larrd(3)

NAME
larrd - larrd: step in stemr, tridiag eig

SYNOPSIS

  Functions

subroutine dlarrd (range, order, n, vl, vu, il, iu, gers, reltol, d, e, e2, pivmin, nsplit, isplit, m, w, werr, wl, wu, iblock, indexw, work, iwork, info)
DLARRD computes the eigenvalues of a symmetric tridiagonal matrix to suitable accuracy.
subroutine slarrd (range, order, n, vl, vu, il, iu, gers, reltol, d, e, e2, pivmin, nsplit, isplit, m, w, werr, wl, wu, iblock, indexw, work, iwork, info)
SLARRD computes the eigenvalues of a symmetric tridiagonal matrix to suitable accuracy.

Detailed Description

Function Documentation

  subroutine dlarrd (character range, character order, integer n, double precision vl, double precision vu, integer il, integer iu, double precision, dimension( * ) gers, double precision reltol, double precision, dimension( * ) d, double precision, dimension( * ) e, double precision, dimension( * ) e2, double precision pivmin, integer nsplit, integer, dimension( * ) isplit, integer m, double precision, dimension( * ) w, double precision, dimension( * ) werr, double precision wl, double precision wu, integer, dimension( * ) iblock, integer, dimension( * ) indexw, double precision, dimension( * ) work, integer, dimension( * ) iwork, integer info)

DLARRD computes the eigenvalues of a symmetric tridiagonal matrix to suitable accuracy.

Purpose:

    !>
    !> DLARRD computes the eigenvalues of a symmetric tridiagonal
    !> matrix T to suitable accuracy. This is an auxiliary code to be
    !> called from DSTEMR.
    !> The user may ask for all eigenvalues, all eigenvalues
    !> in the half-open interval (VL, VU], or the IL-th through IU-th
    !> eigenvalues.
    !>
    !> To avoid overflow, the matrix must be scaled so that its
    !> largest element is no greater than overflow**(1/2) * underflow**(1/4) in absolute value, and for greatest
    !> accuracy, it should not be much smaller than that.
    !>
    !> See W. Kahan , Report CS41, Computer Science Dept., Stanford
    !> University, July 21, 1966.
    !>

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.
    !>

    ORDER

    !>          ORDER is CHARACTER*1
    !>          = 'B': () the eigenvalues will be grouped by
    !>                              split-off block (see IBLOCK, ISPLIT) and
    !>                              ordered from smallest to largest within
    !>                              the block.
    !>          = 'E': ()
    !>                              the eigenvalues for the entire matrix
    !>                              will be ordered from smallest to
    !>                              largest.
    !>

    N

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

    VL

    !>          VL is DOUBLE PRECISION
    !>          If RANGE='V', the lower bound of the interval to
    !>          be searched for eigenvalues.  Eigenvalues less than or equal
    !>          to VL, or greater than VU, will not be returned.  VL < VU.
    !>          Not referenced if RANGE = 'A' or 'I'.
    !>

    VU

    !>          VU is DOUBLE PRECISION
    !>          If RANGE='V', the upper bound of the interval to
    !>          be searched for eigenvalues.  Eigenvalues less than or equal
    !>          to VL, or greater than VU, will not be returned.  VL < VU.
    !>          Not referenced if RANGE = 'A' or 'I'.
    !>

    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'.
    !>

    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'.
    !>

    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)).
    !>

    RELTOL

    !>          RELTOL is DOUBLE PRECISION
    !>          The minimum relative width of an interval.  When an interval
    !>          is narrower than RELTOL times the larger (in
    !>          magnitude) endpoint, then it is considered to be
    !>          sufficiently small, i.e., converged.  Note: this should
    !>          always be at least radix*machine epsilon.
    !>

    D

    !>          D is DOUBLE PRECISION array, dimension (N)
    !>          The n diagonal elements of the tridiagonal matrix T.
    !>

    E

    !>          E is DOUBLE PRECISION array, dimension (N-1)
    !>          The (n-1) off-diagonal elements of the tridiagonal matrix T.
    !>

