*> \brief SSTEVR 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 SSTEVR + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE SSTEVR( JOBZ, RANGE, N, D, E, VL, VU, IL, IU, ABSTOL, * M, W, Z, LDZ, ISUPPZ, WORK, LWORK, IWORK, * LIWORK, INFO ) * * .. Scalar Arguments .. * CHARACTER JOBZ, RANGE * INTEGER IL, INFO, IU, LDZ, LIWORK, LWORK, M, N * REAL ABSTOL, VL, VU * .. * .. Array Arguments .. * INTEGER ISUPPZ( * ), IWORK( * ) * REAL D( * ), E( * ), W( * ), WORK( * ), Z( LDZ, * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> SSTEVR computes selected eigenvalues and, optionally, eigenvectors *> of a real symmetric tridiagonal matrix T. Eigenvalues and *> eigenvectors can be selected by specifying either a range of values *> or a range of indices for the desired eigenvalues. *> *> Whenever possible, SSTEVR calls SSTEMR to compute the *> eigenspectrum using Relatively Robust Representations. SSTEMR *> computes eigenvalues by the dqds algorithm, while orthogonal *> eigenvectors are computed from various "good" L D L^T representations *> (also known as Relatively Robust Representations). Gram-Schmidt *> orthogonalization is avoided as far as possible. More specifically, *> the various steps of the algorithm are as follows. For the i-th *> unreduced block of T, *> (a) Compute T - sigma_i = L_i D_i L_i^T, such that L_i D_i L_i^T *> is a relatively robust representation, *> (b) Compute the eigenvalues, lambda_j, of L_i D_i L_i^T to high *> relative accuracy by the dqds algorithm, *> (c) If there is a cluster of close eigenvalues, "choose" sigma_i *> close to the cluster, and go to step (a), *> (d) Given the approximate eigenvalue lambda_j of L_i D_i L_i^T, *> compute the corresponding eigenvector by forming a *> rank-revealing twisted factorization. *> The desired accuracy of the output can be specified by the input *> parameter ABSTOL. *> *> For more details, see "A new O(n^2) algorithm for the symmetric *> tridiagonal eigenvalue/eigenvector problem", by Inderjit Dhillon, *> Computer Science Division Technical Report No. UCB//CSD-97-971, *> UC Berkeley, May 1997. *> *> *> Note 1 : SSTEVR calls SSTEMR when the full spectrum is requested *> on machines which conform to the ieee-754 floating point standard. *> SSTEVR calls SSTEBZ and SSTEIN on non-ieee machines and *> when partial spectrum requests are made. *> *> Normal execution of SSTEMR may create NaNs and infinities and *> hence may abort due to a floating point exception in environments *> which do not handle NaNs and infinities in the ieee standard default *> manner. *> \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. *> For RANGE = 'V' or 'I' and IU - IL < N - 1, SSTEBZ and *> SSTEIN are called *> \endverbatim *> *> \param[in] N *> \verbatim *> N is INTEGER *> The order of the matrix. N >= 0. *> \endverbatim *> *> \param[in,out] D *> \verbatim *> D is REAL array, dimension (N) *> On entry, the n diagonal elements of the tridiagonal matrix *> A. *> On exit, D may be multiplied by a constant factor chosen *> to avoid over/underflow in computing the eigenvalues. *> \endverbatim *> *> \param[in,out] E *> \verbatim *> E is REAL array, dimension (max(1,N-1)) *> On entry, the (n-1) subdiagonal elements of the tridiagonal *> matrix A in elements 1 to N-1 of E. *> On exit, E may be multiplied by a constant factor chosen *> to avoid over/underflow in computing the eigenvalues. *> \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 A to tridiagonal form. *> *> See "Computing Small Singular Values of Bidiagonal Matrices *> with Guaranteed High Relative Accuracy," by Demmel and *> Kahan, LAPACK Working Note #3. *> *> If high relative accuracy is important, set ABSTOL to *> SLAMCH( 'Safe minimum' ). Doing so will guarantee that *> eigenvalues are computed to high relative accuracy when *> possible in future releases. The current code does not *> make any guarantees about high relative accuracy, but *> future releases will. See J. Barlow and J. Demmel, *> "Computing Accurate Eigensystems of Scaled Diagonally *> Dominant Matrices", LAPACK Working Note #7, for a discussion *> of which matrices define their eigenvalues to high relative *> accuracy. *> \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). *> 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] ISUPPZ *> \verbatim *> ISUPPZ is INTEGER array, dimension ( 2*max(1,M) ) *> The support of the eigenvectors in Z, i.e., the indices *> indicating the nonzero elements in Z. The i-th eigenvector *> is nonzero only in elements ISUPPZ( 2*i-1 ) through *> ISUPPZ( 2*i ). *> Implemented only for RANGE = 'A' or 'I' and IU - IL = N - 1 *> \endverbatim *> *> \param[out] WORK *> \verbatim *> WORK is REAL array, dimension (MAX(1,LWORK)) *> On exit, if INFO = 0, WORK(1) returns the optimal (and *> minimal) LWORK. *> \endverbatim *> *> \param[in] LWORK *> \verbatim *> LWORK is INTEGER *> The dimension of the array WORK. LWORK >= 20*N. *> *> If LWORK = -1, then a workspace query is assumed; the routine *> only calculates the optimal sizes of the WORK and IWORK *> arrays, returns these values as the first entries of the WORK *> and IWORK arrays, and no error message related to LWORK or *> LIWORK is issued by XERBLA. *> \endverbatim *> *> \param[out] IWORK *> \verbatim *> IWORK is INTEGER array, dimension (MAX(1,LIWORK)) *> On exit, if INFO = 0, IWORK(1) returns the optimal (and *> minimal) LIWORK. *> \endverbatim *> *> \param[in] LIWORK *> \verbatim *> LIWORK is INTEGER *> The dimension of the array IWORK. LIWORK >= 10*N. *> *> If LIWORK = -1, then a workspace query is assumed; the *> routine only calculates the optimal sizes of the WORK and *> IWORK arrays, returns these values as the first entries of *> the WORK and IWORK arrays, and no error message related to *> LWORK or LIWORK is issued by XERBLA. *> \endverbatim *> *> \param[out] INFO *> \verbatim *> INFO is INTEGER *> = 0: successful exit *> < 0: if INFO = -i, the i-th argument had an illegal value *> > 0: Internal error *> \endverbatim * * Authors: * ======== * *> \author Univ. of Tennessee *> \author Univ. of California Berkeley *> \author Univ. of Colorado Denver *> \author NAG Ltd. * *> \ingroup realOTHEReigen * *> \par Contributors: * ================== *> *> Inderjit Dhillon, IBM Almaden, USA \n *> Osni Marques, LBNL/NERSC, USA \n *> Ken Stanley, Computer Science Division, University of *> California at Berkeley, USA \n *> Jason Riedy, Computer Science Division, University of *> California at Berkeley, USA \n *> * ===================================================================== SUBROUTINE SSTEVR( JOBZ, RANGE, N, D, E, VL, VU, IL, IU, ABSTOL, $ M, W, Z, LDZ, ISUPPZ, WORK, LWORK, IWORK, $ LIWORK, 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 INTEGER IL, INFO, IU, LDZ, LIWORK, LWORK, M, N REAL ABSTOL, VL, VU * .. * .. Array Arguments .. INTEGER ISUPPZ( * ), IWORK( * ) REAL D( * ), E( * ), W( * ), WORK( * ), Z( LDZ, * ) * .. * * ===================================================================== * * .. Parameters .. REAL ZERO, ONE, TWO PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0, TWO = 2.0E+0 ) * .. * .. Local Scalars .. LOGICAL ALLEIG, INDEIG, TEST, LQUERY, VALEIG, WANTZ, $ TRYRAC CHARACTER ORDER INTEGER I, IEEEOK, IMAX, INDIBL, INDIFL, INDISP, $ INDIWO, ISCALE, J, JJ, LIWMIN, LWMIN, NSPLIT REAL BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA, SMLNUM, $ TMP1, TNRM, VLL, VUU * .. * .. External Functions .. LOGICAL LSAME INTEGER ILAENV REAL SLAMCH, SLANST EXTERNAL LSAME, ILAENV, SLAMCH, SLANST * .. * .. External Subroutines .. EXTERNAL SCOPY, SSCAL, SSTEBZ, SSTEMR, SSTEIN, SSTERF, $ SSWAP, XERBLA * .. * .. Intrinsic Functions .. INTRINSIC MAX, MIN, SQRT * .. * .. Executable Statements .. * * * Test the input parameters. * IEEEOK = ILAENV( 10, 'SSTEVR', 'N', 1, 2, 3, 4 ) * WANTZ = LSAME( JOBZ, 'V' ) ALLEIG = LSAME( RANGE, 'A' ) VALEIG = LSAME( RANGE, 'V' ) INDEIG = LSAME( RANGE, 'I' ) * LQUERY = ( ( LWORK.EQ.