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## sstevr()

subroutine sstevr | ( | character | jobz,
|  | character | range,
|  | integer | n,
|  | real, dimension( * ) | d,
|  | real, dimension( * ) | e,
|  | real | vl,
|  | real | vu,
|  | integer | il,
|  | integer | iu,
|  | real | abstol,
|  | integer | m,
|  | real, dimension( * ) | w,
|  | real, dimension( ldz, * ) | z,
|  | integer | ldz,
|  | integer, dimension( * ) | isuppz,
|  | real, dimension( * ) | work,
|  | integer | lwork,
|  | integer, dimension( * ) | iwork,
|  | integer | liwork,
|  | integer | info )

SSTEVR computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices

Download SSTEVR + dependencies [TGZ] [ZIP] [TXT]

Purpose:

      !>
      !> 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  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,  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 , 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.
      !>

Parameters

  [in]

  JOBZ

      !>          JOBZ is CHARACTER*1
      !>          = 'N':  Compute eigenvalues only;
      !>          = 'V':  Compute eigenvalues and eigenvectors.
      !>

  [in]

  RANGE

      !>          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
      !>

  [in]

  N

      !>          N is INTEGER
      !>          The order of the matrix.  N >= 0.
      !>

  [in,out]

  D

      !>          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.
      !>

  [in,out]

  E

      !>          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.
      !>

  [in]

  VL

      !>          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'.
      !>

  [in]

  VU

      !>          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'.
      !>

  [in]

  IL

      !>          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'.
      !>

  [in]

  IU

      !>          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'.
      !>

  [in]

  ABSTOL

      !>          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  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,
      !>          , LAPACK Working Note #7, for a discussion
      !>          of which matrices define their eigenvalues to high relative
      !>          accuracy.
      !>

  [out]

  M

      !>          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.
      !>

  [out]

  W

      !>          W is REAL array, dimension (N)
      !>          The first M elements contain the selected eigenvalues in
      !>          ascending order.
      !>

  [out]

  Z

      !>          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.
      !>

  [in]

  LDZ

      !>          LDZ is INTEGER
      !>          The leading dimension of the array Z.  LDZ >= 1, and if
      !>          JOBZ = 'V', LDZ >= max(1,N).
      !>

  [out]

  ISUPPZ

      !>          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
      !>

  [out]

  WORK

      !>          WORK is REAL array, dimension (MAX(1,LWORK))
      !>          On exit, if INFO = 0, WORK(1) returns the optimal (and
      !>          minimal) LWORK.
      !>

  [in]

  LWORK

      !>          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.
      !>

  [out]

  IWORK

      !>          IWORK is INTEGER array, dimension (MAX(1,LIWORK))
      !>          On exit, if INFO = 0, IWORK(1) returns the optimal (and
      !>          minimal) LIWORK.
      !>

  [in]

  LIWORK

      !>          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.
      !>

  [out]

  INFO

      !>          INFO is INTEGER
      !>          = 0:  successful exit
      !>          < 0:  if INFO = -i, the i-th argument had an illegal value
      !>          > 0:  Internal error
      !>

Author
  Univ. of Tennessee
  Univ. of California Berkeley
  Univ. of Colorado Denver
  NAG Ltd.

Contributors:
  Inderjit Dhillon, IBM Almaden, USA
  Osni Marques, LBNL/NERSC, USA
  Ken Stanley, Computer Science Division, University of California at Berkeley, USA
  Jason Riedy, Computer Science Division, University of California at Berkeley, USA