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dsbgvx.f
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*> \brief \b DSBGVX
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSBGVX + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsbgvx.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsbgvx.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsbgvx.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DSBGVX( JOBZ, RANGE, UPLO, N, KA, KB, AB, LDAB, BB,
* LDBB, Q, LDQ, VL, VU, IL, IU, ABSTOL, M, W, Z,
* LDZ, WORK, 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 AB( LDAB, * ), BB( LDBB, * ), Q( LDQ, * ),
* $ W( * ), WORK( * ), Z( LDZ, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DSBGVX computes selected eigenvalues, and optionally, eigenvectors
*> of a real generalized symmetric-definite banded eigenproblem, of
*> the form A*x=(lambda)*B*x. Here A and B are assumed to be symmetric
*> 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 DOUBLE PRECISION array, dimension (LDAB, N)
*> On entry, the upper or lower triangle of the symmetric 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 DOUBLE PRECISION array, dimension (LDBB, N)
*> On entry, the upper or lower triangle of the symmetric 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(ka+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**T*S, as returned by DPBSTF.
*> \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 DOUBLE PRECISION 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 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)
*> If INFO = 0, the eigenvalues in ascending order.
*> \endverbatim
*>
*> \param[out] Z
*> \verbatim
*> Z is DOUBLE PRECISION 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 Z**T*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 >= max(1,N).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK 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 (M)
*> 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 eigenvalues 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
*> <= N: if INFO = i, then i eigenvectors failed to converge.
*> Their indices are stored in IFAIL.
*> > N: DPBSTF returned an error code; i.e.,
*> 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 doubleOTHEReigen
*
*> \par Contributors:
* ==================
*>
*> Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
*
* =====================================================================
SUBROUTINE DSBGVX( JOBZ, RANGE, UPLO, N, KA, KB, AB, LDAB, BB,
$ LDBB, Q, LDQ, VL, VU, IL, IU, ABSTOL, M, W, Z,
$ LDZ, WORK, 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, IU, KA, KB, LDAB, LDBB, LDQ, LDZ, M,
$ N
DOUBLE PRECISION ABSTOL, VL, VU
* ..
* .. Array Arguments ..
INTEGER IFAIL( * ), IWORK( * )
DOUBLE PRECISION AB( LDAB, * ), BB( LDBB, * ), Q( LDQ, * ),
$ W( * ), WORK( * ), Z( LDZ, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL ALLEIG, INDEIG, TEST, UPPER, VALEIG, WANTZ
CHARACTER ORDER, VECT
INTEGER I, IINFO, INDD, INDE, INDEE, INDIBL, INDISP,
$ INDIWO, INDWRK, ITMP1, J, JJ, NSPLIT
DOUBLE PRECISION TMP1
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DGEMV, DLACPY, DPBSTF, DSBGST, DSBTRD,
$ DSTEBZ, DSTEIN, DSTEQR, DSTERF, DSWAP, XERBLA
* ..
* .. 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( 'DSBGVX', -INFO )
RETURN
END IF
*
* Quick return if possible
*
M = 0
IF( N.EQ.0 )
$ RETURN
*
* Form a split Cholesky factorization of B.
*
CALL DPBSTF( UPLO, N, KB, BB, LDBB, INFO )
IF( INFO.NE.0 ) THEN
INFO = N + INFO
RETURN
END IF
*
* Transform problem to standard eigenvalue problem.
*
CALL DSBGST( JOBZ, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, Q, LDQ,
$ WORK, IINFO )
*
* Reduce symmetric band matrix to tridiagonal form.
*
INDD = 1
INDE = INDD + N
INDWRK = INDE + N
IF( WANTZ ) THEN
VECT = 'U'
ELSE
VECT = 'N'
END IF
CALL DSBTRD( VECT, UPLO, N, KA, AB, LDAB, WORK( INDD ),
$ WORK( INDE ), Q, LDQ, WORK( INDWRK ), IINFO )
*
* If all eigenvalues are desired and ABSTOL is less than or equal
* to zero, then call DSTERF or SSTEQR. 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, WORK( INDD ), 1, W, 1 )
INDEE = INDWRK + 2*N
CALL DCOPY( N-1, WORK( INDE ), 1, WORK( INDEE ), 1 )
IF( .NOT.WANTZ ) THEN
CALL DSTERF( N, W, WORK( INDEE ), INFO )
ELSE
CALL DLACPY( 'A', N, N, Q, LDQ, Z, LDZ )
CALL DSTEQR( JOBZ, N, W, WORK( INDEE ), Z, LDZ,
$ WORK( INDWRK ), 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 DSTEIN.
*
IF( WANTZ ) THEN
ORDER = 'B'
ELSE
ORDER = 'E'
END IF
INDIBL = 1
INDISP = INDIBL + N
INDIWO = INDISP + N
CALL DSTEBZ( RANGE, ORDER, N, VL, VU, IL, IU, ABSTOL,
$ WORK( INDD ), WORK( INDE ), M, NSPLIT, W,
$ IWORK( INDIBL ), IWORK( INDISP ), WORK( INDWRK ),
$ IWORK( INDIWO ), INFO )
*
IF( WANTZ ) THEN
CALL DSTEIN( N, WORK( INDD ), WORK( INDE ), M, W,
$ IWORK( INDIBL ), IWORK( INDISP ), Z, LDZ,
$ WORK( INDWRK ), IWORK( INDIWO ), IFAIL, INFO )
*
* Apply transformation matrix used in reduction to tridiagonal
* form to eigenvectors returned by DSTEIN.
*
DO 20 J = 1, M
CALL DCOPY( N, Z( 1, J ), 1, WORK( 1 ), 1 )
CALL DGEMV( 'N', N, N, ONE, Q, LDQ, WORK, 1, ZERO,
$ 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 DSWAP( 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 DSBGVX
*
END