    E2

    !>          E2 is DOUBLE PRECISION array, dimension (N-1)
    !>          The (n-1) squared off-diagonal elements of the tridiagonal matrix T.
    !>

    PIVMIN

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

    NSPLIT

    !>          NSPLIT is INTEGER
    !>          The number of diagonal blocks in the matrix T.
    !>          1 <= NSPLIT <= N.
    !>

    ISPLIT

    !>          ISPLIT is INTEGER array, dimension (N)
    !>          The splitting points, at which T breaks up into submatrices.
    !>          The first submatrix 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.
    !>          (Only the first NSPLIT elements will actually be used, but
    !>          since the user cannot know a priori what value NSPLIT will
    !>          have, N words must be reserved for ISPLIT.)
    !>

    M

    !>          M is INTEGER
    !>          The actual number of eigenvalues found. 0 <= M <= N.
    !>          (See also the description of INFO=2,3.)
    !>

    W

    !>          W is DOUBLE PRECISION array, dimension (N)
    !>          On exit, the first M elements of W will contain the
    !>          eigenvalue approximations. DLARRD computes an interval
    !>          I_j = (a_j, b_j] that includes eigenvalue j. The eigenvalue
    !>          approximation is given as the interval midpoint
    !>          W(j)= ( a_j + b_j)/2. The corresponding error is bounded by
    !>          WERR(j) = abs( a_j - b_j)/2
    !>

    WERR

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

    WL

    !>          WL is DOUBLE PRECISION
    !>

    WU

    !>          WU is DOUBLE PRECISION
    !>          The interval (WL, WU] contains all the wanted eigenvalues.
    !>          If RANGE='V', then WL=VL and WU=VU.
    !>          If RANGE='A', then WL and WU are the global Gerschgorin bounds
    !>                        on the spectrum.
    !>          If RANGE='I', then WL and WU are computed by DLAEBZ from the
    !>                        index range specified.
    !>

    IBLOCK

    !>          IBLOCK is INTEGER array, dimension (N)
    !>          At each row/column j where E(j) is zero or small, the
    !>          matrix T is considered to split into a block diagonal
    !>          matrix.  On exit, if INFO = 0, IBLOCK(i) specifies to which
    !>          block (from 1 to the number of blocks) the eigenvalue W(i)
    !>          belongs.  (DLARRD may use the remaining N-M elements as
    !>          workspace.)
    !>

    INDEXW

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

    WORK

    !>          WORK is DOUBLE PRECISION array, dimension (4*N)
    !>

    IWORK

    !>          IWORK is INTEGER array, dimension (3*N)
    !>

    INFO

    !>          INFO is INTEGER
    !>          = 0:  successful exit
    !>          < 0:  if INFO = -i, the i-th argument had an illegal value
    !>          > 0:  some or all of the eigenvalues failed to converge or
    !>                were not computed:
    !>                =1 or 3: Bisection failed to converge for some
    !>                        eigenvalues; these eigenvalues are flagged by a
    !>                        negative block number.  The effect is that the
    !>                        eigenvalues may not be as accurate as the
    !>                        absolute and relative tolerances.  This is
    !>                        generally caused by unexpectedly inaccurate
    !>                        arithmetic.
    !>                =2 or 3: RANGE='I' only: Not all of the eigenvalues
    !>                        IL:IU were found.
    !>                        Effect: M < IU+1-IL
    !>                        Cause:  non-monotonic arithmetic, causing the
    !>                                Sturm sequence to be non-monotonic.
    !>                        Cure:   recalculate, using RANGE='A', and pick
    !>                                out eigenvalues IL:IU.  In some cases,
    !>                                increasing the PARAMETER  may
    !>                                make things work.
    !>                = 4:    RANGE='I', and the Gershgorin interval
    !>                        initially used was too small.  No eigenvalues
    !>                        were computed.
    !>                        Probable cause: your machine has sloppy
    !>                                        floating-point arithmetic.
    !>                        Cure: Increase the PARAMETER ,
    !>                              recompile, and try again.
    !>

