slaebz.f(3)							      LAPACK							       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

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

       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:
	   November 2011

       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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Version 3.4.1							  Sun May 26 2013						       slaebz.f(3)