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/* Ergo, version 3.8.2, a program for linear scaling electronic structure
* calculations.
* Copyright (C) 2023 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_STEBZ_HEADER
#define TEMPLATE_LAPACK_STEBZ_HEADER
template<class Treal>
int template_lapack_stebz(const char *range, const char *order, const integer *n, const Treal
*vl, const Treal *vu, const integer *il, const integer *iu, const Treal *abstol,
const Treal *d__, const Treal *e, integer *m, integer *nsplit,
Treal *w, integer *iblock, integer *isplit, Treal *work,
integer *iwork, integer *info)
{
/* -- LAPACK 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
=======
DSTEBZ computes the eigenvalues of a symmetric tridiagonal
matrix T. The user may ask for all eigenvalues, all eigenvalues
in the half-open interval (VL, VU], or the IL-th through IU-th
eigenvalues.
To avoid overflow, the matrix must be scaled so that its
largest element is no greater than overflow**(1/2) *
underflow**(1/4) in absolute value, and for greatest
accuracy, it should not be much smaller than that.
See W. Kahan "Accurate Eigenvalues of a Symmetric Tridiagonal
Matrix", Report CS41, Computer Science Dept., Stanford
University, July 21, 1966.
Arguments
=========
RANGE (input) CHARACTER
= 'A': ("All") all eigenvalues will be found.
= 'V': ("Value") all eigenvalues in the half-open interval
(VL, VU] will be found.
= 'I': ("Index") the IL-th through IU-th eigenvalues (of the
entire matrix) will be found.
ORDER (input) CHARACTER
= 'B': ("By Block") the eigenvalues will be grouped by
split-off block (see IBLOCK, ISPLIT) and
ordered from smallest to largest within
the block.
= 'E': ("Entire matrix")
the eigenvalues for the entire matrix
will be ordered from smallest to
largest.
N (input) INTEGER
The order of the tridiagonal matrix T. N >= 0.
VL (input) DOUBLE PRECISION
VU (input) DOUBLE PRECISION
If RANGE='V', the lower and upper bounds of the interval to
be searched for eigenvalues. Eigenvalues less than or equal
to VL, or greater than VU, will not be returned. VL < VU.
Not referenced if RANGE = 'A' or 'I'.
IL (input) INTEGER
IU (input) INTEGER
If RANGE='I', the indices (in ascending order) of the
smallest and largest eigenvalues to be returned.
1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
Not referenced if RANGE = 'A' or 'V'.
ABSTOL (input) DOUBLE PRECISION
The absolute tolerance for the eigenvalues. An eigenvalue
(or cluster) is considered to be located if it has been
determined to lie in an interval whose width is ABSTOL or
less. If ABSTOL is less than or equal to zero, then ULP*|T|
will be used, where |T| means the 1-norm of T.
Eigenvalues will be computed most accurately when ABSTOL is
set to twice the underflow threshold 2*DLAMCH('S'), not zero.
D (input) DOUBLE PRECISION array, dimension (N)
The n diagonal elements of the tridiagonal matrix T.
E (input) DOUBLE PRECISION array, dimension (N-1)
The (n-1) off-diagonal elements of the tridiagonal matrix T.
M (output) INTEGER
The actual number of eigenvalues found. 0 <= M <= N.
(See also the description of INFO=2,3.)
NSPLIT (output) INTEGER
The number of diagonal blocks in the matrix T.
1 <= NSPLIT <= N.
W (output) DOUBLE PRECISION array, dimension (N)
On exit, the first M elements of W will contain the
eigenvalues. (DSTEBZ may use the remaining N-M elements as
workspace.)
IBLOCK (output) INTEGER array, dimension (N)
At each row/column j where E(j) is zero or small, the
matrix T is considered to split into a block diagonal
matrix. On exit, if INFO = 0, IBLOCK(i) specifies to which
block (from 1 to the number of blocks) the eigenvalue W(i)
belongs. (DSTEBZ may use the remaining N-M elements as
workspace.)
ISPLIT (output) INTEGER array, dimension (N)
The splitting points, at which T breaks up into submatrices.
The first submatrix consists of rows/columns 1 to ISPLIT(1),
the second of rows/columns ISPLIT(1)+1 through ISPLIT(2),
etc., and the NSPLIT-th consists of rows/columns
ISPLIT(NSPLIT-1)+1 through ISPLIT(NSPLIT)=N.
