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

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( * )

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.

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 magnitude) 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 magnitude) 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.

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.