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SLAEBZ (3lapack)


SYNOPSIS

  SUBROUTINE SLAEBZ( IJOB, NITMAX, N, MMAX, MINP, NBMIN, ABSTOL, RELTOL,
                     PIVMIN, D, E, E2, NVAL, AB, C, MOUT, NAB, WORK, IWORK,
                     INFO )

      INTEGER        IJOB, INFO, MINP, MMAX, MOUT, N, NBMIN, NITMAX

      REAL           ABSTOL, PIVMIN, RELTOL

      INTEGER        IWORK( * ), NAB( MMAX, * ), NVAL( * )

      REAL           AB( MMAX, * ), C( * ), D( * ), E( * ), E2( * ), WORK( *
                     )

PURPOSE

  SLAEBZ contains the iteration loops which compute and use the function
  N(w), which is the count of eigenvalues of a symmetric tridiagonal matrix T
  less than or equal to its argument  w.  It performs a choice of two types
  of loops:

  IJOB=1, followed by
  IJOB=2: It takes as input a list of intervals and returns a list of
          sufficiently small intervals whose union contains the same
          eigenvalues as the union of the original intervals.
          The input intervals are (AB(j,1),AB(j,2)], j=1,...,MINP.
          The output interval (AB(j,1),AB(j,2)] will contain
          eigenvalues NAB(j,1)+1,...,NAB(j,2), where 1 <= j <= MOUT.

  IJOB=3: It performs a binary search in each input interval
          (AB(j,1),AB(j,2)] for a point  w(j)  such that
          N(w(j))=NVAL(j), and uses  C(j)  as the starting point of
          the search.  If such a w(j) is found, then on output
          AB(j,1)=AB(j,2)=w.  If no such w(j) is found, then on output
          (AB(j,1),AB(j,2)] will be a small interval containing the
          point where N(w) jumps through NVAL(j), unless that point
          lies outside the initial interval.

  Note that the intervals are in all cases half-open intervals, i.e., of the
  form  (a,b] , which includes  b  but not  a .

  To avoid underflow, the matrix should be scaled so that its largest element
  is no greater than  overflow**(1/2) * underflow**(1/4) in absolute value.
  To assure the most accurate computation of small eigenvalues, the matrix
  should be scaled to be
  not much smaller than that, either.

  See W. Kahan "Accurate Eigenvalues of a Symmetric Tridiagonal Matrix",
  Report CS41, Computer Science Dept., Stanford
  University, July 21, 1966

  Note: the arguments are, in general, *not* checked for unreasonable values.

ARGUMENTS

  IJOB    (input) INTEGER
          Specifies what is to be done:
          = 1:  Compute NAB for the initial intervals.
          = 2:  Perform bisection iteration to find eigenvalues of T.
          = 3:  Perform bisection iteration to invert N(w), i.e., to find a
          point which has a specified number of eigenvalues of T to its left.
          Other values will cause SLAEBZ to return with INFO=-1.

  NITMAX  (input) INTEGER
          The maximum number of "levels" of bisection to be performed, i.e.,
          an interval of width W will not be made smaller than 2^(-NITMAX) *
          W.  If not all intervals have converged after NITMAX iterations,
          then INFO is set to the number of non-converged intervals.

  N       (input) INTEGER
          The dimension n of the tridiagonal matrix T.  It must be at least
          1.

  MMAX    (input) INTEGER
          The maximum number of intervals.  If more than MMAX intervals are
          generated, then SLAEBZ will quit with INFO=MMAX+1.

  MINP    (input) INTEGER
          The initial number of intervals.  It may not be greater than MMAX.

  NBMIN   (input) INTEGER
          The smallest number of intervals that should be processed using a
          vector loop.  If zero, then only the scalar loop will be used.

  ABSTOL  (input) REAL
          The minimum (absolute) width of an interval.  When an interval is
          narrower than ABSTOL, or than RELTOL times the larger (in
          magnitude) endpoint, then it is considered to be sufficiently
          small, i.e., converged.  This must be at least zero.

  RELTOL  (input) REAL
          The minimum relative width of an interval.  When an interval is
          narrower than ABSTOL, or 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.

  PIVMIN  (input) REAL
          The minimum absolute value of a "pivot" in the Sturm sequence loop.
          This *must* be at least  max |e(j)**2| * safe_min  and at least
          safe_min, where safe_min is at least the smallest number that can
          divide one without overflow.

  D       (input) REAL array, dimension (N)
          The diagonal elements of the tridiagonal matrix T.

  E       (input) REAL array, dimension (N)
          The offdiagonal elements of the tridiagonal matrix T in positions 1
          through N-1.  E(N) is arbitrary.

  E2      (input) REAL array, dimension (N)
          The squares of the offdiagonal elements of the tridiagonal matrix
          T.  E2(N) is ignored.

