dstevx.f

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*> \brief <b> DSTEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices</b>
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
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*
*  Definition:
*  ===========
*
*       SUBROUTINE DSTEVX( JOBZ, RANGE, N, D, E, VL, VU, IL, IU, ABSTOL,
*                          M, W, Z, LDZ, WORK, IWORK, IFAIL, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          JOBZ, RANGE
*       INTEGER            IL, INFO, IU, LDZ, M, N
*       DOUBLE PRECISION   ABSTOL, VL, VU
*       ..
*       .. Array Arguments ..
*       INTEGER            IFAIL( * ), IWORK( * )
*       DOUBLE PRECISION   D( * ), E( * ), W( * ), WORK( * ), Z( LDZ, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DSTEVX computes selected eigenvalues and, optionally, eigenvectors
*> of a real symmetric tridiagonal 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] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*>          D is DOUBLE PRECISION 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 DOUBLE PRECISION 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 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.
*>
*>          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').
*>
*>          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 DOUBLE PRECISION array, dimension (N)
*>          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 = '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 (INFO > 0), 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 DOUBLE PRECISION array, dimension (5*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 stevx
*
*  =====================================================================
      SUBROUTINE dstevx( JOBZ, RANGE, N, D, E, 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
      INTEGER            IL, INFO, IU, LDZ, M, N
      DOUBLE PRECISION   ABSTOL, VL, VU
*     ..
*     .. Array Arguments ..
      INTEGER            IFAIL( * ), IWORK( * )
      DOUBLE PRECISION   D( * ), E( * ), W( * ), WORK( * ), Z( LDZ, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0d0, one = 1.0d0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            ALLEIG, INDEIG, TEST, VALEIG, WANTZ
      CHARACTER          ORDER
      INTEGER            I, IMAX, INDISP, INDIWO, INDWRK,
     $                   iscale, itmp1, j, jj, nsplit
      DOUBLE PRECISION   BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA, SMLNUM,
     $                   TMP1, TNRM, VLL, VUU
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      DOUBLE PRECISION   DLAMCH, DLANST
      EXTERNAL           lsame, dlamch, dlanst
*     ..
*     .. External Subroutines ..
      EXTERNAL           dcopy, dscal, dstebz, dstein, dsteqr,
     $                   dsterf,
     $                   dswap, 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' )
*
      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 ) )
     $      info = -14
      END IF
*
      IF( info.NE.0 ) THEN
         CALL xerbla( 'DSTEVX', -info )
         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 = dlamch( 'Safe minimum' )
      eps = dlamch( '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
      ELSE
         vll = zero
         vuu = zero
      END IF
      tnrm = dlanst( '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 dscal( n, sigma, d, 1 )
         CALL dscal( n-1, sigma, e( 1 ), 1 )
         IF( valeig ) THEN
            vll = vl*sigma
            vuu = vu*sigma
         END IF
      END IF
*
*     If all eigenvalues are desired and ABSTOL is less than 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, d, 1, w, 1 )
         CALL dcopy( n-1, e( 1 ), 1, work( 1 ), 1 )
         indwrk = n + 1
         IF( .NOT.wantz ) THEN
            CALL dsterf( n, w, work, info )
         ELSE
            CALL dsteqr( 'I', n, w, work, 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 20
         END IF
         info = 0
      END IF
*
*     Otherwise, call DSTEBZ and, if eigenvectors are desired, SSTEIN.
*
      IF( wantz ) THEN
         order = 'B'
      ELSE
         order = 'E'
      END IF
      indwrk = 1
      indisp = 1 + n
      indiwo = indisp + n
      CALL dstebz( range, order, n, vll, vuu, il, iu, abstol, d, e,
     $             m,
     $             nsplit, w, iwork( 1 ), iwork( indisp ),
     $             work( indwrk ), iwork( indiwo ), info )
*
      IF( wantz ) THEN
         CALL dstein( n, d, e, m, w, iwork( 1 ), iwork( indisp ),
     $                z, ldz, work( indwrk ), iwork( indiwo ), ifail,
     $                info )
      END IF
*
*     If matrix was scaled, then rescale eigenvalues appropriately.
*
   20 CONTINUE
      IF( iscale.EQ.1 ) THEN
         IF( info.EQ.0 ) THEN
            imax = m
         ELSE
            imax = info - 1
         END IF
         CALL dscal( imax, one / sigma, w, 1 )
      END IF
*
*     If eigenvalues are not in order, then sort them, along with
*     eigenvectors.
*
      IF( wantz ) THEN
         DO 40 j = 1, m - 1
            i = 0
            tmp1 = w( j )
            DO 30 jj = j + 1, m
               IF( w( jj ).LT.tmp1 ) THEN
                  i = jj
                  tmp1 = w( jj )
               END IF
   30       CONTINUE
*
            IF( i.NE.0 ) THEN
               itmp1 = iwork( 1 + i-1 )
               w( i ) = w( j )
               iwork( 1 + i-1 ) = iwork( 1 + j-1 )
               w( j ) = tmp1
               iwork( 1 + 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
   40    CONTINUE
      END IF
*
      RETURN
*
*     End of DSTEVX
*
      SUBROUTINE dstevx( JOBZ, RANGE, N, D, E, VL, VU, IL, IU, …
      END