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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_LAG2_HEADER
#define TEMPLATE_LAPACK_LAG2_HEADER
template<class Treal>
int template_lapack_lag2(const Treal *a, const integer *lda, const Treal *b,
const integer *ldb, const Treal *safmin, Treal *scale1, Treal *
scale2, Treal *wr1, Treal *wr2, Treal *wi)
{
/* -- LAPACK auxiliary routine (version 3.0) --
Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
Courant Institute, Argonne National Lab, and Rice University
March 31, 1993
Purpose
=======
DLAG2 computes the eigenvalues of a 2 x 2 generalized eigenvalue
problem A - w B, with scaling as necessary to avoid over-/underflow.
The scaling factor "s" results in a modified eigenvalue equation
s A - w B
where s is a non-negative scaling factor chosen so that w, w B,
and s A do not overflow and, if possible, do not underflow, either.
Arguments
=========
A (input) DOUBLE PRECISION array, dimension (LDA, 2)
On entry, the 2 x 2 matrix A. It is assumed that its 1-norm
is less than 1/SAFMIN. Entries less than
sqrt(SAFMIN)*norm(A) are subject to being treated as zero.
LDA (input) INTEGER
The leading dimension of the array A. LDA >= 2.
B (input) DOUBLE PRECISION array, dimension (LDB, 2)
On entry, the 2 x 2 upper triangular matrix B. It is
assumed that the one-norm of B is less than 1/SAFMIN. The
diagonals should be at least sqrt(SAFMIN) times the largest
element of B (in absolute value); if a diagonal is smaller
than that, then +/- sqrt(SAFMIN) will be used instead of
that diagonal.
LDB (input) INTEGER
The leading dimension of the array B. LDB >= 2.
SAFMIN (input) DOUBLE PRECISION
The smallest positive number s.t. 1/SAFMIN does not
overflow. (This should always be DLAMCH('S') -- it is an
argument in order to avoid having to call DLAMCH frequently.)
SCALE1 (output) DOUBLE PRECISION
A scaling factor used to avoid over-/underflow in the
eigenvalue equation which defines the first eigenvalue. If
the eigenvalues are complex, then the eigenvalues are
( WR1 +/- WI i ) / SCALE1 (which may lie outside the
exponent range of the machine), SCALE1=SCALE2, and SCALE1
will always be positive. If the eigenvalues are real, then
the first (real) eigenvalue is WR1 / SCALE1 , but this may
overflow or underflow, and in fact, SCALE1 may be zero or
less than the underflow threshhold if the exact eigenvalue
is sufficiently large.
SCALE2 (output) DOUBLE PRECISION
A scaling factor used to avoid over-/underflow in the
eigenvalue equation which defines the second eigenvalue. If
the eigenvalues are complex, then SCALE2=SCALE1. If the
eigenvalues are real, then the second (real) eigenvalue is
WR2 / SCALE2 , but this may overflow or underflow, and in
fact, SCALE2 may be zero or less than the underflow
threshhold if the exact eigenvalue is sufficiently large.
WR1 (output) DOUBLE PRECISION
If the eigenvalue is real, then WR1 is SCALE1 times the
eigenvalue closest to the (2,2) element of A B**(-1). If the
eigenvalue is complex, then WR1=WR2 is SCALE1 times the real
part of the eigenvalues.
WR2 (output) DOUBLE PRECISION
If the eigenvalue is real, then WR2 is SCALE2 times the
other eigenvalue. If the eigenvalue is complex, then
WR1=WR2 is SCALE1 times the real part of the eigenvalues.
WI (output) DOUBLE PRECISION
If the eigenvalue is real, then WI is zero. If the
eigenvalue is complex, then WI is SCALE1 times the imaginary
part of the eigenvalues. WI will always be non-negative.
