SLAEBZ(l)								 )								 SLAEBZ(l)

NAME

       SLAEBZ  - contain 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

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 abso-
       lute 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  magni-
	       tude) 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 magni-
	       tude) 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.

LAPACK version 3.0						   15 June 2000 							 SLAEBZ(l)