(Only the first NSPLIT elements will actually be used, but
since the user cannot know a priori what value NSPLIT will
have, N words must be reserved for ISPLIT.)
WORK (workspace) DOUBLE PRECISION array, dimension (4*N)
IWORK (workspace) INTEGER array, dimension (3*N)
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: some or all of the eigenvalues failed to converge or
were not computed:
=1 or 3: Bisection failed to converge for some
eigenvalues; these eigenvalues are flagged by a
negative block number. The effect is that the
eigenvalues may not be as accurate as the
absolute and relative tolerances. This is
generally caused by unexpectedly inaccurate
arithmetic.
=2 or 3: RANGE='I' only: Not all of the eigenvalues
IL:IU were found.
Effect: M < IU+1-IL
Cause: non-monotonic arithmetic, causing the
Sturm sequence to be non-monotonic.
Cure: recalculate, using RANGE='A', and pick
out eigenvalues IL:IU. In some cases,
increasing the PARAMETER "FUDGE" may
make things work.
= 4: RANGE='I', and the Gershgorin interval
initially used was too small. No eigenvalues
were computed.
Probable cause: your machine has sloppy
floating-point arithmetic.
Cure: Increase the PARAMETER "FUDGE",
recompile, and try again.
Internal Parameters
===================
RELFAC DOUBLE PRECISION, default = 2.0e0
The relative tolerance. An interval (a,b] lies within
"relative tolerance" if b-a < RELFAC*ulp*max(|a|,|b|),
where "ulp" is the machine precision (distance from 1 to
the next larger floating point number.)
FUDGE DOUBLE PRECISION, default = 2
A "fudge factor" to widen the Gershgorin intervals. Ideally,
a value of 1 should work, but on machines with sloppy
arithmetic, this needs to be larger. The default for
publicly released versions should be large enough to handle
the worst machine around. Note that this has no effect
on accuracy of the solution.
=====================================================================
Parameter adjustments */
/* Table of constant values */
integer c__1 = 1;
integer c_n1 = -1;
integer c__3 = 3;
integer c__2 = 2;
integer c__0 = 0;
/* System generated locals */
integer i__1, i__2, i__3;
Treal d__1, d__2, d__3, d__4, d__5;
/* Local variables */
integer iend, ioff, iout, itmp1, j, jdisc;
integer iinfo;
Treal atoli;
integer iwoff;
Treal bnorm;
integer itmax;
Treal wkill, rtoli, tnorm;
integer ib, jb, ie, je, nb;
Treal gl;
integer im, in;
integer ibegin;
Treal gu;
integer iw;
Treal wl;
integer irange, idiscl;
Treal safemn, wu;
integer idumma[1];
integer idiscu, iorder;
logical ncnvrg;
Treal pivmin;
logical toofew;
integer nwl;
Treal ulp, wlu, wul;
integer nwu;
Treal tmp1, tmp2;
--iwork;
--work;
--isplit;
--iblock;
--w;
--e;
--d__;
/* Initialization added by Elias to get rid of compiler warnings. */
wlu = wul = 0;
/* Function Body */
*info = 0;
/* Decode RANGE */
if (template_blas_lsame(range, "A")) {
irange = 1;
} else if (template_blas_lsame(range, "V")) {
irange = 2;
} else if (template_blas_lsame(range, "I")) {
irange = 3;
} else {
irange = 0;
}
/* Decode ORDER */
if (template_blas_lsame(order, "B")) {
iorder = 2;
} else if (template_blas_lsame(order, "E")) {
iorder = 1;
} else {
iorder = 0;
}
/* Check for Errors */
if (irange <= 0) {
*info = -1;
} else if (iorder <= 0) {
*info = -2;
} else if (*n < 0) {
*info = -3;
} else if (irange == 2) {
if (*vl >= *vu) {
*info = -5;
}
} else if (irange == 3 && (*il < 1 || *il > maxMACRO(1,*n))) {
*info = -6;
} else if (irange == 3 && (*iu < minMACRO(*n,*il) || *iu > *n)) {
*info = -7;
}
if (*info != 0) {
i__1 = -(*info);
template_blas_erbla("STEBZ ", &i__1);
return 0;
}
/* Initialize error flags */
*info = 0;
ncnvrg = FALSE_;
toofew = FALSE_;
