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  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 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.
    !>

    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

  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  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 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.
    !>

    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

Author

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