dlaebz(3S)

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( * )

IMPLEMENTATION
     These routines are part of the SCSL Scientific Library and can be loaded
     using either the -lscs or the -lscs_mp option.  The -lscs_mp option
     directs the linker to use the multi-processor version of the library.

     When linking to SCSL with -lscs or -lscs_mp, the default integer size is
     4 bytes (32 bits). Another version of SCSL is available in which integers
     are 8 bytes (64 bits).  This version allows the user access to larger
     memory sizes and helps when porting legacy Cray codes.  It can be loaded
     by using the -lscs_i8 option or the -lscs_i8_mp option. A program may use
     only one of the two versions; 4-byte integer and 8-byte integer library
     calls cannot be mixed.

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.


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.


SEE ALSO
     INTRO_LAPACK(3S), INTRO_SCSL(3S)

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