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OpenBLAS/lapack-netlib/SRC/dspgvx.f

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*> \brief \b DSPGVX
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSPGVX + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dspgvx.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dspgvx.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dspgvx.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DSPGVX( ITYPE, JOBZ, RANGE, UPLO, N, AP, BP, VL, VU,
* IL, IU, ABSTOL, M, W, Z, LDZ, WORK, IWORK,
* IFAIL, INFO )
*
* .. Scalar Arguments ..
* CHARACTER JOBZ, RANGE, UPLO
* INTEGER IL, INFO, ITYPE, IU, LDZ, M, N
* DOUBLE PRECISION ABSTOL, VL, VU
* ..
* .. Array Arguments ..
* INTEGER IFAIL( * ), IWORK( * )
* DOUBLE PRECISION AP( * ), BP( * ), W( * ), WORK( * ),
* $ Z( LDZ, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DSPGVX 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, stored in packed storage, 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] AP
*> \verbatim
*> AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*> On entry, the upper or lower triangle of the symmetric matrix
*> A, packed columnwise in a linear array. The j-th column of A
*> is stored in the array AP as follows:
*> if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
*> if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
*>
*> On exit, the contents of AP are destroyed.
*> \endverbatim
*>
*> \param[in,out] BP
*> \verbatim
*> BP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*> On entry, the upper or lower triangle of the symmetric matrix
*> B, packed columnwise in a linear array. The j-th column of B
*> is stored in the array BP as follows:
*> if UPLO = 'U', BP(i + (j-1)*j/2) = B(i,j) for 1<=i<=j;
*> if UPLO = 'L', BP(i + (j-1)*(2*n-j)/2) = B(i,j) for j<=i<=n.
*>
*> On exit, the triangular factor U or L from the Cholesky
*> factorization B = U**T*U or B = L*L**T, in the same storage
*> format as B.
*> \endverbatim
*>
*> \param[in] VL
*> \verbatim
*> VL is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in] VU
*> \verbatim
*> VU is DOUBLE PRECISION
*>
*> 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 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 A 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)
*> 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 (8*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: DPPTRF or DSPEVX returned an error code:
*> <= N: if INFO = i, DSPEVX 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 November 2015
*
*> \ingroup doubleOTHEReigen
*
*> \par Contributors:
* ==================
*>
*> Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
*
* =====================================================================
SUBROUTINE DSPGVX( ITYPE, JOBZ, RANGE, UPLO, N, AP, BP, VL, VU,
$ IL, IU, ABSTOL, M, W, Z, LDZ, WORK, IWORK,
$ IFAIL, INFO )
*
* -- LAPACK driver routine (version 3.6.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2015
*
* .. Scalar Arguments ..
CHARACTER JOBZ, RANGE, UPLO
INTEGER IL, INFO, ITYPE, IU, LDZ, M, N
DOUBLE PRECISION ABSTOL, VL, VU
* ..
* .. Array Arguments ..
INTEGER IFAIL( * ), IWORK( * )
DOUBLE PRECISION AP( * ), BP( * ), W( * ), WORK( * ),
$ Z( LDZ, * )
* ..
*
* =====================================================================
*
* .. Local Scalars ..
LOGICAL ALLEIG, INDEIG, UPPER, VALEIG, WANTZ
CHARACTER TRANS
INTEGER J
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL DPPTRF, DSPEVX, DSPGST, DTPMV, DTPSV, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC 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' )
*
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( VALEIG ) THEN
IF( N.GT.0 .AND. VU.LE.VL ) THEN
INFO = -9
END IF
ELSE IF( INDEIG ) THEN
IF( IL.LT.1 ) THEN
INFO = -10
ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN
INFO = -11
END IF
END IF
END IF
IF( INFO.EQ.0 ) THEN
IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
INFO = -16
END IF
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DSPGVX', -INFO )
RETURN
END IF
*
* Quick return if possible
*
M = 0
IF( N.EQ.0 )
$ RETURN
*
* Form a Cholesky factorization of B.
*
CALL DPPTRF( UPLO, N, BP, INFO )
IF( INFO.NE.0 ) THEN
INFO = N + INFO
RETURN
END IF
*
* Transform problem to standard eigenvalue problem and solve.
*
CALL DSPGST( ITYPE, UPLO, N, AP, BP, INFO )
CALL DSPEVX( JOBZ, RANGE, UPLO, N, AP, VL, VU, IL, IU, ABSTOL, M,
$ W, Z, LDZ, WORK, 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
*
DO 10 J = 1, M
CALL DTPSV( UPLO, TRANS, 'Non-unit', N, BP, Z( 1, J ),
$ 1 )
10 CONTINUE
*
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
*
DO 20 J = 1, M
CALL DTPMV( UPLO, TRANS, 'Non-unit', N, BP, Z( 1, J ),
$ 1 )
20 CONTINUE
END IF
END IF
*
RETURN
*
* End of DSPGVX
*
END
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