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

          IJOB is 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

          NITMAX is 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

          N is INTEGER


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


          least 1.

MMAX

          MMAX is INTEGER


          The maximum number of intervals.  If more than MMAX intervals


          are generated, then SLAEBZ will quit with INFO=MMAX+1.

MINP

          MINP is INTEGER


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


          MMAX.

NBMIN

          NBMIN is 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

          ABSTOL is 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

          RELTOL is 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

          PIVMIN is 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

          D is REAL array, dimension (N)


          The diagonal elements of the tridiagonal matrix T.

E

          E is REAL array, dimension (N)


          The offdiagonal elements of the tridiagonal matrix T in


          positions 1 through N-1.  E(N) is arbitrary.

E2

          E2 is REAL array, dimension (N)


          The squares of the offdiagonal elements of the tridiagonal


          matrix T.  E2(N) is ignored.

NVAL

          NVAL is 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

          AB is 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

          C is 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

          MOUT is 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

          NAB is 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

          WORK is REAL array, dimension (MMAX)


          Workspace.

IWORK

          IWORK is INTEGER array, dimension (MMAX)


          Workspace.

INFO

          INFO is INTEGER


          = 0:       All intervals converged.


          = 1--MMAX: The last INFO intervals did not converge.


          = MMAX+1:  More than MMAX intervals were generated.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

December 2016

      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.