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## dstebz()

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

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

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

  [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]

  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.
      !>

  [in]

  N

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

  [in]

  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'.
      !>

  [in]

  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'.
      !>

  [in]

  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'.
      !>

  [in]

  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'.
      !>

  [in]

  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.
      !>

  [in]

  D

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

  [in]

  E

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

  [out]

  M

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

  [out]

  NSPLIT

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

  [out]

  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.)
      !>

  [out]

  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.)
      !>

  [out]

  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.)
      !>

  [out]

  WORK

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

  [out]

  IWORK

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

  [out]

  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.