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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_LAEV2_HEADER
#define TEMPLATE_LAPACK_LAEV2_HEADER
template<class Treal>
int template_lapack_laev2(Treal *a, Treal *b, Treal *c__,
Treal *rt1, Treal *rt2, Treal *cs1, Treal *sn1)
{
/* -- LAPACK auxiliary routine (version 3.0) --
Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
Courant Institute, Argonne National Lab, and Rice University
October 31, 1992
Purpose
=======
DLAEV2 computes the eigendecomposition of a 2-by-2 symmetric matrix
[ A B ]
[ B C ].
On return, RT1 is the eigenvalue of larger absolute value, RT2 is the
eigenvalue of smaller absolute value, and (CS1,SN1) is the unit right
eigenvector for RT1, giving the decomposition
[ CS1 SN1 ] [ A B ] [ CS1 -SN1 ] = [ RT1 0 ]
[-SN1 CS1 ] [ B C ] [ SN1 CS1 ] [ 0 RT2 ].
Arguments
=========
A (input) DOUBLE PRECISION
The (1,1) element of the 2-by-2 matrix.
B (input) DOUBLE PRECISION
The (1,2) element and the conjugate of the (2,1) element of
the 2-by-2 matrix.
C (input) DOUBLE PRECISION
The (2,2) element of the 2-by-2 matrix.
RT1 (output) DOUBLE PRECISION
The eigenvalue of larger absolute value.
RT2 (output) DOUBLE PRECISION
The eigenvalue of smaller absolute value.
CS1 (output) DOUBLE PRECISION
SN1 (output) DOUBLE PRECISION
The vector (CS1, SN1) is a unit right eigenvector for RT1.
Further Details
===============
RT1 is accurate to a few ulps barring over/underflow.
RT2 may be inaccurate if there is massive cancellation in the
determinant A*C-B*B; higher precision or correctly rounded or
correctly truncated arithmetic would be needed to compute RT2
accurately in all cases.
CS1 and SN1 are accurate to a few ulps barring over/underflow.
Overflow is possible only if RT1 is within a factor of 5 of overflow.
Underflow is harmless if the input data is 0 or exceeds
underflow_threshold / macheps.
=====================================================================
Compute the eigenvalues */
/* System generated locals */
Treal d__1;
/* Local variables */
Treal acmn, acmx, ab, df, cs, ct, tb, sm, tn, rt, adf, acs;
integer sgn1, sgn2;
sm = *a + *c__;
df = *a - *c__;
adf = absMACRO(df);
tb = *b + *b;
ab = absMACRO(tb);
if (absMACRO(*a) > absMACRO(*c__)) {
acmx = *a;
acmn = *c__;
} else {
acmx = *c__;
acmn = *a;
}
if (adf > ab) {
/* Computing 2nd power */
d__1 = ab / adf;
rt = adf * template_blas_sqrt(d__1 * d__1 + 1.);
} else if (adf < ab) {
/* Computing 2nd power */
d__1 = adf / ab;
rt = ab * template_blas_sqrt(d__1 * d__1 + 1.);
} else {
/* Includes case AB=ADF=0 */
rt = ab * template_blas_sqrt(2.);
}
if (sm < 0.) {
*rt1 = (sm - rt) * .5;
sgn1 = -1;
/* Order of execution important.
To get fully accurate smaller eigenvalue,
next line needs to be executed in higher precision. */
*rt2 = acmx / *rt1 * acmn - *b / *rt1 * *b;
} else if (sm > 0.) {
*rt1 = (sm + rt) * .5;
sgn1 = 1;
/* Order of execution important.
To get fully accurate smaller eigenvalue,
next line needs to be executed in higher precision. */
*rt2 = acmx / *rt1 * acmn - *b / *rt1 * *b;
} else {
/* Includes case RT1 = RT2 = 0 */
*rt1 = rt * .5;
*rt2 = rt * -.5;
sgn1 = 1;
}
/* Compute the eigenvector */
if (df >= 0.) {
cs = df + rt;
sgn2 = 1;
} else {
cs = df - rt;
sgn2 = -1;
}
acs = absMACRO(cs);
if (acs > ab) {
ct = -tb / cs;
*sn1 = 1. / template_blas_sqrt(ct * ct + 1.);
*cs1 = ct * *sn1;
} else {
if (ab == 0.) {
*cs1 = 1.;
*sn1 = 0.;
} else {
tn = -cs / tb;
*cs1 = 1. / template_blas_sqrt(tn * tn + 1.);
*sn1 = tn * *cs1;
}
}
if (sgn1 == sgn2) {
tn = *cs1;
*cs1 = -(*sn1);
*sn1 = tn;
}
return 0;
/* End of DLAEV2 */
} /* dlaev2_ */
#endif
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