/* Quick return if possible */
*m = 0;
if (*n == 0) {
return 0;
}
/* Simplifications: */
if (irange == 3 && *il == 1 && *iu == *n) {
irange = 1;
}
/* Get machine constants
NB is the minimum vector length for vector bisection, or 0
if only scalar is to be done. */
safemn = template_lapack_lamch("S", (Treal)0);
ulp = template_lapack_lamch("P", (Treal)0);
rtoli = ulp * 2.;
nb = template_lapack_ilaenv(&c__1, "DSTEBZ", " ", n, &c_n1, &c_n1, &c_n1, (ftnlen)6, (
ftnlen)1);
if (nb <= 1) {
nb = 0;
}
/* Special Case when N=1 */
if (*n == 1) {
*nsplit = 1;
isplit[1] = 1;
if (irange == 2 && (*vl >= d__[1] || *vu < d__[1])) {
*m = 0;
} else {
w[1] = d__[1];
iblock[1] = 1;
*m = 1;
}
return 0;
}
/* Compute Splitting Points */
*nsplit = 1;
work[*n] = 0.;
pivmin = 1.;
/* DIR$ NOVECTOR */
i__1 = *n;
for (j = 2; j <= i__1; ++j) {
/* Computing 2nd power */
d__1 = e[j - 1];
tmp1 = d__1 * d__1;
/* Computing 2nd power */
d__2 = ulp;
if ((d__1 = d__[j] * d__[j - 1], absMACRO(d__1)) * (d__2 * d__2) + safemn
> tmp1) {
isplit[*nsplit] = j - 1;
++(*nsplit);
work[j - 1] = 0.;
} else {
work[j - 1] = tmp1;
pivmin = maxMACRO(pivmin,tmp1);
}
/* L10: */
}
isplit[*nsplit] = *n;
pivmin *= safemn;
/* Compute Interval and ATOLI */
if (irange == 3) {
/* RANGE='I': Compute the interval containing eigenvalues
IL through IU.
Compute Gershgorin interval for entire (split) matrix
and use it as the initial interval */
gu = d__[1];
gl = d__[1];
tmp1 = 0.;
i__1 = *n - 1;
for (j = 1; j <= i__1; ++j) {
tmp2 = template_blas_sqrt(work[j]);
/* Computing MAX */
d__1 = gu, d__2 = d__[j] + tmp1 + tmp2;
gu = maxMACRO(d__1,d__2);
/* Computing MIN */
d__1 = gl, d__2 = d__[j] - tmp1 - tmp2;
gl = minMACRO(d__1,d__2);
tmp1 = tmp2;
/* L20: */
}
/* Computing MAX */
d__1 = gu, d__2 = d__[*n] + tmp1;
gu = maxMACRO(d__1,d__2);
/* Computing MIN */
d__1 = gl, d__2 = d__[*n] - tmp1;
gl = minMACRO(d__1,d__2);
/* Computing MAX */
d__1 = absMACRO(gl), d__2 = absMACRO(gu);
tnorm = maxMACRO(d__1,d__2);
gl = gl - tnorm * 2. * ulp * *n - pivmin * 4.;
gu = gu + tnorm * 2. * ulp * *n + pivmin * 2.;
/* Compute Iteration parameters */
itmax = (integer) ((template_blas_log(tnorm + pivmin) - template_blas_log(pivmin)) / template_blas_log(2.)) + 2;
if (*abstol <= 0.) {
atoli = ulp * tnorm;
} else {
atoli = *abstol;
}
work[*n + 1] = gl;
work[*n + 2] = gl;
work[*n + 3] = gu;
work[*n + 4] = gu;
work[*n + 5] = gl;
work[*n + 6] = gu;
iwork[1] = -1;
iwork[2] = -1;
iwork[3] = *n + 1;
iwork[4] = *n + 1;
iwork[5] = *il - 1;
iwork[6] = *iu;
template_lapack_laebz(&c__3, &itmax, n, &c__2, &c__2, &nb, &atoli, &rtoli, &pivmin,
&d__[1], &e[1], &work[1], &iwork[5], &work[*n + 1], &work[*n
+ 5], &iout, &iwork[1], &w[1], &iblock[1], &iinfo);
if (iwork[6] == *iu) {
wl = work[*n + 1];
wlu = work[*n + 3];
nwl = iwork[1];
wu = work[*n + 4];
wul = work[*n + 2];
nwu = iwork[4];
} else {
wl = work[*n + 2];
wlu = work[*n + 4];
nwl = iwork[2];
wu = work[*n + 3];
wul = work[*n + 1];
nwu = iwork[3];
}
if (nwl < 0 || nwl >= *n || nwu < 1 || nwu > *n) {
*info = 4;
return 0;
}
} else {
/* RANGE='A' or 'V' -- Set ATOLI
Computing MAX */
d__3 = absMACRO(d__[1]) + absMACRO(e[1]), d__4 = (d__1 = d__[*n], absMACRO(d__1)) + (
d__2 = e[*n - 1], absMACRO(d__2));
tnorm = maxMACRO(d__3,d__4);
i__1 = *n - 1;
for (j = 2; j <= i__1; ++j) {
/* Computing MAX */
d__4 = tnorm, d__5 = (d__1 = d__[j], absMACRO(d__1)) + (d__2 = e[j - 1]
, absMACRO(d__2)) + (d__3 = e[j], absMACRO(d__3));
tnorm = maxMACRO(d__4,d__5);
/* L30: */
}
if (*abstol <= 0.) {
atoli = ulp * tnorm;
} else {
atoli = *abstol;
}
if (irange == 2) {
wl = *vl;
wu = *vu;
} else {
wl = 0.;
wu = 0.;
}
}
/* Find Eigenvalues -- Loop Over Blocks and recompute NWL and NWU.
