stebz(3)

NAME
stebz - stebz: eig, Kahan

SYNOPSIS

  Functions

subroutine dstebz (range, order, n, vl, vu, il, iu, abstol, d, e, m, nsplit, w, iblock, isplit, work, iwork, info)
DSTEBZ
subroutine sstebz (range, order, n, vl, vu, il, iu, abstol, d, e, m, nsplit, w, iblock, isplit, work, iwork, info)
SSTEBZ

Detailed Description

Function Documentation

  subroutine dstebz (character range, character order, integer n, double precision vl, double precision vu, integer il, integer iu, double precision abstol, double precision, dimension( * ) d, double precision, dimension( * ) e, integer m, integer nsplit, double precision, dimension( * ) w, integer, dimension( * ) iblock, integer, dimension( * ) isplit, double precision, dimension( * ) work, integer, dimension( * ) iwork, integer info)

DSTEBZ

Purpose:

    !>
    !> DSTEBZ computes the eigenvalues of a symmetric tridiagonal
    !> matrix T.  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'.
    !>

    ABSTOL

    !>          ABSTOL is DOUBLE PRECISION
    !>          The absolute tolerance for the eigenvalues.  An eigenvalue
    !>          (or cluster) is considered to be located if it has been
    !>          determined to lie in an interval whose width is ABSTOL or
    !>          less.  If ABSTOL is less than or equal to zero, then ULP*|T|
    !>          will be used, where |T| means the 1-norm of T.
    !>
    !>          Eigenvalues will be computed most accurately when ABSTOL is
    !>          set to twice the underflow threshold 2*DLAMCH('S'), not zero.
    !>

    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.
    !>

    M

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

    NSPLIT

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

    W

    !>          W is DOUBLE PRECISION array, dimension (N)
    !>          On exit, the first M elements of W will contain the
    !>          eigenvalues.  (DSTEBZ may use the remaining N-M elements as
    !>          workspace.)
    !>

    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.  (DSTEBZ may use the remaining N-M elements as
    !>          workspace.)
    !>

    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.)
    !>

    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:

    !>  RELFAC  DOUBLE PRECISION, default = 2.0e0
    !>          The relative tolerance.  An interval (a,b] lies within
    !>           if  b-a < RELFAC*ulp*max(|a|,|b|),
    !>          where  is the machine precision (distance from 1 to
    !>          the next larger floating point number.)
    !>
    !>  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.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

  subroutine sstebz (character range, character order, integer n, real vl, real vu, integer il, integer iu, real abstol, real, dimension( * ) d, real, dimension( * ) e, integer m, integer nsplit, real, dimension( * ) w, integer, dimension( * ) iblock, integer, dimension( * ) isplit, real, dimension( * ) work, integer, dimension( * ) iwork, integer info)

SSTEBZ

Purpose:

    !>
    !> SSTEBZ computes the eigenvalues of a symmetric tridiagonal
    !> matrix T.  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'.
    !>

    ABSTOL

    !>          ABSTOL is REAL
    !>          The absolute tolerance for the eigenvalues.  An eigenvalue
    !>          (or cluster) is considered to be located if it has been
    !>          determined to lie in an interval whose width is ABSTOL or
    !>          less.  If ABSTOL is less than or equal to zero, then ULP*|T|
    !>          will be used, where |T| means the 1-norm of T.
    !>
    !>          Eigenvalues will be computed most accurately when ABSTOL is
    !>          set to twice the underflow threshold 2*SLAMCH('S'), not zero.
    !>

    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.
    !>

    M

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

    NSPLIT

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

    W

    !>          W is REAL array, dimension (N)
    !>          On exit, the first M elements of W will contain the
    !>          eigenvalues.  (SSTEBZ may use the remaining N-M elements as
    !>          workspace.)
    !>

    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.  (SSTEBZ may use the remaining N-M elements as
    !>          workspace.)
    !>

    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.)
    !>

    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:

    !>  RELFAC  REAL, default = 2.0e0
    !>          The relative tolerance.  An interval (a,b] lies within
    !>           if  b-a < RELFAC*ulp*max(|a|,|b|),
    !>          where  is the machine precision (distance from 1 to
    !>          the next larger floating point number.)
    !>
    !>  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.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Author

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