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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
argument 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
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 contain-
ing 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 inter-
vals, i.e., of the form (a,b] , which includes b but not
a .
See W. Kahan "Accurate Eigenvalues of a Symmetric Tridiago-
nal 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 eigen-
values 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 pro-
cessed 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
considered to be sufficiently small, i.e., con-
verged. 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
considered to be sufficiently small, i.e., con-
verged. 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.
To avoid underflow, the matrix should be scaled so
that its largest entry is no greater than over-
flow**(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.
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. To
avoid underflow, the matrix should be scaled so that
its largest entry is no greater than over-
flow**(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.
E2 (input) DOUBLE PRECISION array, dimension (N)
The squares of the offdiagonal elements of the tri-
diagonal 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) 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.
(MMAX)
C (input/workspace) DOUBLE PRECISION array, dimension
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 inter-
vals. If IJOB=2 or 3, the number of intervals out-
put. 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. Usu-
ally, NAB(i,j)=N(AB(i,j)), from a previous call to
DLAEBZ with IJOB=1. On output, NAB(i,j) will con-
tain 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 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 modi-
fied, 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. Nor-
mally, 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 interface 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 entry 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.