slaebz(3)

NAME

SLAEBZ - contain the iteration loops which compute and use

the function N(w), which is the count of eigenvalues of a symmet

ric 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 symmet

ric 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 inter

vals.
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 out

put
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 con

taining 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 .

To avoid underflow, the matrix should be scaled so that

its largest element is no greater than overflow**(1/2) * under

flow**(1/4) in absolute value. To assure the most accurate com

putation of small eigenvalues, the matrix should be scaled to be
not much smaller than that, either.

See W. Kahan "Accurate Eigenvalues of a Symmetric Tridiag

onal Matrix", Report CS41, Computer Science Dept., Stanford
University, July 21, 1966

Note: the arguments are, in general, *not* checked for un

reasonable 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 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-con

verged 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 IN

FO=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 magnitude) endpoint, then it is considered to be suf

ficiently 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 magnitude) endpoint, then it is considered to be suf

ficiently 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 inter

vals. 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 in

put 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 con

verge.
= MMAX+1: More than MMAX intervals were generat

ed.

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) de

creased.
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(3)