*> \brief \b DSYGVX * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download DSYGVX + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE DSYGVX( ITYPE, JOBZ, RANGE, UPLO, N, A, LDA, B, LDB, * VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, * LWORK, IWORK, IFAIL, INFO ) * * .. Scalar Arguments .. * CHARACTER JOBZ, RANGE, UPLO * INTEGER IL, INFO, ITYPE, IU, LDA, LDB, LDZ, LWORK, M, N * DOUBLE PRECISION ABSTOL, VL, VU * .. * .. Array Arguments .. * INTEGER IFAIL( * ), IWORK( * ) * DOUBLE PRECISION A( LDA, * ), B( LDB, * ), W( * ), WORK( * ), * \$ Z( LDZ, * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> DSYGVX computes selected eigenvalues, and optionally, eigenvectors *> of a real generalized symmetric-definite eigenproblem, of the form *> A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x. Here A *> and B are assumed to be symmetric and B is also positive definite. *> 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] ITYPE *> \verbatim *> ITYPE is INTEGER *> Specifies the problem type to be solved: *> = 1: A*x = (lambda)*B*x *> = 2: A*B*x = (lambda)*x *> = 3: B*A*x = (lambda)*x *> \endverbatim *> *> \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 and B are stored; *> = 'L': Lower triangle of A and B are stored. *> \endverbatim *> *> \param[in] N *> \verbatim *> N is INTEGER *> The order of the matrix pencil (A,B). N >= 0. *> \endverbatim *> *> \param[in,out] A *> \verbatim *> A is DOUBLE PRECISION array, dimension (LDA, N) *> On entry, the symmetric 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,out] B *> \verbatim *> B is DOUBLE PRECISION array, dimension (LDB, N) *> On entry, the symmetric matrix B. If UPLO = 'U', the *> leading N-by-N upper triangular part of B contains the *> upper triangular part of the matrix B. If UPLO = 'L', *> the leading N-by-N lower triangular part of B contains *> the lower triangular part of the matrix B. *> *> On exit, if INFO <= N, the part of B containing the matrix is *> overwritten by the triangular factor U or L from the Cholesky *> factorization B = U**T*U or B = L*L**T. *> \endverbatim *> *> \param[in] LDB *> \verbatim *> LDB is INTEGER *> The leading dimension of the array B. LDB >= 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 C to tridiagonal form, where C is the symmetric *> matrix of the standard symmetric problem to which the *> generalized problem is transformed. *> *> 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) *> On normal exit, the first M elements contain the selected *> eigenvalues in ascending order. *> \endverbatim *> *> \param[out] Z *> \verbatim *> Z is DOUBLE PRECISION array, dimension (LDZ, max(1,M)) *> If JOBZ = 'N', then Z is not referenced. *> 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). *> The eigenvectors are normalized as follows: *> if ITYPE = 1 or 2, Z**T*B*Z = I; *> if ITYPE = 3, Z**T*inv(B)*Z = 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. *> 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 DOUBLE PRECISION 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 >= max(1,8*N). *> For optimal efficiency, LWORK >= (NB+3)*N, *> where NB is the blocksize for DSYTRD 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] 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: DPOTRF or DSYEVX returned an error code: *> <= N: if INFO = i, DSYEVX failed to converge; *> i eigenvectors failed to converge. Their indices *> are stored in array IFAIL. *> > N: if INFO = N + i, for 1 <= i <= N, then the leading *> minor of order i of 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 doubleSYeigen * *> \par Contributors: * ================== *> *> Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA * * ===================================================================== SUBROUTINE DSYGVX( ITYPE, JOBZ, RANGE, UPLO, N, A, LDA, B, LDB, \$ VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, \$ LWORK, IWORK, IFAIL, 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..