZHBGST, ZHBTRD, ZLACPY, $ ZPBSTF, ZSTEDC * .. * .. Executable Statements .. * * Test the input parameters. * WANTZ = LSAME( JOBZ, 'V' ) UPPER = LSAME( UPLO, 'U' ) LQUERY = ( LWORK.EQ.-1 .OR. LRWORK.EQ.-1 .OR. LIWORK.EQ.-1 ) * INFO = 0 IF( N.LE.1 ) THEN LWMIN = 1+N LRWMIN = 1+N 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 IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN INFO = -1 ELSE IF( .NOT.( UPPER .OR. LSAME( UPLO, 'L' ) ) ) THEN INFO = -2 ELSE IF( N.LT.0 ) THEN INFO = -3 ELSE IF( KA.LT.0 ) THEN INFO = -4 ELSE IF( KB.LT.0 .OR. KB.GT.KA ) THEN INFO = -5 ELSE IF( LDAB.LT.KA+1 ) THEN INFO = -7 ELSE IF( LDBB.LT.KB+1 ) THEN INFO = -9 ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN INFO = -12 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 = -14 ELSE IF( LRWORK.LT.LRWMIN .AND. .NOT.LQUERY ) THEN INFO = -16 ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN INFO = -18 END IF END IF * IF( INFO.NE.0 ) THEN CALL XERBLA( 'ZHBGVD', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF * * Quick return if possible * IF( N.EQ.0 ) $ RETURN * * Form a split Cholesky factorization of B. * CALL ZPBSTF( UPLO, N, KB, BB, LDBB, INFO ) IF( INFO.NE.0 ) THEN INFO = N + INFO RETURN END IF * * Transform problem to standard eigenvalue problem. * INDE = 1 INDWRK = INDE + N INDWK2 = 1 + N*N LLWK2 = LWORK - INDWK2 + 2 LLRWK = LRWORK - INDWRK + 2 CALL ZHBGST( JOBZ, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, Z, LDZ, $ WORK, RWORK, IINFO ) * * Reduce Hermitian band matrix to tridiagonal form. * IF( WANTZ ) THEN VECT = 'U' ELSE VECT = 'N' END IF CALL ZHBTRD( VECT, UPLO, N, KA, 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 * WORK( 1 ) = LWMIN RWORK( 1 ) = LRWMIN IWORK( 1 ) = LIWMIN RETURN * * End of ZHBGVD * END