sstevr.f

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*> \brief <b> SSTEVR computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices</b>
*
*  =========== DOCUMENTATION ===========
*
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*
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*
*  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 stevr
*
*> \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, SROUNDUP_LWORK
      EXTERNAL           lsame, ilaenv, slamch, slanst,
     $                   sroundup_lwork
*     ..
*     .. 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 ) = sroundup_lwork(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*real( 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 ) = sroundup_lwork(lwmin)
      iwork( 1 ) = liwmin
      RETURN
*
*     End of SSTEVR
*
      SUBROUTINE sstevr( JOBZ, RANGE, N, D, E, VL, VU, IL, IU, …
      END