working version"
parent
16f97d7f6b
commit
ea2a6a66c5
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import numpy, scipy, gridlock, fdfd_tools
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from fdfd_tools import bloch
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from numpy.linalg import norm
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import logging
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logging.basicConfig(level=logging.DEBUG)
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logger = logging.getLogger(__name__)
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dx = 40
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x_period = 400
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y_period = z_period = 2000
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g = gridlock.Grid([numpy.arange(-x_period/2, x_period/2, dx),
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numpy.arange(-1000, 1000, dx),
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numpy.arange(-1000, 1000, dx)],
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shifts=numpy.array([[0,0,0]]),
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initial=1.445**2,
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periodic=True)
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g.draw_cuboid([0,0,0], [200e8, 220, 220], eps=3.47**2)
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#x_period = y_period = z_period = 13000
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#g = gridlock.Grid([numpy.arange(3), ]*3,
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# shifts=numpy.array([[0, 0, 0]]),
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# initial=2.0**2,
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# periodic=True)
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g2 = g.copy()
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g2.shifts = numpy.zeros((6,3))
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g2.grids = [numpy.zeros(g.shape) for _ in range(6)]
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epsilon = [g.grids[0],] * 3
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reciprocal_lattice = numpy.diag(1000/numpy.array([x_period, y_period, z_period])) #cols are vectors
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#print('Finding k at 1550nm')
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#k, f = bloch.find_k(frequency=1/1550,
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# tolerance=(1/1550 - 1/1551),
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# direction=[1, 0, 0],
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# G_matrix=reciprocal_lattice,
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# epsilon=epsilon,
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# band=0)
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#
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#print("k={}, f={}, 1/f={}, k/f={}".format(k, f, 1/f, norm(reciprocal_lattice @ k) / f ))
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print('Finding f at [0.25, 0, 0]')
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for k0x in [.25]:
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k0 = numpy.array([k0x, 0, 0])
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kmag = norm(reciprocal_lattice @ k0)
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tolerance = 1e-4 * (1/1550) # df = f * dn_eff / n
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logger.info('tolerance {}'.format(tolerance))
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n, v = bloch.eigsolve(4, k0, G_matrix=reciprocal_lattice, epsilon=epsilon, tolerance=tolerance)
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v2e = bloch.hmn_2_exyz(k0, G_matrix=reciprocal_lattice, epsilon=epsilon)
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v2h = bloch.hmn_2_hxyz(k0, G_matrix=reciprocal_lattice, epsilon=epsilon)
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ki = bloch.generate_kmn(k0, reciprocal_lattice, g.shape)
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z = 0
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e = v2e(v[0])
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for i in range(3):
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g2.grids[i] += numpy.real(e[i])
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g2.grids[i+3] += numpy.imag(e[i])
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f = numpy.sqrt(numpy.real(numpy.abs(n))) # TODO
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print('k0x = {:3g}\n eigval = {}\n f = {}\n'.format(k0x, n, f))
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n_eff = norm(reciprocal_lattice @ k0) / f
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print('kmag/f = n_eff = {} \n wl = {}\n'.format(n_eff, 1/f ))
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@ -0,0 +1,412 @@
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/* Copyright (C) 1999-2014 Massachusetts Institute of Technology.
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*
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* This program is free software; you can redistribute it and/or modify
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* it under the terms of the GNU General Public License as published by
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* the Free Software Foundation; either version 2 of the License, or
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* (at your option) any later version.
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*
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* This program is distributed in the hope that it will be useful,
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* but WITHOUT ANY WARRANTY; without even the implied warranty of
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* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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* GNU General Public License for more details.
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*
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* You should have received a copy of the GNU General Public License
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* along with this program; if not, write to the Free Software
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* Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
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*/
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#include <stdlib.h>
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#include <stdio.h>
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#include <math.h>
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#include "config.h"
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#include <mpiglue.h>
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#include <mpi_utils.h>
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#include <check.h>
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#include <scalar.h>
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#include <matrices.h>
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#include <blasglue.h>
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#include "eigensolver.h"
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#include "linmin.h"
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extern void eigensolver_get_eigenvals_aux(evectmatrix Y, real *eigenvals,
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evectoperator A, void *Adata,
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evectmatrix Work1, evectmatrix Work2,
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sqmatrix U, sqmatrix Usqrt,
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sqmatrix Uwork);
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#define STRINGIZEx(x) #x /* a hack so that we can stringize macro values */
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#define STRINGIZE(x) STRINGIZEx(x)
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#define K_PI 3.141592653589793238462643383279502884197
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#define MIN2(a,b) ((a) < (b) ? (a) : (b))
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#define MAX2(a,b) ((a) > (b) ? (a) : (b))
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#if defined(SCALAR_LONG_DOUBLE_PREC)
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# define fabs fabsl
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# define cos cosl
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# define sin sinl
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# define sqrt sqrtl
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# define atan atanl
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# define atan2 atan2l
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#endif
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/* Evalutate op, and set t to the elapsed time (in seconds). */
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#define TIME_OP(t, op) { \
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mpiglue_clock_t xxx_time_op_start_time = MPIGLUE_CLOCK; \
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{ \
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op; \
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} \
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(t) = MPIGLUE_CLOCK_DIFF(MPIGLUE_CLOCK, xxx_time_op_start_time); \
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}
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/**************************************************************************/
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#define EIGENSOLVER_MAX_ITERATIONS 100000
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#define FEEDBACK_TIME 4.0 /* elapsed time before we print progress feedback */
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/* Number of iterations after which to reset conjugate gradient
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direction to steepest descent. (Picked after some experimentation.
