larre(3)

NAME
larre - larre: step in stemr

SYNOPSIS

  Functions
subroutine dlarre (range, n, vl, vu, il, iu, d, e, e2, rtol1, rtol2, spltol, nsplit, isplit, m, w, werr, wgap, iblock, indexw, gers, pivmin, work, iwork, info)
DLARRE given the tridiagonal matrix T, sets small off-diagonal elements to zero and for each unreduced block Ti, finds base representations and eigenvalues.
subroutine slarre (range, n, vl, vu, il, iu, d, e, e2, rtol1, rtol2, spltol, nsplit, isplit, m, w, werr, wgap, iblock, indexw, gers, pivmin, work, iwork, 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.

Detailed Description

Function Documentation

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

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

Purpose:

     To find the desired eigenvalues of a given real symmetric
     tridiagonal matrix T, DLARRE sets any 'small' 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
     DSTEMR 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 DLASQ2) to
     compute all and then discard any unwanted one.
     As an added benefit, DLARRE also outputs the n
     Gerschgorin intervals for the matrices L_i D_i L_i^T.

Parameters
    RANGE

              RANGE is CHARACTER*1
              = 'A': ('All')   all eigenvalues will be found.
              = 'V': ('Value') all eigenvalues in the half-open interval
                               (VL, VU] will be found.
              = 'I': ('Index') the IL-th through IU-th eigenvalues (of the
                               entire matrix) will be found.

    N

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

    VL

              VL is DOUBLE PRECISION
              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', DLARRE computes bounds on the desired
              part of the spectrum.

    VU

              VU is DOUBLE PRECISION
              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', DLARRE computes bounds on the desired
              part of the spectrum.

    IL

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

    IU

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

    D

              D is DOUBLE PRECISION 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.

    E

              E is DOUBLE PRECISION 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.

    E2

              E2 is DOUBLE PRECISION 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

    RTOL1

              RTOL1 is DOUBLE PRECISION

    RTOL2

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

    SPLTOL

              SPLTOL is DOUBLE PRECISION
              The threshold for splitting.

    NSPLIT

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

    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.

    M

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

    W

              W is DOUBLE PRECISION 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 ( DLARRE may use the
              remaining N-M elements as workspace).

    WERR

              WERR is DOUBLE PRECISION array, dimension (N)
              The error bound on the corresponding eigenvalue in W.

    WGAP

              WGAP is DOUBLE PRECISION 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

    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.

    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

    GERS

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

    PIVMIN

              PIVMIN is DOUBLE PRECISION
              The minimum pivot in the Sturm sequence for T.

    WORK

              WORK is DOUBLE PRECISION array, dimension (6*N)
              Workspace.

    IWORK

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

    INFO

              INFO is INTEGER
              = 0:  successful exit
              > 0:  A problem occurred in DLARRE.
              < 0:  One of the called subroutines signaled an internal problem.
                    Needs inspection of the corresponding parameter IINFO
                    for further information.
              =-1:  Problem in DLARRD.
              = 2:  No base representation could be found in MAXTRY iterations.
                    Increasing MAXTRY and recompilation might be a remedy.
              =-3:  Problem in DLARRB when computing the refined root
                    representation for DLASQ2.
              =-4:  Problem in DLARRB when preforming bisection on the
                    desired part of the spectrum.
              =-5:  Problem in DLASQ2.
              =-6:  Problem in DLASQ2.

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

Definition at line 301 of file dlarre.f.

  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.

Purpose:

     To find the desired eigenvalues of a given real symmetric
     tridiagonal matrix T, SLARRE sets any 'small' 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
    RANGE

              RANGE is CHARACTER*1
              = 'A': ('All')   all eigenvalues will be found.
              = 'V': ('Value') all eigenvalues in the half-open interval
                               (VL, VU] will be found.
              = 'I': ('Index') the IL-th through IU-th eigenvalues (of the
                               entire matrix) will be found.

    N

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

    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.

    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.

    IL

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

    IU

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

    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.

    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.

    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

    RTOL1

              RTOL1 is REAL

    RTOL2

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

    SPLTOL

              SPLTOL is REAL
              The threshold for splitting.

    NSPLIT

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

    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.

    M

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

    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).

    WERR

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

    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

    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.

    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

    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)).

    PIVMIN

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

    WORK

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

    IWORK

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

    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

Definition at line 301 of file slarre.f.

Author

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