Internal Parameters:

    !>  FUDGE   DOUBLE PRECISION, default = 2
    !>          A  to widen the Gershgorin intervals.  Ideally,
    !>          a value of 1 should work, but on machines with sloppy
    !>          arithmetic, this needs to be larger.  The default for
    !>          publicly released versions should be large enough to handle
    !>          the worst machine around.  Note that this has no effect
    !>          on accuracy of the solution.
    !>

Contributors:
    W. Kahan, University of California, Berkeley, USA
    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
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

  subroutine slarrd (character range, character order, integer n, real vl, real vu, integer il, integer iu, real, dimension( * ) gers, real reltol, real, dimension( * ) d, real, dimension( * ) e, real, dimension( * ) e2, real pivmin, integer nsplit, integer, dimension( * ) isplit, integer m, real, dimension( * ) w, real, dimension( * ) werr, real wl, real wu, integer, dimension( * ) iblock, integer, dimension( * ) indexw, real, dimension( * ) work, integer, dimension( * ) iwork, integer info)

SLARRD computes the eigenvalues of a symmetric tridiagonal matrix to suitable accuracy.

Purpose:

    !>
    !> SLARRD computes the eigenvalues of a symmetric tridiagonal
    !> matrix T to suitable accuracy. This is an auxiliary code to be
    !> called from SSTEMR.
    !> The user may ask for all eigenvalues, all eigenvalues
    !> in the half-open interval (VL, VU], or the IL-th through IU-th
    !> eigenvalues.
    !>
    !> To avoid overflow, the matrix must be scaled so that its
    !> largest element is no greater than overflow**(1/2) * underflow**(1/4) in absolute value, and for greatest
    !> accuracy, it should not be much smaller than that.
    !>
    !> See W. Kahan , Report CS41, Computer Science Dept., Stanford
    !> University, July 21, 1966.
    !>

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.
    !>

    ORDER

    !>          ORDER is CHARACTER*1
    !>          = 'B': () the eigenvalues will be grouped by
    !>                              split-off block (see IBLOCK, ISPLIT) and
    !>                              ordered from smallest to largest within
    !>                              the block.
    !>          = 'E': ()
    !>                              the eigenvalues for the entire matrix
    !>                              will be ordered from smallest to
    !>                              largest.
    !>

    N

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

    VL

    !>          VL is REAL
    !>          If RANGE='V', the lower bound of the interval to
    !>          be searched for eigenvalues.  Eigenvalues less than or equal
    !>          to VL, or greater than VU, will not be returned.  VL < VU.
    !>          Not referenced if RANGE = 'A' or 'I'.
    !>

    VU

    !>          VU is REAL
    !>          If RANGE='V', the upper bound of the interval to
    !>          be searched for eigenvalues.  Eigenvalues less than or equal
    !>          to VL, or greater than VU, will not be returned.  VL < VU.
    !>          Not referenced if RANGE = 'A' or 'I'.
    !>

    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'.
    !>

    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'.
    !>

    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)).
    !>

    RELTOL

    !>          RELTOL is REAL
    !>          The minimum relative width of an interval.  When an interval
    !>          is narrower than RELTOL times the larger (in
    !>          magnitude) endpoint, then it is considered to be
    !>          sufficiently small, i.e., converged.  Note: this should
    !>          always be at least radix*machine epsilon.
    !>

    D

    !>          D is REAL array, dimension (N)
    !>          The n diagonal elements of the tridiagonal matrix T.
    !>

    E

    !>          E is REAL array, dimension (N-1)
    !>          The (n-1) off-diagonal elements of the tridiagonal matrix T.
    !>

    E2

    !>          E2 is REAL array, dimension (N-1)
    !>          The (n-1) squared off-diagonal elements of the tridiagonal matrix T.
    !>

    PIVMIN

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

    NSPLIT

    !>          NSPLIT is INTEGER
    !>          The number of diagonal blocks in the matrix T.
    !>          1 <= NSPLIT <= N.
    !>