NWL accumulates the number of eigenvalues .le. WL,
NWU accumulates the number of eigenvalues .le. WU */
*m = 0;
iend = 0;
*info = 0;
nwl = 0;
nwu = 0;
i__1 = *nsplit;
for (jb = 1; jb <= i__1; ++jb) {
ioff = iend;
ibegin = ioff + 1;
iend = isplit[jb];
in = iend - ioff;
if (in == 1) {
/* Special Case -- IN=1 */
if (irange == 1 || wl >= d__[ibegin] - pivmin) {
++nwl;
}
if (irange == 1 || wu >= d__[ibegin] - pivmin) {
++nwu;
}
if (irange == 1 || ( wl < d__[ibegin] - pivmin && wu >= d__[ibegin]
- pivmin ) ) {
++(*m);
w[*m] = d__[ibegin];
iblock[*m] = jb;
}
} else {
/* General Case -- IN > 1
Compute Gershgorin Interval
and use it as the initial interval */
gu = d__[ibegin];
gl = d__[ibegin];
tmp1 = 0.;
i__2 = iend - 1;
for (j = ibegin; j <= i__2; ++j) {
tmp2 = (d__1 = e[j], absMACRO(d__1));
/* Computing MAX */
d__1 = gu, d__2 = d__[j] + tmp1 + tmp2;
gu = maxMACRO(d__1,d__2);
/* Computing MIN */
d__1 = gl, d__2 = d__[j] - tmp1 - tmp2;
gl = minMACRO(d__1,d__2);
tmp1 = tmp2;
/* L40: */
}
/* Computing MAX */
d__1 = gu, d__2 = d__[iend] + tmp1;
gu = maxMACRO(d__1,d__2);
/* Computing MIN */
d__1 = gl, d__2 = d__[iend] - tmp1;
gl = minMACRO(d__1,d__2);
/* Computing MAX */
d__1 = absMACRO(gl), d__2 = absMACRO(gu);
bnorm = maxMACRO(d__1,d__2);
gl = gl - bnorm * 2. * ulp * in - pivmin * 2.;
gu = gu + bnorm * 2. * ulp * in + pivmin * 2.;
/* Compute ATOLI for the current submatrix */
if (*abstol <= 0.) {
/* Computing MAX */
d__1 = absMACRO(gl), d__2 = absMACRO(gu);
atoli = ulp * maxMACRO(d__1,d__2);
} else {
atoli = *abstol;
}
if (irange > 1) {
if (gu < wl) {
nwl += in;
nwu += in;
goto L70;
}
gl = maxMACRO(gl,wl);
gu = minMACRO(gu,wu);
if (gl >= gu) {
goto L70;
}
}
/* Set Up Initial Interval */
work[*n + 1] = gl;
work[*n + in + 1] = gu;
template_lapack_laebz(&c__1, &c__0, &in, &in, &c__1, &nb, &atoli, &rtoli, &
pivmin, &d__[ibegin], &e[ibegin], &work[ibegin], idumma, &
work[*n + 1], &work[*n + (in << 1) + 1], &im, &iwork[1], &
w[*m + 1], &iblock[*m + 1], &iinfo);
nwl += iwork[1];
nwu += iwork[in + 1];
iwoff = *m - iwork[1];
/* Compute Eigenvalues */
itmax = (integer) ((template_blas_log(gu - gl + pivmin) - template_blas_log(pivmin)) / template_blas_log(2.)