-1 ) .OR. ( LIWORK.EQ.-1 ) ) LWMIN = MAX( 1, 20*N ) LIWMIN = MAX(1, 10*N ) * * 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( N.LT.0 ) THEN INFO = -3 ELSE IF( VALEIG ) THEN IF( N.GT.0 .AND. VU.LE.VL ) $ INFO = -7 ELSE IF( INDEIG ) THEN IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN INFO = -8 ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN INFO = -9 END IF END IF END IF IF( INFO.EQ.0 ) THEN IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN INFO = -14 END IF END IF * IF( INFO.EQ.0 ) THEN WORK( 1 ) = LWMIN IWORK( 1 ) = LIWMIN * IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN INFO = -17 ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN INFO = -19 END IF END IF * IF( INFO.NE.0 ) THEN CALL XERBLA( 'SSTEVR', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF * * Quick return if possible * M = 0 IF( N.EQ.0 ) $ RETURN * IF( N.EQ.1 ) THEN IF( ALLEIG .OR. INDEIG ) THEN M = 1 W( 1 ) = D( 1 ) ELSE IF( VL.LT.D( 1 ) .AND. VU.GE.D( 1 ) ) THEN M = 1 W( 1 ) = D( 1 ) END IF END IF IF( WANTZ ) $ Z( 1, 1 ) = ONE 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 IF( VALEIG ) THEN VLL = VL VUU = VU END IF * TNRM = SLANST( 'M', N, D, E ) IF( TNRM.GT.ZERO .AND. TNRM.LT.RMIN ) THEN ISCALE = 1 SIGMA = RMIN / TNRM ELSE IF( TNRM.GT.RMAX ) THEN ISCALE = 1 SIGMA = RMAX / TNRM END IF IF( ISCALE.EQ.1 ) THEN CALL SSCAL( N, SIGMA, D, 1 ) CALL SSCAL( N-1, SIGMA, E( 1 ), 1 ) IF( VALEIG ) THEN VLL = VL*SIGMA VUU = VU*SIGMA END IF END IF * Initialize indices into workspaces. Note: These indices are used only * if SSTERF or SSTEMR fail. * IWORK(INDIBL:INDIBL+M-1) corresponds to IBLOCK in SSTEBZ and * stores the block indices of each of the M<=N eigenvalues. INDIBL = 1 * IWORK(INDISP:INDISP+NSPLIT-1) corresponds to ISPLIT in SSTEBZ and * stores the starting and finishing indices of each block. INDISP = INDIBL + N * IWORK(INDIFL:INDIFL+N-1) stores the indices of eigenvectors * that corresponding to eigenvectors that fail to converge in * SSTEIN. This information is discarded; if any fail, the driver * returns INFO > 0. INDIFL = INDISP + N * INDIWO is the offset of the remaining integer workspace. INDIWO = INDISP + N * * If all eigenvalues are desired, then * call SSTERF or SSTEMR. 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. IEEEOK.EQ.1 ) THEN CALL SCOPY( N-1, E( 1 ), 1, WORK( 1 ), 1 ) IF( .NOT.WANTZ ) THEN CALL SCOPY( N, D, 1, W, 1 ) CALL SSTERF( N, W, WORK, INFO ) ELSE CALL SCOPY( N, D, 1, WORK( N+1 ), 1 ) IF (ABSTOL .LE. TWO*N*EPS) THEN TRYRAC = .TRUE. ELSE TRYRAC = .FALSE. END IF CALL SSTEMR( JOBZ, 'A', N, WORK( N+1 ), WORK, VL, VU, IL, $ IU, M, W, Z, LDZ, N, ISUPPZ, TRYRAC, $ WORK( 2*N+1 ), LWORK-2*N, IWORK, LIWORK, INFO ) * END IF IF( INFO.EQ.0 ) THEN M = N GO TO 10 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 CALL SSTEBZ( RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTOL, D, E, M, $ NSPLIT, W, IWORK( INDIBL ), IWORK( INDISP ), WORK, $ IWORK( INDIWO ), INFO ) * IF( WANTZ ) THEN CALL SSTEIN( N, D, E, M, W, IWORK( INDIBL ), IWORK( INDISP ), $ Z, LDZ, WORK, IWORK( INDIWO ), IWORK( INDIFL ), $ INFO ) END IF * * If matrix was scaled, then rescale eigenvalues appropriately. * 10 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 30 J = 1, M - 1 I = 0 TMP1 = W( J ) DO 20 JJ = J + 1, M IF( W( JJ ).LT.TMP1 ) THEN I = JJ TMP1 = W( JJ ) END IF 20 CONTINUE * IF( I.NE.0 ) THEN W( I ) = W( J ) W( J ) = TMP1 CALL SSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 ) END IF 30 CONTINUE END IF * * Causes problems with tests 19 & 20: * IF (wantz .and. INDEIG ) Z( 1,1) = Z(1,1) / 1.002 + .002 * * WORK( 1 ) = LWMIN IWORK( 1 ) = LIWMIN RETURN * * End of SSTEVR * END