*> \brief SSBEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download SSBEVX + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE SSBEVX( JOBZ, RANGE, UPLO, N, KD, AB, LDAB, 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, KD, LDAB, LDQ, LDZ, M, N * REAL ABSTOL, VL, VU * .. * .. Array Arguments .. * INTEGER IFAIL( * ), IWORK( * ) * REAL AB( LDAB, * ), Q( LDQ, * ), W( * ), WORK( * ), * \$ Z( LDZ, * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> SSBEVX computes selected eigenvalues and, optionally, eigenvectors *> of a real symmetric band matrix A. 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] 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 is stored; *> = 'L': Lower triangle of A is stored. *> \endverbatim *> *> \param[in] N *> \verbatim *> N is INTEGER *> The order of the matrix A. N >= 0. *> \endverbatim *> *> \param[in] KD *> \verbatim *> KD is INTEGER *> The number of superdiagonals of the matrix A if UPLO = 'U', *> or the number of subdiagonals if UPLO = 'L'. KD >= 0. *> \endverbatim *> *> \param[in,out] AB *> \verbatim *> AB is REAL array, dimension (LDAB, N) *> On entry, the upper or lower triangle of the symmetric band *> matrix A, stored in the first KD+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(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; *> if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd). *> *> On exit, AB is overwritten by values generated during the *> reduction to tridiagonal form. If UPLO = 'U', the first *> superdiagonal and the diagonal of the tridiagonal matrix T *> are returned in rows KD and KD+1 of AB, and if UPLO = 'L', *> the diagonal and first subdiagonal of T are returned in the *> first two rows of AB. *> \endverbatim *> *> \param[in] LDAB *> \verbatim *> LDAB is INTEGER *> The leading dimension of the array AB. LDAB >= KD + 1. *> \endverbatim *> *> \param[out] Q *> \verbatim *> Q is REAL array, dimension (LDQ, N) *> If JOBZ = 'V', the N-by-N orthogonal matrix used in the *> reduction 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 = 'V', then *> LDQ >= max(1,N). *> \endverbatim *> *> \param[in] VL *> \verbatim *> VL is REAL *> 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 REAL *> 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 REAL *> 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 AB to tridiagonal form. *> *> Eigenvalues will be computed most accurately when ABSTOL is *> set to twice the underflow threshold 2*SLAMCH('S'), not zero. *> If this routine returns with INFO>0, indicating that some *> eigenvectors did not converge, try setting ABSTOL to *> 2*SLAMCH('S'). *> *> See "Computing Small Singular Values of Bidiagonal Matrices *> with Guaranteed High Relative Accuracy," by Demmel and *> Kahan, LAPACK Working Note #3. *> \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 REAL array, dimension (N) *> The first M elements contain the selected eigenvalues in *> ascending order. *> \endverbatim *> *> \param[out] Z *> \verbatim *> Z is REAL array, dimension (LDZ, max(1,M)) *> 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). *> 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. *> If JOBZ = 'N', then Z is not referenced. *> 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 REAL 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, then i eigenvectors failed to converge. *> Their indices are stored in array IFAIL. *> \endverbatim * * Authors: * ======== * *> \author Univ. of Tennessee *> \author Univ. of California Berkeley *> \author Univ. of Colorado Denver *> \author NAG Ltd. * *> \ingroup realOTHEReigen * * ===================================================================== SUBROUTINE SSBEVX( JOBZ, RANGE, UPLO, N, KD, AB, LDAB, Q, LDQ, VL, \$ VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, IWORK, \$ IFAIL, INFO ) * * -- LAPACK driver routine -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * * .. Scalar Arguments .. CHARACTER JOBZ, RANGE, UPLO INTEGER IL, INFO, IU, KD, LDAB, LDQ, LDZ, M, N REAL ABSTOL, VL, VU * .. * .. Array Arguments .. INTEGER IFAIL( * ), IWORK( * ) REAL AB( LDAB, * ), Q( LDQ, * ), W( * ), WORK( * ), \$ Z( LDZ, * ) * .. * * ===================================================================== * * .. Parameters .. REAL ZERO, ONE PARAMETER ( ZERO = 0.0E0, ONE = 1.0E0 ) * .. * .. Local Scalars .. LOGICAL ALLEIG, INDEIG, LOWER, TEST, VALEIG, WANTZ CHARACTER ORDER INTEGER I, IINFO, IMAX, INDD, INDE, INDEE, INDIBL, \$ INDISP, INDIWO, INDWRK, ISCALE, ITMP1, J, JJ, \$ NSPLIT REAL ABSTLL, ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN, \$ SIGMA, SMLNUM, TMP1, VLL, VUU * .. * .. External Functions .. LOGICAL LSAME REAL SLAMCH, SLANSB EXTERNAL LSAME, SLAMCH, SLANSB * .. * .. External Subroutines .. EXTERNAL SCOPY, SGEMV, SLACPY, SLASCL, SSBTRD, SSCAL, \$ SSTEBZ, SSTEIN, SSTEQR, SSTERF, SSWAP, XERBLA * .. * .. Intrinsic Functions .. INTRINSIC MAX, MIN, SQRT * .. * .. Executable Statements .. * * Test the input parameters. * WANTZ = LSAME( JOBZ, 'V' ) ALLEIG = LSAME( RANGE, 'A' ) VALEIG = LSAME( RANGE, 'V' ) INDEIG = LSAME( RANGE, 'I' ) LOWER = LSAME( UPLO, 'L' ) * 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.( LOWER .OR. LSAME( UPLO, 'U' ) ) ) THEN INFO = -3 ELSE IF( N.LT.0 ) THEN INFO = -4 ELSE IF( KD.LT.0 ) THEN INFO = -5 ELSE IF( LDAB.LT.KD+1 ) THEN INFO = -7 ELSE IF( WANTZ .AND. LDQ.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 ) ) \$ INFO = -18 END IF * IF( INFO.NE.0 ) THEN CALL XERBLA( 'SSBEVX', -INFO ) RETURN END IF * * Quick return if possible * M = 0 IF( N.EQ.0 ) \$ RETURN * IF( N.EQ.1 ) THEN M = 1 IF( LOWER ) THEN TMP1 = AB( 1, 1 ) ELSE TMP1 = AB( KD+1, 1 ) END IF IF( VALEIG ) THEN IF( .NOT.( VL.LT.TMP1 .AND. VU.GE.TMP1 ) ) \$ M = 0 END IF IF( M.EQ.1 ) THEN W( 1 ) = TMP1 IF( WANTZ ) \$ Z( 1, 1 ) = ONE END IF RETURN END IF * * Get machine constants. * SAFMIN = SLAMCH( 'Safe minimum' ) EPS = SLAMCH( 'Precision' ) SMLNUM = SAFMIN / EPS BIGNUM = ONE / SMLNUM RMIN = SQRT( SMLNUM ) RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) ) * * Scale matrix to allowable range, if necessary. * ISCALE = 0 ABSTLL = ABSTOL IF ( VALEIG ) THEN VLL = VL VUU = VU ELSE VLL = ZERO VUU = ZERO ENDIF ANRM = SLANSB( 'M', UPLO, N, KD, AB, LDAB, WORK ) IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN ISCALE = 1 SIGMA = RMIN / ANRM ELSE IF( ANRM.GT.RMAX ) THEN ISCALE = 1 SIGMA = RMAX / ANRM END IF IF( ISCALE.EQ.1 ) THEN IF( LOWER ) THEN CALL SLASCL( 'B', KD, KD, ONE, SIGMA, N, N, AB, LDAB, INFO ) ELSE CALL SLASCL( 'Q', KD, KD, ONE, SIGMA, N, N, AB, LDAB, INFO ) END IF IF( ABSTOL.GT.0 ) \$ ABSTLL = ABSTOL*SIGMA IF( VALEIG ) THEN VLL = VL*SIGMA VUU = VU*SIGMA END IF END IF * * Call SSBTRD to reduce symmetric band matrix to tridiagonal form. * INDD = 1 INDE = INDD + N INDWRK = INDE + N CALL SSBTRD( JOBZ, UPLO, N, KD, 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 SSTERF or SSTEQR. If this fails for some * eigenvalue, then try SSTEBZ. * 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 SCOPY( N, WORK( INDD ), 1, W, 1 ) INDEE = INDWRK + 2*N IF( .NOT.WANTZ ) THEN CALL SCOPY( N-1, WORK( INDE ), 1, WORK( INDEE ), 1 ) CALL SSTERF( N, W, WORK( INDEE ), INFO ) ELSE CALL SLACPY( 'A', N, N, Q, LDQ, Z, LDZ ) CALL SCOPY( N-1, WORK( INDE ), 1, WORK( INDEE ), 1 ) CALL SSTEQR( 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 SSTEBZ and, if eigenvectors are desired, SSTEIN. * IF( WANTZ ) THEN ORDER = 'B' ELSE ORDER = 'E' END IF INDIBL = 1 INDISP = INDIBL + N INDIWO = INDISP + N CALL SSTEBZ( RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTLL, \$ WORK( INDD ), WORK( INDE ), M, NSPLIT, W, \$ IWORK( INDIBL ), IWORK( INDISP ), WORK( INDWRK ), \$ IWORK( INDIWO ), INFO ) * IF( WANTZ ) THEN CALL SSTEIN( N, WORK( INDD ), WORK( INDE ), M, W, \$ IWORK( INDIBL ), IWORK( INDISP ), Z, LDZ, \$ WORK( INDWRK ), IWORK( INDIWO ), IFAIL, INFO ) * * Apply orthogonal matrix used in reduction to tridiagonal * form to eigenvectors returned by SSTEIN. * DO 20 J = 1, M CALL SCOPY( N, Z( 1, J ), 1, WORK( 1 ), 1 ) CALL SGEMV( 'N', N, N, ONE, Q, LDQ, WORK, 1, ZERO, \$ Z( 1, J ), 1 ) 20 CONTINUE END IF * * If matrix was scaled, then rescale eigenvalues appropriately. * 30 CONTINUE IF( ISCALE.EQ.1 ) THEN IF( INFO.EQ.0 ) THEN IMAX = M ELSE IMAX = INFO - 1 END IF CALL SSCAL( IMAX, ONE / SIGMA, W, 1 ) END IF * * 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 SSWAP( 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 SSBEVX * END