=====================================================================
Parameter adjustments */
/* System generated locals */
integer a_dim1, a_offset, b_dim1, b_offset;
Treal d__1, d__2, d__3, d__4, d__5, d__6;
/* Local variables */
Treal diff, bmin, wbig, wabs, wdet, r__, binv11, binv22,
discr, anorm, bnorm, bsize, shift, c1, c2, c3, c4, c5, rtmin,
rtmax, wsize, s1, s2, a11, a12, a21, a22, b11, b12, b22, ascale,
bscale, pp, qq, ss, wscale, safmax, wsmall, as11, as12, as22, sum,
abi22;
#define a_ref(a_1,a_2) a[(a_2)*a_dim1 + a_1]
#define b_ref(a_1,a_2) b[(a_2)*b_dim1 + a_1]
a_dim1 = *lda;
a_offset = 1 + a_dim1 * 1;
a -= a_offset;
b_dim1 = *ldb;
b_offset = 1 + b_dim1 * 1;
b -= b_offset;
/* Function Body */
rtmin = template_blas_sqrt(*safmin);
rtmax = 1. / rtmin;
safmax = 1. / *safmin;
/* Scale A
Computing MAX */
d__5 = (d__1 = a_ref(1, 1), absMACRO(d__1)) + (d__2 = a_ref(2, 1), absMACRO(d__2)),
d__6 = (d__3 = a_ref(1, 2), absMACRO(d__3)) + (d__4 = a_ref(2, 2), absMACRO(
d__4)), d__5 = maxMACRO(d__5,d__6);
anorm = maxMACRO(d__5,*safmin);
ascale = 1. / anorm;
a11 = ascale * a_ref(1, 1);
a21 = ascale * a_ref(2, 1);
a12 = ascale * a_ref(1, 2);
a22 = ascale * a_ref(2, 2);
/* Perturb B if necessary to insure non-singularity */
b11 = b_ref(1, 1);
b12 = b_ref(1, 2);
b22 = b_ref(2, 2);
/* Computing MAX */
d__1 = absMACRO(b11), d__2 = absMACRO(b12), d__1 = maxMACRO(d__1,d__2), d__2 = absMACRO(b22),
d__1 = maxMACRO(d__1,d__2);
bmin = rtmin * maxMACRO(d__1,rtmin);
if (absMACRO(b11) < bmin) {
b11 = template_lapack_d_sign(&bmin, &b11);
}
if (absMACRO(b22) < bmin) {
b22 = template_lapack_d_sign(&bmin, &b22);
}
/* Scale B
Computing MAX */
d__1 = absMACRO(b11), d__2 = absMACRO(b12) + absMACRO(b22), d__1 = maxMACRO(d__1,d__2);
bnorm = maxMACRO(d__1,*safmin);
/* Computing MAX */
d__1 = absMACRO(b11), d__2 = absMACRO(b22);
bsize = maxMACRO(d__1,d__2);
bscale = 1. / bsize;
b11 *= bscale;
b12 *= bscale;
b22 *= bscale;
/* Compute larger eigenvalue by method described by C. van Loan
( AS is A shifted by -SHIFT*B ) */
binv11 = 1. / b11;
binv22 = 1. / b22;
s1 = a11 * binv11;
s2 = a22 * binv22;
if (absMACRO(s1) <= absMACRO(s2)) {
as12 = a12 - s1 * b12;
as22 = a22 - s1 * b22;
ss = a21 * (binv11 * binv22);
abi22 = as22 * binv22 - ss * b12;
pp = abi22 * .5;
shift = s1;
} else {
as12 = a12 - s2 * b12;
as11 = a11 - s2 * b11;
ss = a21 * (binv11 * binv22);
abi22 = -ss * b12;
pp = (as11 * binv11 + abi22) * .5;
shift = s2;
}
qq = ss * as12;
if ((d__1 = pp * rtmin, absMACRO(d__1)) >= 1.) {
/* Computing 2nd power */
d__1 = rtmin * pp;
discr = d__1 * d__1 + qq * *safmin;
r__ = template_blas_sqrt((absMACRO(discr))) * rtmax;
} else {
/* Computing 2nd power */
d__1 = pp;
if (d__1 * d__1 + absMACRO(qq) <= *safmin) {
/* Computing 2nd power */
d__1 = rtmax * pp;
discr = d__1 * d__1 + qq * safmax;
r__ = template_blas_sqrt((absMACRO(discr))) * rtmin;
} else {
/* Computing 2nd power */
d__1 = pp;
discr = d__1 * d__1 + qq;
r__ = template_blas_sqrt((absMACRO(discr)));
}
}
/* Note: the test of R in the following IF is to cover the case when
DISCR is small and negative and is flushed to zero during
the calculation of R. On machines which have a consistent
flush-to-zero threshhold and handle numbers above that
threshhold correctly, it would not be necessary. */
if (discr >= 0. || r__ == 0.) {
sum = pp + template_lapack_d_sign(&r__, &pp);
diff = pp - template_lapack_d_sign(&r__, &pp);
wbig = shift + sum;
/* Compute smaller eigenvalue */
wsmall = shift + diff;
/* Computing MAX */
d__1 = absMACRO(wsmall);
if (absMACRO(wbig) * .5 > maxMACRO(d__1,*safmin)) {
wdet = (a11 * a22 - a12 * a21) * (binv11 * binv22);
wsmall = wdet / wbig;
}
/* Choose (real) eigenvalue closest to 2,2 element of A*B**(-1)
for WR1. */
if (pp > abi22) {
*wr1 = minMACRO(wbig,wsmall);
*wr2 = maxMACRO(wbig,wsmall);
} else {
*wr1 = maxMACRO(wbig,wsmall);
*wr2 = minMACRO(wbig,wsmall);
}
*wi = 0.;
} else {
/* Complex eigenvalues */
*wr1 = shift + pp;
*wr2 = *wr1;
*wi = r__;
}
/* Further scaling to avoid underflow and overflow in computing
SCALE1 and overflow in computing w*B.
