# ?laebz

Computes the number of eigenvalues of a real symmetric tridiagonal matrix which are less than or equal to a given value, and performs other tasks required by the routine ?stebz.

## Syntax

call slaebz( ijob, nitmax, n, mmax, minp, nbmin, abstol, reltol, pivmin, d, e, e2, nval, ab, c, mout, nab, work, iwork, info )

call dlaebz( ijob, nitmax, n, mmax, minp, nbmin, abstol, reltol, pivmin, d, e, e2, nval, ab, c, mout, nab, work, iwork, info )

## Description

The routine ?laebz 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, that is, 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 `overflow1/2 * overflow1/4` 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.

### Note

In general, the arguments are not checked for unreasonable values.

## Input Parameters

ijob

  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 ?laebz to return with `info=-1`.

nitmax

  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

  INTEGER. The dimension n of the tridiagonal matrix T. It must be at least 1.

mmax

  INTEGER. The maximum number of intervals. If more than mmax intervals are generated, then ?laebz will quit with `info=mmax+1`.

minp

  INTEGER. The initial number of intervals. It may not be greater than mmax.

nbmin

  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

  REAL for slaebz

  DOUBLE PRECISION for dlaebz.

  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

  REAL for slaebz

  DOUBLE PRECISION for dlaebz.

  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

  REAL for slaebz

  DOUBLE PRECISION for dlaebz.

  The minimum absolute value of a "pivot" in the Sturm sequence loop. This value 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, e, e2

  REAL for slaebz

  DOUBLE PRECISION for dlaebz.

  Arrays, dimension (n) each. The array d contains the diagonal elements of the tridiagonal matrix T.

  The array e contains the off-diagonal elements of the tridiagonal matrix T in positions 1 through n-1. e(n)vis arbitrary.

  The array e2 contains the squares of the off-diagonal elements of the tridiagonal matrix T. e2(n) is ignored.

nval

  INTEGER.

  Array, dimension (minp).

  If ijob=1 or 2, not referenced.

  If ijob=3, the desired values of n(w).

ab

  REAL for slaebz

  DOUBLE PRECISION for dlaebz.

  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.

c

  REAL for slaebz

  DOUBLE PRECISION for dlaebz.

  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.

nab

  INTEGER.

  Array, dimension (mmax,2)

  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)` are considered. Usually, `nab(i,j)=n(ab(i,j))`, from a previous call to ?laebz with ijob=1.

  If ijob=3, normally, nab should be set to some distinctive value(s) before ?laebz is called.

work

  REAL for slaebz

  DOUBLE PRECISION for dlaebz.

  Workspace array, dimension (mmax).

iwork

  INTEGER.

  Workspace array, dimension (mmax).

## Output Parameters

nval

  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

  The input intervals will, in general, be modified, split, and reordered by the calculation.

mout

  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

  If ijob=1, then on output nab(i,j) will be set to N(ab(i,j)).

  If ijob=2, then 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 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.

info

  INTEGER.

  If `info = 0` - all intervals converged

  If `info = 1--mmax` - the last info interval did not converge.

  If `info = mmax+1` - more than mmax intervals were generated

## Application Notes

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, ?laebz should have one or more initial intervals set up in ab, and ?laebz 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. ?laebz 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, respectively. 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). ?laebz 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.

Parent topic: Auxiliary Routines