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RedHat 9 (Linux i386) - man page for dlaebz (redhat section l)

DLAEBZ(l)					)					DLAEBZ(l)

NAME
       DLAEBZ - 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  argu-
       ment w

SYNOPSIS
       SUBROUTINE DLAEBZ( 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

	   DOUBLE	  PRECISION ABSTOL, PIVMIN, RELTOL

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

	   DOUBLE	  PRECISION AB( MMAX, * ), C( * ), D( * ), E( * ), E2( * ), WORK( * )

PURPOSE
       DLAEBZ 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 argu-
       ment 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, Com-
       puter 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
	       DLAEBZ 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
	       DLAEBZ 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) DOUBLE PRECISION
	       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  con-
	       sidered to be sufficiently small, i.e., converged.  This must be at least zero.

       RELTOL  (input) DOUBLE PRECISION
	       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  con-
	       sidered to be sufficiently small, i.e., converged.  Note: this should always be at
	       least radix*machine epsilon.

       PIVMIN  (input) DOUBLE PRECISION
	       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) DOUBLE PRECISION array, dimension (N)
	       The diagonal elements of the tridiagonal matrix T.

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

       E2      (input) DOUBLE PRECISION 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  ele-
	       ments  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) DOUBLE PRECISION 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) DOUBLE PRECISION 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 eigenval-
	       ues NAB(i,1)+1,...,NAB(i,2) will  be  considered.   Usually,  NAB(i,j)=N(AB(i,j)),
	       from  a	previous  call	to  DLAEBZ 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)  con-
	       tains  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 DLAEBZ is called.

       WORK    (workspace) DOUBLE PRECISION 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 inter-
       face is less user-friendly.  It is intended for two purposes:

       (a) finding eigenvalues.  In this case, DLAEBZ should have one or
	   more initial intervals set up in AB, and DLAEBZ 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.	DLAEBZ 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).  DLAEBZ 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 				DLAEBZ(l)


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