slaebz.f(3)

NAME
slaebz.f -

SYNOPSIS

  Functions/Subroutines
subroutine slaebz (IJOB, NITMAX, N, MMAX, MINP, NBMIN, ABSTOL, RELTOL, PIVMIN, D, E, E2, NVAL, AB, C, MOUT, NAB, WORK, IWORK, INFO)
SLAEBZ computes the number of eigenvalues of a real symmetric tridiagonal matrix which are less than or equal to a given value, and performs other tasks required by the routine sstebz.

Function/Subroutine Documentation

  subroutine slaebz (integerIJOB, integerNITMAX, integerN, integerMMAX, integerMINP, integerNBMIN, realABSTOL, realRELTOL, realPIVMIN, real, dimension( * )D, real, dimension( * )E, real, dimension( * )E2, integer, dimension( * )NVAL, real, dimension( mmax, * )AB, real, dimension( * )C, integerMOUT, integer, dimension( mmax, * )NAB, real, dimension( * )WORK, integer, dimension( * )IWORK, integerINFO)

SLAEBZ computes the number of eigenvalues of a real symmetric tridiagonal matrix which are less than or equal to a given value, and performs other tasks required by the routine sstebz.

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.

Parameters:
    IJOB

              IJOB is 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

              NITMAX is 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

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

    MMAX

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

    MINP

              MINP is INTEGER
              The initial number of intervals.  It may not be greater than
              MMAX.

    NBMIN

              NBMIN is 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

              ABSTOL is 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

              RELTOL is 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

              PIVMIN is 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

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

    E

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

    E2

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

    NVAL

              NVAL is 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

              AB is 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

              C is 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

              MOUT is 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

              NAB is 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

              WORK is REAL array, dimension (MMAX)
              Workspace.

    IWORK

              IWORK is INTEGER array, dimension (MMAX)
              Workspace.

    INFO

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

Author:
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date:
    September 2012

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.

Definition at line 318 of file slaebz.f.

Author

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