!>
!> 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 , 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 DLAEBZ to return with INFO=-1.
!> 

NITMAX

!>          NITMAX is INTEGER
!>          The maximum number of  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 DLAEBZ 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 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

!>          RELTOL is 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

!>          PIVMIN is DOUBLE PRECISION
!>          The minimum absolute value of a  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 DOUBLE PRECISION array, dimension (N)
!>          The diagonal elements of the tridiagonal matrix T.
!> 

E

!>          E is DOUBLE PRECISION array, dimension (N)
!>          The offdiagonal elements of the tridiagonal matrix T in
!>          positions 1 through N-1.  E(N) is arbitrary.
!> 

E2

!>          E2 is DOUBLE PRECISION 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 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

!>          C is 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

!>          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 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

!>          WORK is DOUBLE PRECISION 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.
!> 

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NAG Ltd.

!>
!>      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.
!>