/* -- translated by f2c (version 20100827).
You must link the resulting object file with libf2c:
on Microsoft Windows system, link with libf2c.lib;
on Linux or Unix systems, link with .../path/to/libf2c.a -lm
or, if you install libf2c.a in a standard place, with -lf2c -lm
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Source for libf2c is in /netlib/f2c/libf2c.zip, e.g.,
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*/
#include "f2c.h"
/* > \brief \b 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.
=========== DOCUMENTATION ===========
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Definition:
===========
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( * )
> \par Purpose:
=============
>
> \verbatim
>
> 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.
> \endverbatim
Arguments:
==========
> \param[in] IJOB
> \verbatim
> 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.
> \endverbatim
>
> \param[in] NITMAX
> \verbatim
> 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.
> \endverbatim
>
> \param[in] N
> \verbatim
> N is INTEGER
> The dimension n of the tridiagonal matrix T. It must be at
> least 1.
> \endverbatim
>
> \param[in] MMAX
> \verbatim
> MMAX is INTEGER
> The maximum number of intervals. If more than MMAX intervals
> are generated, then DLAEBZ will quit with INFO=MMAX+1.
> \endverbatim
>
> \param[in] MINP
> \verbatim
> MINP is INTEGER
> The initial number of intervals. It may not be greater than
> MMAX.
> \endverbatim
>
> \param[in] NBMIN
> \verbatim
> 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.
> \endverbatim
>
> \param[in] ABSTOL
> \verbatim
> 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.
> \endverbatim
>
> \param[in] RELTOL
> \verbatim
> 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.
> \endverbatim
>
> \param[in] PIVMIN
> \verbatim
> PIVMIN is 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.
> \endverbatim
>
> \param[in] D
> \verbatim
> D is DOUBLE PRECISION array, dimension (N)
> The diagonal elements of the tridiagonal matrix T.
> \endverbatim
>
> \param[in] E
> \verbatim
> 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.
> \endverbatim
>
> \param[in] E2
> \verbatim
> E2 is DOUBLE PRECISION array, dimension (N)
> The squares of the offdiagonal elements of the tridiagonal
> matrix T. E2(N) is ignored.
> \endverbatim
>
> \param[in,out] NVAL
> \verbatim
> 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.
> \endverbatim
>
> \param[in,out] AB
> \verbatim
> 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.
> \endverbatim
>
> \param[in,out] C
> \verbatim
> 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.
> \endverbatim
>
> \param[out] MOUT
> \verbatim
> 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.
> \endverbatim
>
> \param[in,out] NAB
> \verbatim
> 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.
> \endverbatim
>
> \param[out] WORK
> \verbatim
> WORK is DOUBLE PRECISION array, dimension (MMAX)
> Workspace.
> \endverbatim
>
> \param[out] IWORK
> \verbatim
> IWORK is INTEGER array, dimension (MMAX)
> Workspace.
> \endverbatim
>
> \param[out] INFO
> \verbatim
> INFO is INTEGER
> = 0: All intervals converged.
> = 1--MMAX: The last INFO intervals did not converge.
> = MMAX+1: More than MMAX intervals were generated.
> \endverbatim
Authors:
========
> \author Univ. of Tennessee
> \author Univ. of California Berkeley
> \author Univ. of Colorado Denver
> \author NAG Ltd.
> \date September 2012
> \ingroup auxOTHERauxiliary
> \par Further Details:
=====================
>
> \verbatim
>
> 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.
> \endverbatim
>
=====================================================================
Subroutine */ int igraphdlaebz_(integer *ijob, integer *nitmax, integer *n,
integer *mmax, integer *minp, integer *nbmin, doublereal *abstol,
doublereal *reltol, doublereal *pivmin, doublereal *d__, doublereal *
e, doublereal *e2, integer *nval, doublereal *ab, doublereal *c__,
integer *mout, integer *nab, doublereal *work, integer *iwork,
integer *info)
{
/* System generated locals */
integer nab_dim1, nab_offset, ab_dim1, ab_offset, i__1, i__2, i__3, i__4,
i__5, i__6;
doublereal d__1, d__2, d__3, d__4;
/* Local variables */
integer j, kf, ji, kl, jp, jit;
doublereal tmp1, tmp2;
integer itmp1, itmp2, kfnew, klnew;
/* -- LAPACK auxiliary routine (version 3.4.2) --
-- LAPACK is a software package provided by Univ. of Tennessee, --
-- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
September 2012
=====================================================================
Check for Errors
Parameter adjustments */
nab_dim1 = *mmax;
nab_offset = 1 + nab_dim1;
nab -= nab_offset;
ab_dim1 = *mmax;
ab_offset = 1 + ab_dim1;
ab -= ab_offset;
--d__;
--e;
--e2;
--nval;
--c__;
--work;
--iwork;
/* Function Body */
*info = 0;
if (*ijob < 1 || *ijob > 3) {
*info = -1;
return 0;
}
/* Initialize NAB */
if (*ijob == 1) {
/* Compute the number of eigenvalues in the initial intervals. */
*mout = 0;
i__1 = *minp;
for (ji = 1; ji <= i__1; ++ji) {
for (jp = 1; jp <= 2; ++jp) {
tmp1 = d__[1] - ab[ji + jp * ab_dim1];
if (abs(tmp1) < *pivmin) {
tmp1 = -(*pivmin);
}
nab[ji + jp * nab_dim1] = 0;
if (tmp1 <= 0.) {
nab[ji + jp * nab_dim1] = 1;
}
i__2 = *n;
for (j = 2; j <= i__2; ++j) {
tmp1 = d__[j] - e2[j - 1] / tmp1 - ab[ji + jp * ab_dim1];
if (abs(tmp1) < *pivmin) {
tmp1 = -(*pivmin);
}
if (tmp1 <= 0.) {
++nab[ji + jp * nab_dim1];
}
/* L10: */
}
/* L20: */
}
*mout = *mout + nab[ji + (nab_dim1 << 1)] - nab[ji + nab_dim1];
/* L30: */
}
return 0;
}
/* Initialize for loop
KF and KL have the following meaning:
Intervals 1,...,KF-1 have converged.
