SLAEBZ(3)      LAPACK routine of NEC Numeric Library Collection      SLAEBZ(3)



NAME
       SLAEBZ

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



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.


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




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

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

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

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

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

           NVAL      (input/output)
                     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        (input/output)
                     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         (input/output)
                     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      (output)
                     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       (input/output)
                     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      (output)
                     WORK is REAL array, dimension (MMAX)
                     Workspace.

           IWORK     (output)
                     IWORK is INTEGER array, dimension (MMAX)
                     Workspace.

           INFO      (output)
                     INFO is 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.



LAPACK routine                  31 October 2017                      SLAEBZ(3)