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|
/* Ergo, version 3.8, a program for linear scaling electronic structure
* calculations.
* Copyright (C) 2019 Elias Rudberg, Emanuel H. Rubensson, Pawel Salek,
* and Anastasia Kruchinina.
*
* This program is free software: you can redistribute it and/or modify
* it under the terms of the GNU General Public License as published by
* the Free Software Foundation, either version 3 of the License, or
* (at your option) any later version.
*
* This program is distributed in the hope that it will be useful,
* but WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
* GNU General Public License for more details.
*
* You should have received a copy of the GNU General Public License
* along with this program. If not, see <http://www.gnu.org/licenses/>.
*
* Primary academic reference:
* Ergo: An open-source program for linear-scaling electronic structure
* calculations,
* Elias Rudberg, Emanuel H. Rubensson, Pawel Salek, and Anastasia
* Kruchinina,
* SoftwareX 7, 107 (2018),
* <http://dx.doi.org/10.1016/j.softx.2018.03.005>
*
* For further information about Ergo, see <http://www.ergoscf.org>.
*/
/* This file belongs to the template_lapack part of the Ergo source
* code. The source files in the template_lapack directory are modified
* versions of files originally distributed as CLAPACK, see the
* Copyright/license notice in the file template_lapack/COPYING.
*/
#ifndef TEMPLATE_LAPACK_LAEBZ_HEADER
#define TEMPLATE_LAPACK_LAEBZ_HEADER
template<class Treal>
int template_lapack_laebz(const integer *ijob, const integer *nitmax, const integer *n,
const integer *mmax, const integer *minp, const integer *nbmin, const Treal *abstol,
const Treal *reltol, const Treal *pivmin, const Treal *d__, const Treal *
e, const Treal *e2, integer *nval, Treal *ab, Treal *c__,
integer *mout, integer *nab, Treal *work, integer *iwork,
integer *info)
{
/* -- LAPACK auxiliary routine (version 3.0) --
Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
Courant Institute, Argonne National Lab, and Rice University
June 30, 1999
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.
=====================================================================
Check for Errors
Parameter adjustments */
/* System generated locals */
integer nab_dim1, nab_offset, ab_dim1, ab_offset, i__1, i__2, i__3, i__4,
i__5, i__6;
Treal d__1, d__2, d__3, d__4;
/* Local variables */
integer itmp1, itmp2, j, kfnew, klnew, kf, ji, kl, jp, jit;
Treal tmp1, tmp2;
#define ab_ref(a_1,a_2) ab[(a_2)*ab_dim1 + a_1]
#define nab_ref(a_1,a_2) nab[(a_2)*nab_dim1 + a_1]
nab_dim1 = *mmax;
nab_offset = 1 + nab_dim1 * 1;
nab -= nab_offset;
ab_dim1 = *mmax;
ab_offset = 1 + ab_dim1 * 1;
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;
/* DIR$ NOVECTOR */
i__1 = *minp;
for (ji = 1; ji <= i__1; ++ji) {
for (jp = 1; jp <= 2; ++jp) {
tmp1 = d__[1] - ab_ref(ji, jp);
if (absMACRO(tmp1) < *pivmin) {
tmp1 = -(*pivmin);
}
nab_ref(ji, jp) = 0;
if (tmp1 <= 0.) {
nab_ref(ji, jp) = 1;
}
i__2 = *n;
for (j = 2; j <= i__2; ++j) {
tmp1 = d__[j] - e2[j - 1] / tmp1 - ab_ref(ji, jp);
if (absMACRO(tmp1) < *pivmin) {
tmp1 = -(*pivmin);
}
if (tmp1 <= 0.) {
nab_ref(ji, jp) = nab_ref(ji, jp) + 1;
}
/* L10: */
}
/* L20: */
}
*mout = *mout + nab_ref(ji, 2) - nab_ref(ji, 1);
/* 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_ref(ji, 1) + ab_ref(ji, 2)) * .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] = minMACRO(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] = minMACRO(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_ref(ji, 1), i__6 = iwork[ji];