    ISPLIT

    !>          ISPLIT is INTEGER array, dimension (N)
    !>          The splitting points, at which T breaks up into submatrices.
    !>          The first submatrix 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.
    !>          (Only the first NSPLIT elements will actually be used, but
    !>          since the user cannot know a priori what value NSPLIT will
    !>          have, N words must be reserved for ISPLIT.)
    !>

    M

    !>          M is INTEGER
    !>          The actual number of eigenvalues found. 0 <= M <= N.
    !>          (See also the description of INFO=2,3.)
    !>

    W

    !>          W is REAL array, dimension (N)
    !>          On exit, the first M elements of W will contain the
    !>          eigenvalue approximations. SLARRD computes an interval
    !>          I_j = (a_j, b_j] that includes eigenvalue j. The eigenvalue
    !>          approximation is given as the interval midpoint
    !>          W(j)= ( a_j + b_j)/2. The corresponding error is bounded by
    !>          WERR(j) = abs( a_j - b_j)/2
    !>

    WERR

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

    WL

    !>          WL is REAL
    !>

    WU

    !>          WU is REAL
    !>          The interval (WL, WU] contains all the wanted eigenvalues.
    !>          If RANGE='V', then WL=VL and WU=VU.
    !>          If RANGE='A', then WL and WU are the global Gerschgorin bounds
    !>                        on the spectrum.
    !>          If RANGE='I', then WL and WU are computed by SLAEBZ from the
    !>                        index range specified.
    !>

    IBLOCK

    !>          IBLOCK is INTEGER array, dimension (N)
    !>          At each row/column j where E(j) is zero or small, the
    !>          matrix T is considered to split into a block diagonal
    !>          matrix.  On exit, if INFO = 0, IBLOCK(i) specifies to which
    !>          block (from 1 to the number of blocks) the eigenvalue W(i)
    !>          belongs.  (SLARRD may use the remaining N-M elements as
    !>          workspace.)
    !>

    INDEXW

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

    WORK

    !>          WORK is REAL array, dimension (4*N)
    !>

    IWORK

    !>          IWORK is INTEGER array, dimension (3*N)
    !>

    INFO

    !>          INFO is INTEGER
    !>          = 0:  successful exit
    !>          < 0:  if INFO = -i, the i-th argument had an illegal value
    !>          > 0:  some or all of the eigenvalues failed to converge or
    !>                were not computed:
    !>                =1 or 3: Bisection failed to converge for some
    !>                        eigenvalues; these eigenvalues are flagged by a
    !>                        negative block number.  The effect is that the
    !>                        eigenvalues may not be as accurate as the
    !>                        absolute and relative tolerances.  This is
    !>                        generally caused by unexpectedly inaccurate
    !>                        arithmetic.
    !>                =2 or 3: RANGE='I' only: Not all of the eigenvalues
    !>                        IL:IU were found.
    !>                        Effect: M < IU+1-IL
    !>                        Cause:  non-monotonic arithmetic, causing the
    !>                                Sturm sequence to be non-monotonic.
    !>                        Cure:   recalculate, using RANGE='A', and pick
    !>                                out eigenvalues IL:IU.  In some cases,
    !>                                increasing the PARAMETER  may
    !>                                make things work.
    !>                = 4:    RANGE='I', and the Gershgorin interval
    !>                        initially used was too small.  No eigenvalues
    !>                        were computed.
    !>                        Probable cause: your machine has sloppy
    !>                                        floating-point arithmetic.
    !>                        Cure: Increase the PARAMETER ,
    !>                              recompile, and try again.
    !>

Internal Parameters:

    !>  FUDGE   REAL, default = 2
    !>          A  to widen the Gershgorin intervals.  Ideally,
    !>          a value of 1 should work, but on machines with sloppy
    !>          arithmetic, this needs to be larger.  The default for
    !>          publicly released versions should be large enough to handle
    !>          the worst machine around.  Note that this has no effect
    !>          on accuracy of the solution.
    !>

Contributors:
    W. Kahan, University of California, Berkeley, USA
    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
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Author

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