*> \brief \b ZHBGVX * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download ZHBGVX + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE ZHBGVX( JOBZ, RANGE, UPLO, N, KA, KB, AB, LDAB, BB, * LDBB, Q, LDQ, VL, VU, IL, IU, ABSTOL, M, W, Z, * LDZ, WORK, RWORK, IWORK, IFAIL, INFO ) * * .. Scalar Arguments .. * CHARACTER JOBZ, RANGE, UPLO * INTEGER IL, INFO, IU, KA, KB, LDAB, LDBB, LDQ, LDZ, M, * $ N * DOUBLE PRECISION ABSTOL, VL, VU * .. * .. Array Arguments .. * INTEGER IFAIL( * ), IWORK( * ) * DOUBLE PRECISION RWORK( * ), W( * ) * COMPLEX*16 AB( LDAB, * ), BB( LDBB, * ), Q( LDQ, * ), * $ WORK( * ), Z( LDZ, * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> ZHBGVX 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. Eigenvalues and *> eigenvectors can be selected by specifying either all eigenvalues, *> 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 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] Q *> \verbatim *> Q is COMPLEX*16 array, dimension (LDQ, N) *> If JOBZ = 'V', the n-by-n matrix used in the reduction of *> A*x = (lambda)*B*x to standard form, i.e. C*x = (lambda)*x, *> and consequently C to tridiagonal form. *> If JOBZ = 'N', the array Q is not referenced. *> \endverbatim *> *> \param[in] LDQ *> \verbatim *> LDQ is INTEGER *> The leading dimension of the array Q. If JOBZ = 'N', *> LDQ >= 1. If JOBZ = 'V', LDQ >= max(1,N). *> \endverbatim *> *> \param[in] VL *> \verbatim *> VL is DOUBLE PRECISION *> *> If RANGE='V', the lower bound of the interval to *> be searched for eigenvalues. VL < VU. *> Not referenced if RANGE = 'A' or 'I'. *> \endverbatim *> *> \param[in] VU *> \verbatim *> VU is DOUBLE PRECISION *> *> If RANGE='V', the upper bound 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 *> *> If RANGE='I', the index of the *> smallest eigenvalue 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] IU *> \verbatim *> IU is INTEGER *> *> If RANGE='I', the index of the *> largest eigenvalue 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 DOUBLE PRECISION *> 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 AP to tridiagonal form. *> *> Eigenvalues will be computed most accurately when ABSTOL is *> set to twice the underflow threshold 2*DLAMCH('S'), not zero. *> If this routine returns with INFO>0, indicating that some *> eigenvectors did not converge, try setting ABSTOL to *> 2*DLAMCH('S'). *> \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 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 (N) *> \endverbatim *> *> \param[out] RWORK *> \verbatim *> RWORK is DOUBLE PRECISION 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, and i is: *> <= N: then i eigenvectors failed to converge. Their *> indices are stored in array IFAIL. *> > 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 ZHBGVX( JOBZ, RANGE, UPLO, N, KA, KB, AB, LDAB, BB, $ LDBB, Q, LDQ, VL, VU, IL, IU, ABSTOL, M, W, Z, $ LDZ, WORK, RWORK, IWORK, IFAIL, 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, RANGE, UPLO INTEGER IL, INFO, IU, KA, KB, LDAB, LDBB, LDQ, LDZ, M, $ N DOUBLE PRECISION ABSTOL, VL, VU * .. * .. Array Arguments .. INTEGER IFAIL( * ), IWORK( * ) DOUBLE PRECISION RWORK( * ), W( * ) COMPLEX*16 AB( LDAB, * ), BB( LDBB, * ), Q( LDQ, * ), $ WORK( * ), Z( LDZ, * ) * .. * * ===================================================================== * * .. Parameters .. DOUBLE PRECISION ZERO PARAMETER ( ZERO = 0.0D+0 ) COMPLEX*16 CZERO, CONE PARAMETER ( CZERO = ( 0.0D+0, 0.0D+0 ), $ CONE = ( 1.0D+0, 0.0D+0 ) ) * .. * .. Local Scalars .. LOGICAL ALLEIG, INDEIG, TEST, UPPER, VALEIG, WANTZ CHARACTER ORDER, VECT INTEGER I, IINFO, INDD, INDE, INDEE, INDIBL, INDISP, $ INDIWK, INDRWK, INDWRK, ITMP1, J, JJ, NSPLIT DOUBLE PRECISION TMP1 * .. * .. External Functions .. LOGICAL LSAME EXTERNAL LSAME * .. * .. External Subroutines .. EXTERNAL DCOPY, DSTEBZ, DSTERF, XERBLA, ZCOPY, ZGEMV, $ ZHBGST, ZHBTRD, ZLACPY, ZPBSTF, ZSTEIN, ZSTEQR, $ ZSWAP * .. * .. Intrinsic Functions .. INTRINSIC MIN * .. * .. Executable Statements .. * * Test the input parameters. * WANTZ = LSAME( JOBZ, 'V' ) UPPER = LSAME( UPLO, 'U' ) ALLEIG = LSAME( RANGE, 'A' ) VALEIG = LSAME( RANGE, 'V' ) INDEIG = LSAME( RANGE, 'I' ) * 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.( UPPER .OR. LSAME( UPLO, 'L' ) ) ) THEN INFO = -3 ELSE IF( N.LT.0 ) THEN INFO = -4 ELSE IF( KA.LT.0 ) THEN INFO = -5 ELSE IF( KB.LT.0 .OR. KB.GT.KA ) THEN INFO = -6 ELSE IF( LDAB.LT.KA+1 ) THEN INFO = -8 ELSE IF( LDBB.LT.KB+1 ) THEN INFO = -10 ELSE IF( LDQ.LT.1 .OR. ( WANTZ .AND. LDQ.LT.N ) ) THEN INFO = -12 ELSE IF( VALEIG ) THEN IF( N.GT.0 .AND. VU.LE.VL ) $ INFO = -14 ELSE IF( INDEIG ) THEN IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN INFO = -15 ELSE IF ( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN INFO = -16 END IF END IF END IF IF( INFO.EQ.0) THEN IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN INFO = -21 END IF END IF * IF( INFO.NE.0 ) THEN CALL XERBLA( 'ZHBGVX', -INFO ) RETURN END IF * * Quick return if possible * M = 0 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. * CALL ZHBGST( JOBZ, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, Q, LDQ, $ WORK, RWORK, IINFO ) * * Solve the standard eigenvalue problem. * Reduce Hermitian band matrix to tridiagonal form. * INDD = 1 INDE = INDD + N INDRWK = INDE + N INDWRK = 1 IF( WANTZ ) THEN VECT = 'U' ELSE VECT = 'N' END IF CALL ZHBTRD( VECT, UPLO, N, KA, AB, LDAB, RWORK( INDD ), $ RWORK( INDE ), Q, LDQ, WORK( INDWRK ), IINFO ) * * If all eigenvalues are desired and ABSTOL is less than or equal * to zero, then call DSTERF or ZSTEQR. If this fails for some * eigenvalue, then try DSTEBZ. * 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 DCOPY( N, RWORK( INDD ), 1, W, 1 ) INDEE = INDRWK + 2*N CALL DCOPY( N-1, RWORK( INDE ), 1, RWORK( INDEE ), 1 ) IF( .NOT.WANTZ ) THEN CALL DSTERF( N, W, RWORK( INDEE ), INFO ) ELSE CALL ZLACPY( 'A', N, N, Q, LDQ, Z, LDZ ) CALL ZSTEQR( JOBZ, N, W, RWORK( INDEE ), Z, LDZ, $ RWORK( INDRWK ), INFO ) IF( INFO.EQ.0 ) THEN DO 10 I = 1, N IFAIL( I ) = 0 10 CONTINUE END IF END IF IF( INFO.EQ.0 ) THEN M = N GO TO 30 END IF INFO = 0 END IF * * Otherwise, call DSTEBZ and, if eigenvectors are desired, * call ZSTEIN. * IF( WANTZ ) THEN ORDER = 'B' ELSE ORDER = 'E' END IF INDIBL = 1 INDISP = INDIBL + N INDIWK = INDISP + N CALL DSTEBZ( RANGE, ORDER, N, VL, VU, IL, IU, ABSTOL, $ RWORK( INDD ), RWORK( INDE ), M, NSPLIT, W, $ IWORK( INDIBL ), IWORK( INDISP ), RWORK( INDRWK ), $ IWORK( INDIWK ), INFO ) * IF( WANTZ ) THEN CALL ZSTEIN( 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 ZSTEIN. * DO 20 J = 1, M CALL ZCOPY( N, Z( 1, J ), 1, WORK( 1 ), 1 ) CALL ZGEMV( 'N', N, N, CONE, Q, LDQ, WORK, 1, CZERO, $ Z( 1, J ), 1 ) 20 CONTINUE END IF * 30 CONTINUE * * If eigenvalues are not in order, then sort them, along with * eigenvectors. * IF( WANTZ ) THEN DO 50 J = 1, M - 1 I = 0 TMP1 = W( J ) DO 40 JJ = J + 1, M IF( W( JJ ).LT.TMP1 ) THEN I = JJ TMP1 = W( JJ ) END IF 40 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 ZSWAP( 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 50 CONTINUE END IF * RETURN * * End of ZHBGVX * END