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## slarre()

subroutine slarre | ( | character | range,
|  | integer | n,
|  | real | vl,
|  | real | vu,
|  | integer | il,
|  | integer | iu,
|  | real, dimension( * ) | d,
|  | real, dimension( * ) | e,
|  | real, dimension( * ) | e2,
|  | real | rtol1,
|  | real | rtol2,
|  | real | spltol,
|  | integer | nsplit,
|  | integer, dimension( * ) | isplit,
|  | integer | m,
|  | real, dimension( * ) | w,
|  | real, dimension( * ) | werr,
|  | real, dimension( * ) | wgap,
|  | integer, dimension( * ) | iblock,
|  | integer, dimension( * ) | indexw,
|  | real, dimension( * ) | gers,
|  | real | pivmin,
|  | real, dimension( * ) | work,
|  | integer, dimension( * ) | iwork,
|  | integer | info )

SLARRE given the tridiagonal matrix T, sets small off-diagonal elements to zero and for each unreduced block Ti, finds base representations and eigenvalues.

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

Purpose:

      !>
      !> To find the desired eigenvalues of a given real symmetric
      !> tridiagonal matrix T, SLARRE sets any  off-diagonal
      !> elements to zero, and for each unreduced block T_i, it finds
      !> (a) a suitable shift at one end of the block's spectrum,
      !> (b) the base representation, T_i - sigma_i I = L_i D_i L_i^T, and
      !> (c) eigenvalues of each L_i D_i L_i^T.
      !> The representations and eigenvalues found are then used by
      !> SSTEMR to compute the eigenvectors of T.
      !> The accuracy varies depending on whether bisection is used to
      !> find a few eigenvalues or the dqds algorithm (subroutine SLASQ2) to
      !> compute all and then discard any unwanted one.
      !> As an added benefit, SLARRE also outputs the n
      !> Gerschgorin intervals for the matrices L_i D_i L_i^T.
      !>

Parameters

  [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 (of the
      !>                           entire matrix) will be found.
      !>

  [in]

  N

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

  [in,out]

  VL

      !>          VL is REAL
      !>          If RANGE='V', the lower bound for the eigenvalues.
      !>          Eigenvalues less than or equal to VL, or greater than VU,
      !>          will not be returned.  VL < VU.
      !>          If RANGE='I' or ='A', SLARRE computes bounds on the desired
      !>          part of the spectrum.
      !>

  [in,out]

  VU

      !>          VU is REAL
      !>          If RANGE='V', the upper bound for the eigenvalues.
      !>          Eigenvalues less than or equal to VL, or greater than VU,
      !>          will not be returned.  VL < VU.
      !>          If RANGE='I' or ='A', SLARRE computes bounds on the desired
      !>          part of the spectrum.
      !>

  [in]

  IL

      !>          IL is INTEGER
      !>          If RANGE='I', the index of the
      !>          smallest eigenvalue to be returned.
      !>          1 <= IL <= IU <= N.
      !>

  [in]

  IU

      !>          IU is INTEGER
      !>          If RANGE='I', the index of the
      !>          largest eigenvalue to be returned.
      !>          1 <= IL <= IU <= N.
      !>

  [in,out]

  D

      !>          D is REAL array, dimension (N)
      !>          On entry, the N diagonal elements of the tridiagonal
      !>          matrix T.
      !>          On exit, the N diagonal elements of the diagonal
      !>          matrices D_i.
      !>

  [in,out]

  E

      !>          E is REAL array, dimension (N)
      !>          On entry, the first (N-1) entries contain the subdiagonal
      !>          elements of the tridiagonal matrix T; E(N) need not be set.
      !>          On exit, E contains the subdiagonal elements of the unit
      !>          bidiagonal matrices L_i. The entries E( ISPLIT( I ) ),
      !>          1 <= I <= NSPLIT, contain the base points sigma_i on output.
      !>

  [in,out]

  E2

      !>          E2 is REAL array, dimension (N)
      !>          On entry, the first (N-1) entries contain the SQUARES of the
      !>          subdiagonal elements of the tridiagonal matrix T;
      !>          E2(N) need not be set.
      !>          On exit, the entries E2( ISPLIT( I ) ),
      !>          1 <= I <= NSPLIT, have been set to zero
      !>

  [in]

  RTOL1

      !>          RTOL1 is REAL
      !>

  [in]