) + 2;
template_lapack_laebz(&c__2, &itmax, &in, &in, &c__1, &nb, &atoli, &rtoli, &
pivmin, &d__[ibegin], &e[ibegin], &work[ibegin], idumma, &
work[*n + 1], &work[*n + (in << 1) + 1], &iout, &iwork[1],
&w[*m + 1], &iblock[*m + 1], &iinfo);
/* Copy Eigenvalues Into W and IBLOCK
Use -JB for block number for unconverged eigenvalues. */
i__2 = iout;
for (j = 1; j <= i__2; ++j) {
tmp1 = (work[j + *n] + work[j + in + *n]) * .5;
/* Flag non-convergence. */
if (j > iout - iinfo) {
ncnvrg = TRUE_;
ib = -jb;
} else {
ib = jb;
}
i__3 = iwork[j + in] + iwoff;
for (je = iwork[j] + 1 + iwoff; je <= i__3; ++je) {
w[je] = tmp1;
iblock[je] = ib;
/* L50: */
}
/* L60: */
}
*m += im;
}
L70:
;
}
/* If RANGE='I', then (WL,WU) contains eigenvalues NWL+1,...,NWU
If NWL+1 < IL or NWU > IU, discard extra eigenvalues. */
if (irange == 3) {
im = 0;
idiscl = *il - 1 - nwl;
idiscu = nwu - *iu;
if (idiscl > 0 || idiscu > 0) {
i__1 = *m;
for (je = 1; je <= i__1; ++je) {
if (w[je] <= wlu && idiscl > 0) {
--idiscl;
} else if (w[je] >= wul && idiscu > 0) {
--idiscu;
} else {
++im;
w[im] = w[je];
iblock[im] = iblock[je];
}
/* L80: */
}
*m = im;
}
if (idiscl > 0 || idiscu > 0) {
/* Code to deal with effects of bad arithmetic:
Some low eigenvalues to be discarded are not in (WL,WLU],
or high eigenvalues to be discarded are not in (WUL,WU]
so just kill off the smallest IDISCL/largest IDISCU
eigenvalues, by simply finding the smallest/largest
eigenvalue(s).
(If N(w) is monotone non-decreasing, this should never
happen.) */
if (idiscl > 0) {
wkill = wu;
i__1 = idiscl;
for (jdisc = 1; jdisc <= i__1; ++jdisc) {
iw = 0;
i__2 = *m;
for (je = 1; je <= i__2; ++je) {
if (iblock[je] != 0 && (w[je] < wkill || iw == 0)) {
iw = je;
wkill = w[je];
}
/* L90: */
}
iblock[iw] = 0;
/* L100: */
}
}
if (idiscu > 0) {
wkill = wl;
i__1 = idiscu;
for (jdisc = 1; jdisc <= i__1; ++jdisc) {
iw = 0;
i__2 = *m;
for (je = 1; je <= i__2; ++je) {
if (iblock[je] != 0 && (w[je] > wkill || iw == 0)) {
iw = je;
wkill = w[je];
}
/* L110: */
}
iblock[iw] = 0;
/* L120: */
}
}
im = 0;
i__1 = *m;
for (je = 1; je <= i__1; ++je) {
if (iblock[je] != 0) {
++im;
w[im] = w[je];
iblock[im] = iblock[je];
}
/* L130: */
}
*m = im;
}
if (idiscl < 0 || idiscu < 0) {
toofew = TRUE_;
}
}
/* If ORDER='B', do nothing -- the eigenvalues are already sorted
by block.
If ORDER='E', sort the eigenvalues from smallest to largest */
if (iorder == 1 && *nsplit > 1) {
i__1 = *m - 1;
for (je = 1; je <= i__1; ++je) {
ie = 0;
tmp1 = w[je];
i__2 = *m;
for (j = je + 1; j <= i__2; ++j) {
if (w[j] < tmp1) {
ie = j;
tmp1 = w[j];
}
/* L140: */
}
if (ie != 0) {
itmp1 = iblock[ie];
w[ie] = w[je];
iblock[ie] = iblock[je];
w[je] = tmp1;
iblock[je] = itmp1;
}
/* L150: */
}
}
*info = 0;
if (ncnvrg) {
++(*info);
}
if (toofew) {
*info += 2;
}
return 0;
/* End of DSTEBZ */
} /* dstebz_ */
#endif
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