i__3 = nab_ref(ji, 2), i__4 = maxMACRO(i__5,i__6);
iwork[ji] = minMACRO(i__3,i__4);
/* Update the Queue -- add intervals if both halves
contain eigenvalues. */
if (iwork[ji] == nab_ref(ji, 2)) {
/* No eigenvalue in the upper interval:
just use the lower interval. */
ab_ref(ji, 2) = c__[ji];
} else if (iwork[ji] == nab_ref(ji, 1)) {
/* No eigenvalue in the lower interval:
just use the upper interval. */
ab_ref(ji, 1) = c__[ji];
} else {
++klnew;
if (klnew <= *mmax) {
/* Eigenvalue in both intervals -- add upper to
queue. */
ab_ref(klnew, 2) = ab_ref(ji, 2);
nab_ref(klnew, 2) = nab_ref(ji, 2);
ab_ref(klnew, 1) = c__[ji];
nab_ref(klnew, 1) = iwork[ji];
ab_ref(ji, 2) = c__[ji];
nab_ref(ji, 2) = 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_ref(ji, 1) = c__[ji];
nab_ref(ji, 1) = iwork[ji];
}
if (iwork[ji] >= nval[ji]) {
ab_ref(ji, 2) = c__[ji];
nab_ref(ji, 2) = 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 = minMACRO(d__1,d__2);
}
/* A series of compiler directives to defeat vectorization
for the next loop
$PL$ CMCHAR=' '
DIR$ NEXTSCALAR
$DIR SCALAR
DIR$ NEXT SCALAR
VD$L NOVECTOR
DEC$ NOVECTOR
VD$ NOVECTOR
VDIR NOVECTOR
VOCL LOOP,SCALAR
IBM PREFER SCALAR
$PL$ CMCHAR='*' */
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 = minMACRO(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_ref(ji, 1);
i__3 = nab_ref(ji, 2), i__4 = maxMACRO(i__5,itmp1);
itmp1 = minMACRO(i__3,i__4);
/* Update the Queue -- add intervals if both halves
contain eigenvalues. */
if (itmp1 == nab_ref(ji, 2)) {
/* No eigenvalue in the upper interval:
just use the lower interval. */
ab_ref(ji, 2) = tmp1;
} else if (itmp1 == nab_ref(ji, 1)) {
/* No eigenvalue in the lower interval:
just use the upper interval. */
ab_ref(ji, 1) = tmp1;
} else if (klnew < *mmax) {
/* Eigenvalue in both intervals -- add upper to queue. */
++klnew;
ab_ref(klnew, 2) = ab_ref(ji, 2);
nab_ref(klnew, 2) = nab_ref(ji, 2);
ab_ref(klnew, 1) = tmp1;
nab_ref(klnew, 1) = itmp1;
ab_ref(ji, 2) = tmp1;
nab_ref(ji, 2) = 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_ref(ji, 1) = tmp1;
nab_ref(ji, 1) = itmp1;
}
if (itmp1 >= nval[ji]) {
ab_ref(ji, 2) = tmp1;
nab_ref(ji, 2) = itmp1;
}
}
/* L100: */
}
kl = klnew;
/* End of Serial Version of the loop */
}
/* Check for convergence */
kfnew = kf;
i__2 = kl;
for (ji = kf; ji <= i__2; ++ji) {
tmp1 = (d__1 = ab_ref(ji, 2) - ab_ref(ji, 1), absMACRO(d__1));
/* Computing MAX */
d__3 = (d__1 = ab_ref(ji, 2), absMACRO(d__1)), d__4 = (d__2 = ab_ref(
ji, 1), absMACRO(d__2));
tmp2 = maxMACRO(d__3,d__4);
/* Computing MAX */
d__1 = maxMACRO(*abstol,*pivmin), d__2 = *reltol * tmp2;
if (tmp1 < maxMACRO(d__1,d__2) || nab_ref(ji, 1) >= nab_ref(ji, 2)) {
/* Converged -- Swap with position KFNEW,
then increment KFNEW */
if (ji > kfnew) {
tmp1 = ab_ref(ji, 1);
tmp2 = ab_ref(ji, 2);
itmp1 = nab_ref(ji, 1);
itmp2 = nab_ref(ji, 2);
ab_ref(ji, 1) = ab_ref(kfnew, 1);
ab_ref(ji, 2) = ab_ref(kfnew, 2);
nab_ref(ji, 1) = nab_ref(kfnew, 1);
nab_ref(ji, 2) = nab_ref(kfnew, 2);
ab_ref(kfnew, 1) = tmp1;
ab_ref(kfnew, 2) = tmp2;
nab_ref(kfnew, 1) = itmp1;
nab_ref(kfnew, 2) = 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_ref(ji, 1) + ab_ref(ji, 2)) * .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 = maxMACRO(i__1,0);
*mout = kl;
return 0;
/* End of DLAEBZ */
} /* dlaebz_ */
#undef nab_ref
#undef ab_ref
#endif
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