Intervals KF,...,KL still need to be refined. */
kf = 1;
kl = *minp;
/* If IJOB=2, initialize C.
If IJOB=3, use the user-supplied starting point. */
if (*ijob == 2) {
i__1 = *minp;
for (ji = 1; ji <= i__1; ++ji) {
c__[ji] = (ab[ji + ab_dim1] + ab[ji + (ab_dim1 << 1)]) * .5;
/* L40: */
}
}
/* Iteration loop */
i__1 = *nitmax;
for (jit = 1; jit <= i__1; ++jit) {
/* Loop over intervals */
if (kl - kf + 1 >= *nbmin && *nbmin > 0) {
/* Begin of Parallel Version of the loop */
i__2 = kl;
for (ji = kf; ji <= i__2; ++ji) {
/* Compute N(c), the number of eigenvalues less than c */
work[ji] = d__[1] - c__[ji];
iwork[ji] = 0;
if (work[ji] <= *pivmin) {
iwork[ji] = 1;
/* Computing MIN */
d__1 = work[ji], d__2 = -(*pivmin);
work[ji] = min(d__1,d__2);
}
i__3 = *n;
for (j = 2; j <= i__3; ++j) {
work[ji] = d__[j] - e2[j - 1] / work[ji] - c__[ji];
if (work[ji] <= *pivmin) {
++iwork[ji];
/* Computing MIN */
d__1 = work[ji], d__2 = -(*pivmin);
work[ji] = min(d__1,d__2);
}
/* L50: */
}
/* L60: */
}
if (*ijob <= 2) {
/* IJOB=2: Choose all intervals containing eigenvalues. */
klnew = kl;
i__2 = kl;
for (ji = kf; ji <= i__2; ++ji) {
/* Insure that N(w) is monotone
Computing MIN
Computing MAX */
i__5 = nab[ji + nab_dim1], i__6 = iwork[ji];
i__3 = nab[ji + (nab_dim1 << 1)], i__4 = max(i__5,i__6);
iwork[ji] = min(i__3,i__4);
/* Update the Queue -- add intervals if both halves
contain eigenvalues. */
if (iwork[ji] == nab[ji + (nab_dim1 << 1)]) {
/* No eigenvalue in the upper interval:
just use the lower interval. */
ab[ji + (ab_dim1 << 1)] = c__[ji];
} else if (iwork[ji] == nab[ji + nab_dim1]) {
/* No eigenvalue in the lower interval:
just use the upper interval. */
ab[ji + ab_dim1] = c__[ji];
} else {
++klnew;
if (klnew <= *mmax) {
/* Eigenvalue in both intervals -- add upper to
queue. */
ab[klnew + (ab_dim1 << 1)] = ab[ji + (ab_dim1 <<
1)];
nab[klnew + (nab_dim1 << 1)] = nab[ji + (nab_dim1
<< 1)];
ab[klnew + ab_dim1] = c__[ji];
nab[klnew + nab_dim1] = iwork[ji];
ab[ji + (ab_dim1 << 1)] = c__[ji];
nab[ji + (nab_dim1 << 1)] = iwork[ji];
} else {
*info = *mmax + 1;
}
}
/* L70: */
}
if (*info != 0) {
return 0;
}
kl = klnew;
} else {
/* IJOB=3: Binary search. Keep only the interval containing
w s.t. N(w) = NVAL */
i__2 = kl;
for (ji = kf; ji <= i__2; ++ji) {
if (iwork[ji] <= nval[ji]) {
ab[ji + ab_dim1] = c__[ji];
nab[ji + nab_dim1] = iwork[ji];
}
if (iwork[ji] >= nval[ji]) {
ab[ji + (ab_dim1 << 1)] = c__[ji];
nab[ji + (nab_dim1 << 1)] = iwork[ji];
}
/* L80: */
}
}
} else {
/* End of Parallel Version of the loop
Begin of Serial Version of the loop */
klnew = kl;
i__2 = kl;
for (ji = kf; ji <= i__2; ++ji) {
/* Compute N(w), the number of eigenvalues less than w */
tmp1 = c__[ji];
tmp2 = d__[1] - tmp1;
itmp1 = 0;
if (tmp2 <= *pivmin) {
itmp1 = 1;
/* Computing MIN */
d__1 = tmp2, d__2 = -(*pivmin);
tmp2 = min(d__1,d__2);
}
i__3 = *n;
for (j = 2; j <= i__3; ++j) {
tmp2 = d__[j] - e2[j - 1] / tmp2 - tmp1;
if (tmp2 <= *pivmin) {
++itmp1;
/* Computing MIN */
d__1 = tmp2, d__2 = -(*pivmin);
tmp2 = min(d__1,d__2);
}
/* L90: */
}
if (*ijob <= 2) {
/* IJOB=2: Choose all intervals containing eigenvalues.
