laebz(3)

NAME
laebz - laebz: counts eigvals <= value

SYNOPSIS

  Functions

subroutine dlaebz (ijob, nitmax, n, mmax, minp, nbmin, abstol, reltol, pivmin, d, e, e2, nval, ab, c, mout, nab, work, iwork, info)
DLAEBZ 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 sstebz.
subroutine slaebz (ijob, nitmax, n, mmax, minp, nbmin, abstol, reltol, pivmin, d, e, e2, nval, ab, c, mout, nab, work, iwork, info)
SLAEBZ 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 sstebz.

Detailed Description

Function Documentation

  subroutine dlaebz (integer ijob, integer nitmax, integer n, integer mmax, integer minp, integer nbmin, double precision abstol, double precision reltol, double precision pivmin, double precision, dimension( * ) d, double precision, dimension( * ) e, double precision, dimension( * ) e2, integer, dimension( * ) nval, double precision, dimension( mmax, * ) ab, double precision, dimension( * ) c, integer mout, integer, dimension( mmax, * ) nab, double precision, dimension( * ) work, integer, dimension( * ) iwork, integer info)

DLAEBZ 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 sstebz.

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 , Report CS41, Computer Science Dept., Stanford
    !> University, July 21, 1966
    !>
    !> Note: the arguments are, in general, *not* checked for unreasonable
    !> values.
    !>

Parameters
    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.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

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.
    !>

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

SLAEBZ 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 sstebz.

Purpose:

    !>
    !> 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 , Report CS41, Computer Science Dept., Stanford
    !> University, July 21, 1966
    !>
    !> Note: the arguments are, in general, *not* checked for unreasonable
    !> values.
    !>

Parameters
    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  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  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.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

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, 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.
    !>

Author

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