This scale factor (WSCALE) is bounded from above using C1 and C2,
and from below using C3 and C4.
C1 implements the condition s A must never overflow.
C2 implements the condition w B must never overflow.
C3, with C2,
implement the condition that s A - w B must never overflow.
C4 implements the condition s should not underflow.
C5 implements the condition max(s,|w|) should be at least 2. */
c1 = bsize * (*safmin * maxMACRO(1.,ascale));
c2 = *safmin * maxMACRO(1.,bnorm);
c3 = bsize * *safmin;
if (ascale <= 1. && bsize <= 1.) {
/* Computing MIN */
d__1 = 1., d__2 = ascale / *safmin * bsize;
c4 = minMACRO(d__1,d__2);
} else {
c4 = 1.;
}
if (ascale <= 1. || bsize <= 1.) {
/* Computing MIN */
d__1 = 1., d__2 = ascale * bsize;
c5 = minMACRO(d__1,d__2);
} else {
c5 = 1.;
}
/* Scale first eigenvalue */
wabs = absMACRO(*wr1) + absMACRO(*wi);
/* Computing MAX
Computing MIN */
d__3 = c4, d__4 = maxMACRO(wabs,c5) * .5;
d__1 = maxMACRO(*safmin,c1), d__2 = (wabs * c2 + c3) * 1.0000100000000001,
d__1 = maxMACRO(d__1,d__2), d__2 = minMACRO(d__3,d__4);
wsize = maxMACRO(d__1,d__2);
if (wsize != 1.) {
wscale = 1. / wsize;
if (wsize > 1.) {
*scale1 = maxMACRO(ascale,bsize) * wscale * minMACRO(ascale,bsize);
} else {
*scale1 = minMACRO(ascale,bsize) * wscale * maxMACRO(ascale,bsize);
}
*wr1 *= wscale;
if (*wi != 0.) {
*wi *= wscale;
*wr2 = *wr1;
*scale2 = *scale1;
}
} else {
*scale1 = ascale * bsize;
*scale2 = *scale1;
}
/* Scale second eigenvalue (if real) */
if (*wi == 0.) {
/* Computing MAX
Computing MIN
Computing MAX */
d__5 = absMACRO(*wr2);
d__3 = c4, d__4 = maxMACRO(d__5,c5) * .5;
d__1 = maxMACRO(*safmin,c1), d__2 = (absMACRO(*wr2) * c2 + c3) *
1.0000100000000001, d__1 = maxMACRO(d__1,d__2), d__2 = minMACRO(d__3,
d__4);
wsize = maxMACRO(d__1,d__2);
if (wsize != 1.) {
wscale = 1. / wsize;
if (wsize > 1.) {
*scale2 = maxMACRO(ascale,bsize) * wscale * minMACRO(ascale,bsize);
} else {
*scale2 = minMACRO(ascale,bsize) * wscale * maxMACRO(ascale,bsize);
}
*wr2 *= wscale;
} else {
*scale2 = ascale * bsize;
}
}
/* End of DLAG2 */
return 0;
} /* dlag2_ */
#undef b_ref
#undef a_ref
#endif
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