  RTOL2

      !>          RTOL2 is REAL
      !>           Parameters for bisection.
      !>           An interval [LEFT,RIGHT] has converged if
      !>           RIGHT-LEFT < MAX( RTOL1*GAP, RTOL2*MAX(|LEFT|,|RIGHT|) )
      !>

  [in]

  SPLTOL

      !>          SPLTOL is REAL
      !>          The threshold for splitting.
      !>

  [out]

  NSPLIT

      !>          NSPLIT is INTEGER
      !>          The number of blocks T splits into. 1 <= NSPLIT <= N.
      !>

  [out]

  ISPLIT

      !>          ISPLIT is INTEGER array, dimension (N)
      !>          The splitting points, at which T breaks up into blocks.
      !>          The first block consists of rows/columns 1 to ISPLIT(1),
      !>          the second of rows/columns ISPLIT(1)+1 through ISPLIT(2),
      !>          etc., and the NSPLIT-th consists of rows/columns
      !>          ISPLIT(NSPLIT-1)+1 through ISPLIT(NSPLIT)=N.
      !>

  [out]

  M

      !>          M is INTEGER
      !>          The total number of eigenvalues (of all L_i D_i L_i^T)
      !>          found.
      !>

  [out]

  W

      !>          W is REAL array, dimension (N)
      !>          The first M elements contain the eigenvalues. The
      !>          eigenvalues of each of the blocks, L_i D_i L_i^T, are
      !>          sorted in ascending order ( SLARRE may use the
      !>          remaining N-M elements as workspace).
      !>

  [out]

  WERR

      !>          WERR is REAL array, dimension (N)
      !>          The error bound on the corresponding eigenvalue in W.
      !>

  [out]

  WGAP

      !>          WGAP is REAL array, dimension (N)
      !>          The separation from the right neighbor eigenvalue in W.
      !>          The gap is only with respect to the eigenvalues of the same block
      !>          as each block has its own representation tree.
      !>          Exception: at the right end of a block we store the left gap
      !>

  [out]

  IBLOCK

      !>          IBLOCK is INTEGER array, dimension (N)
      !>          The indices of the blocks (submatrices) associated with the
      !>          corresponding eigenvalues in W; IBLOCK(i)=1 if eigenvalue
      !>          W(i) belongs to the first block from the top, =2 if W(i)
      !>          belongs to the second block, etc.
      !>

  [out]

  INDEXW

      !>          INDEXW is INTEGER array, dimension (N)
      !>          The indices of the eigenvalues within each block (submatrix);
      !>          for example, INDEXW(i)= 10 and IBLOCK(i)=2 imply that the
      !>          i-th eigenvalue W(i) is the 10-th eigenvalue in block 2
      !>

  [out]

  GERS

      !>          GERS is REAL array, dimension (2*N)
      !>          The N Gerschgorin intervals (the i-th Gerschgorin interval
      !>          is (GERS(2*i-1), GERS(2*i)).
      !>

  [out]

  PIVMIN

      !>          PIVMIN is REAL
      !>          The minimum pivot in the Sturm sequence for T.
      !>

  [out]

  WORK

      !>          WORK is REAL array, dimension (6*N)
      !>          Workspace.
      !>

  [out]

  IWORK

      !>          IWORK is INTEGER array, dimension (5*N)
      !>          Workspace.
      !>

  [out]

  INFO

      !>          INFO is INTEGER
      !>          = 0:  successful exit
      !>          > 0:  A problem occurred in SLARRE.
      !>          < 0:  One of the called subroutines signaled an internal problem.
      !>                Needs inspection of the corresponding parameter IINFO
      !>                for further information.
      !>
      !>          =-1:  Problem in SLARRD.
      !>          = 2:  No base representation could be found in MAXTRY iterations.
      !>                Increasing MAXTRY and recompilation might be a remedy.
      !>          =-3:  Problem in SLARRB when computing the refined root
      !>                representation for SLASQ2.
      !>          =-4:  Problem in SLARRB when preforming bisection on the
      !>                desired part of the spectrum.
      !>          =-5:  Problem in SLASQ2.
      !>          =-6:  Problem in SLASQ2.
      !>

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

Further Details:

      !>
      !>  The base representations are required to suffer very little
      !>  element growth and consequently define all their eigenvalues to
      !>  high relative accuracy.
      !>

Contributors:
  Beresford Parlett, University of California, Berkeley, USA
  Jim Demmel, University of California, Berkeley, USA
  Inderjit Dhillon, University of Texas, Austin, USA
  Osni Marques, LBNL/NERSC, USA
  Christof Voemel, University of California, Berkeley, USA