Insure that N(w) is monotone
Computing MIN
Computing MAX */
i__5 = nab[ji + nab_dim1];
i__3 = nab[ji + (nab_dim1 << 1)], i__4 = max(i__5,itmp1);
itmp1 = min(i__3,i__4);
/* Update the Queue -- add intervals if both halves
contain eigenvalues. */
if (itmp1 == nab[ji + (nab_dim1 << 1)]) {
/* No eigenvalue in the upper interval:
just use the lower interval. */
ab[ji + (ab_dim1 << 1)] = tmp1;
} else if (itmp1 == nab[ji + nab_dim1]) {
/* No eigenvalue in the lower interval:
just use the upper interval. */
ab[ji + ab_dim1] = tmp1;
} else if (klnew < *mmax) {
/* Eigenvalue in both intervals -- add upper to queue. */
++klnew;
ab[klnew + (ab_dim1 << 1)] = ab[ji + (ab_dim1 << 1)];
nab[klnew + (nab_dim1 << 1)] = nab[ji + (nab_dim1 <<
1)];
ab[klnew + ab_dim1] = tmp1;
nab[klnew + nab_dim1] = itmp1;
ab[ji + (ab_dim1 << 1)] = tmp1;
nab[ji + (nab_dim1 << 1)] = itmp1;
} else {
*info = *mmax + 1;
return 0;
}
} else {
/* IJOB=3: Binary search. Keep only the interval
containing w s.t. N(w) = NVAL */
if (itmp1 <= nval[ji]) {
ab[ji + ab_dim1] = tmp1;
nab[ji + nab_dim1] = itmp1;
}
if (itmp1 >= nval[ji]) {
ab[ji + (ab_dim1 << 1)] = tmp1;
nab[ji + (nab_dim1 << 1)] = itmp1;
}
}
/* L100: */
}
kl = klnew;
}
/* Check for convergence */
kfnew = kf;
i__2 = kl;
for (ji = kf; ji <= i__2; ++ji) {
tmp1 = (d__1 = ab[ji + (ab_dim1 << 1)] - ab[ji + ab_dim1], abs(
d__1));
/* Computing MAX */
d__3 = (d__1 = ab[ji + (ab_dim1 << 1)], abs(d__1)), d__4 = (d__2 =
ab[ji + ab_dim1], abs(d__2));
tmp2 = max(d__3,d__4);
/* Computing MAX */
d__1 = max(*abstol,*pivmin), d__2 = *reltol * tmp2;
if (tmp1 < max(d__1,d__2) || nab[ji + nab_dim1] >= nab[ji + (
nab_dim1 << 1)]) {
/* Converged -- Swap with position KFNEW,
then increment KFNEW */
if (ji > kfnew) {
tmp1 = ab[ji + ab_dim1];
tmp2 = ab[ji + (ab_dim1 << 1)];
itmp1 = nab[ji + nab_dim1];
itmp2 = nab[ji + (nab_dim1 << 1)];
ab[ji + ab_dim1] = ab[kfnew + ab_dim1];
ab[ji + (ab_dim1 << 1)] = ab[kfnew + (ab_dim1 << 1)];
nab[ji + nab_dim1] = nab[kfnew + nab_dim1];
nab[ji + (nab_dim1 << 1)] = nab[kfnew + (nab_dim1 << 1)];
ab[kfnew + ab_dim1] = tmp1;
ab[kfnew + (ab_dim1 << 1)] = tmp2;
nab[kfnew + nab_dim1] = itmp1;
nab[kfnew + (nab_dim1 << 1)] = itmp2;
if (*ijob == 3) {
itmp1 = nval[ji];
nval[ji] = nval[kfnew];
nval[kfnew] = itmp1;
}
}
++kfnew;
}
/* L110: */
}
kf = kfnew;
/* Choose Midpoints */
i__2 = kl;
for (ji = kf; ji <= i__2; ++ji) {
c__[ji] = (ab[ji + ab_dim1] + ab[ji + (ab_dim1 << 1)]) * .5;
/* L120: */
}
/* If no more intervals to refine, quit. */
if (kf > kl) {
goto L140;
}
/* L130: */
}
/* Converged */
L140:
/* Computing MAX */
i__1 = kl + 1 - kf;
*info = max(i__1,0);
*mout = kl;
return 0;
/* End of DLAEBZ */
} /* igraphdlaebz_ */