DLAEBZ
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 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.
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.
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 considered 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 considered 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 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. |
| 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 eigenvalues 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) 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 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 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.
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 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.