-- * June 2016 * * .. Scalar Arguments .. CHARACTER JOBZ, RANGE, UPLO INTEGER IL, INFO, ITYPE, IU, LDA, LDB, LDZ, LWORK, M, N DOUBLE PRECISION ABSTOL, VL, VU * .. * .. Array Arguments .. INTEGER IFAIL( * ), IWORK( * ) DOUBLE PRECISION A( LDA, * ), B( LDB, * ), W( * ), WORK( * ), \$ Z( LDZ, * ) * .. * * ===================================================================== * * .. Parameters .. DOUBLE PRECISION ONE PARAMETER ( ONE = 1.0D+0 ) * .. * .. Local Scalars .. LOGICAL ALLEIG, INDEIG, LQUERY, UPPER, VALEIG, WANTZ CHARACTER TRANS INTEGER LWKMIN, LWKOPT, NB * .. * .. External Functions .. LOGICAL LSAME INTEGER ILAENV EXTERNAL LSAME, ILAENV * .. * .. External Subroutines .. EXTERNAL DPOTRF, DSYEVX, DSYGST, DTRMM, DTRSM, XERBLA * .. * .. Intrinsic Functions .. INTRINSIC MAX, MIN * .. * .. Executable Statements .. * * Test the input parameters. * UPPER = LSAME( UPLO, 'U' ) WANTZ = LSAME( JOBZ, 'V' ) ALLEIG = LSAME( RANGE, 'A' ) VALEIG = LSAME( RANGE, 'V' ) INDEIG = LSAME( RANGE, 'I' ) LQUERY = ( LWORK.EQ.-1 ) * INFO = 0 IF( ITYPE.LT.1 .OR. ITYPE.GT.3 ) THEN INFO = -1 ELSE IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN INFO = -2 ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN INFO = -3 ELSE IF( .NOT.( UPPER .OR. LSAME( UPLO, 'L' ) ) ) THEN INFO = -4 ELSE IF( N.LT.0 ) THEN INFO = -5 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -7 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -9 ELSE IF( VALEIG ) THEN IF( N.GT.0 .AND. VU.LE.VL ) \$ INFO = -11 ELSE IF( INDEIG ) THEN IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN INFO = -12 ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN INFO = -13 END IF END IF END IF IF (INFO.EQ.0) THEN IF (LDZ.LT.1 .OR. (WANTZ .AND. LDZ.LT.N)) THEN INFO = -18 END IF END IF * IF( INFO.EQ.0 ) THEN LWKMIN = MAX( 1, 8*N ) NB = ILAENV( 1, 'DSYTRD', UPLO, N, -1, -1, -1 ) LWKOPT = MAX( LWKMIN, ( NB + 3 )*N ) WORK( 1 ) = LWKOPT * IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY ) THEN INFO = -20 END IF END IF * IF( INFO.NE.0 ) THEN CALL XERBLA( 'DSYGVX', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF * * Quick return if possible * M = 0 IF( N.EQ.0 ) THEN RETURN END IF * * Form a Cholesky factorization of B. * CALL DPOTRF( UPLO, N, B, LDB, INFO ) IF( INFO.NE.0 ) THEN INFO = N + INFO RETURN END IF * * Transform problem to standard eigenvalue problem and solve. * CALL DSYGST( ITYPE, UPLO, N, A, LDA, B, LDB, INFO ) CALL DSYEVX( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU, ABSTOL, \$ M, W, Z, LDZ, WORK, LWORK, IWORK, IFAIL, INFO ) * IF( WANTZ ) THEN * * Backtransform eigenvectors to the original problem. * IF( INFO.GT.0 ) \$ M = INFO - 1 IF( ITYPE.EQ.1 .OR. ITYPE.EQ.2 ) THEN * * For A*x=(lambda)*B*x and A*B*x=(lambda)*x; * backtransform eigenvectors: x = inv(L)**T*y or inv(U)*y * IF( UPPER ) THEN TRANS = 'N' ELSE TRANS = 'T' END IF * CALL DTRSM( 'Left', UPLO, TRANS, 'Non-unit', N, M, ONE, B, \$ LDB, Z, LDZ ) * ELSE IF( ITYPE.EQ.3 ) THEN * * For B*A*x=(lambda)*x; * backtransform eigenvectors: x = L*y or U**T*y * IF( UPPER ) THEN TRANS = 'T' ELSE TRANS = 'N' END IF * CALL DTRMM( 'Left', UPLO, TRANS, 'Non-unit', N, M, ONE, B, \$ LDB, Z, LDZ ) END IF END IF * * Set WORK(1) to optimal workspace size. * WORK( 1 ) = LWKOPT * RETURN * * End of DSYGVX * END