  NVAL    (input/output) INTEGER array, dimension (MINP)
          If IJOB=1 or 2, not referenced.  If IJOB=3, the desired values of
          N(w).  The elements of NVAL will be reordered to correspond with
          the intervals in AB.  Thus, NVAL(j) on output will not, in general
          be the same as NVAL(j) on input, but it will correspond with the
          interval (AB(j,1),AB(j,2)] on output.

  AB      (input/output) REAL array, dimension (MMAX,2)
          The endpoints of the intervals.  AB(j,1) is  a(j), the left
          endpoint of the j-th interval, and AB(j,2) is b(j), the right
          endpoint of the j-th interval.  The input intervals will, in
          general, be modified, split, and reordered by the calculation.

  C       (input/output) REAL array, dimension (MMAX)
          If IJOB=1, ignored.  If IJOB=2, workspace.  If IJOB=3, then on
          input C(j) should be initialized to the first search point in the
          binary search.

  MOUT    (output) INTEGER
          If IJOB=1, the number of eigenvalues in the intervals.  If IJOB=2
          or 3, the number of intervals output.  If IJOB=3, MOUT will equal
          MINP.

  NAB     (input/output) INTEGER array, dimension (MMAX,2)
          If IJOB=1, then on output NAB(i,j) will be set to N(AB(i,j)).  If
          IJOB=2, then on input, NAB(i,j) should be set.  It must satisfy the
          condition: N(AB(i,1)) <= NAB(i,1) <= NAB(i,2) <= N(AB(i,2)), which
          means that in interval i only eigenvalues NAB(i,1)+1,...,NAB(i,2)
          will be considered.  Usually, NAB(i,j)=N(AB(i,j)), from a previous
          call to SLAEBZ with IJOB=1.  On output, NAB(i,j) will contain
          max(na(k),min(nb(k),N(AB(i,j)))), where k is the index of the input
          interval that the output interval (AB(j,1),AB(j,2)] came from, and
          na(k) and nb(k) are the the input values of NAB(k,1) and NAB(k,2).
          If IJOB=3, then on output, NAB(i,j) contains N(AB(i,j)), unless
          N(w) > NVAL(i) for all search points  w , in which case NAB(i,1)
          will not be modified, i.e., the output value will be the same as
          the input value (modulo reorderings -- see NVAL and AB), or unless
          N(w) < NVAL(i) for all search points  w , in which case NAB(i,2)
          will not be modified.  Normally, NAB should be set to some
          distinctive value(s) before SLAEBZ is called.

  WORK    (workspace) REAL array, dimension (MMAX)
          Workspace.

  IWORK   (workspace) INTEGER array, dimension (MMAX)
          Workspace.

  INFO    (output) INTEGER
          = 0:       All intervals converged.
          = 1--MMAX: The last INFO intervals did not converge.
          = MMAX+1:  More than MMAX intervals were generated.

FURTHER DETAILS

      This routine is intended to be called only by other LAPACK routines,
  thus the interface is less user-friendly.  It is intended for two purposes:

  (a) finding eigenvalues.  In this case, SLAEBZ should have one or
      more initial intervals set up in AB, and SLAEBZ should be called
      with IJOB=1.  This sets up NAB, and also counts the eigenvalues.
      Intervals with no eigenvalues would usually be thrown out at
      this point.  Also, if not all the eigenvalues in an interval i
      are desired, NAB(i,1) can be increased or NAB(i,2) decreased.
      For example, set NAB(i,1)=NAB(i,2)-1 to get the largest
      eigenvalue.  SLAEBZ is then called with IJOB=2 and MMAX
      no smaller than the value of MOUT returned by the call with
      IJOB=1.  After this (IJOB=2) call, eigenvalues NAB(i,1)+1
      through NAB(i,2) are approximately AB(i,1) (or AB(i,2)) to the
      tolerance specified by ABSTOL and RELTOL.

  (b) finding an interval (a',b'] containing eigenvalues w(f),...,w(l).
      In this case, start with a Gershgorin interval  (a,b).  Set up
      AB to contain 2 search intervals, both initially (a,b).  One
      NVAL element should contain  f-1  and the other should contain  l
      , while C should contain a and b, resp.  NAB(i,1) should be -1
      and NAB(i,2) should be N+1, to flag an error if the desired
      interval does not lie in (a,b).  SLAEBZ is then called with
      IJOB=3.  On exit, if w(f-1) < w(f), then one of the intervals --
      j -- will have AB(j,1)=AB(j,2) and NAB(j,1)=NAB(j,2)=f-1, while
      if, to the specified tolerance, w(f-k)=...=w(f+r), k > 0 and r
      >= 0, then the interval will have  N(AB(j,1))=NAB(j,1)=f-k and
      N(AB(j,2))=NAB(j,2)=f+r.  The cases w(l) < w(l+1) and
      w(l-r)=...=w(l+k) are handled similarly.

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