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Is there a better basis? Should this change with the problem
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size?) */
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#define CG_RESET_ITERS 70
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/* Threshold for trace(1/YtBY) = trace(U) before we reorthogonalize: */
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#define EIGS_TRACE_U_THRESHOLD 1e8
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/**************************************************************************/
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/* estimated times/iteration for different iteration schemes, based
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on the measure times for various operations and the operation counts: */
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#define EXACT_LINMIN_TIME(t_AZ, t_KZ, t_ZtW, t_ZS, t_ZtZ, t_linmin) \
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((t_AZ)*2 + (t_KZ) + (t_ZtW)*4 + (t_ZS)*2 + (t_ZtZ)*2 + (t_linmin))
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#define APPROX_LINMIN_TIME(t_AZ, t_KZ, t_ZtW, t_ZS, t_ZtZ) \
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((t_AZ)*2 + (t_KZ) + (t_ZtW)*2 + (t_ZS)*2 + (t_ZtZ)*2)
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/* Guess for the convergence slowdown factor due to the approximate
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line minimization. It is probably best to be conservative, as the
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exact line minimization is more reliable and we only want to
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abandon it if there is a big speed gain. */
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#define APPROX_LINMIN_SLOWDOWN_GUESS 2.0
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/* We also don't want to use the approximate line minimization if
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the exact line minimization makes a big difference in the value
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of the trace that's achieved (i.e. if one step of Newton's method
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on the trace derivative does not do a good job). The following
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is the maximum improvement by the exact line minimization (over
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one step of Newton) at which we'll allow the use of approximate line
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minimization. */
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#define APPROX_LINMIN_IMPROVEMENT_THRESHOLD 0.05
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/**************************************************************************/
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typedef struct {
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sqmatrix YtAY, DtAD, symYtAD, YtBY, DtBD, symYtBD, S1, S2, S3;
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real lag, d_lag, trace_YtLY, trace_DtLD, trace_YtLD;
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} trace_func_data;
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static linmin_real trace_func(linmin_real theta, linmin_real *trace_deriv, void *data)
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{
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linmin_real trace;
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trace_func_data *d = (trace_func_data *) data;
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linmin_real c = cos(theta), s = sin(theta);
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YDNi = c*c * YtY + s*s * DtD + 2*s*c * symYtD
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YDNi.inv()
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if not YDNi.inv():
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/* if c or s is small, we sometimes run into numerical
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difficulties, and it is better to use the Taylor expansion */
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if c < 1e-4 * s and c != 0:
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YDNi = DtD.inv()
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S3 = (YDNi @ symYtD.H) @ YDNi.H
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YDNi = 1/(s*s) * YDNi - 2*c/(s*s*s) * S3
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elif s < 1e-4 * c and s != 0:
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YDNi = YtY.inv()
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S3 = (YDNi @ symYtD.H) @ YDNi.H
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YDNi = 1/(c*c) * YDNi - 2*s/(c*c*c) * S3
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else:
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CHECK(0, "inexplicable singularity in linmin trace_func")
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YADN = c*c * YtAY + s*s * DtAD + 2*s*c * smYtAD
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trace = real(trace(YADN.H @ YDNi)) + (c*c * trace_YtLY + s*s * trace_DtLD + 2*s*c * trace_YtLD) * (c * lag + s * d_lag)
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if (trace_deriv) {
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c2 = cos(2*theta)
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s2 = sin(2*theta);
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S3 = -0.5 * s2 * (YtAY - DtAD) + c2 * symYtAD
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*trace_deriv = real(trace(YDNi.H @ S3))
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S2 = (YDNi @ YADN.H) @ YDNi.H
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S3 = -0.5 * s2 * (YtY - DtD) + c2 * symYtD
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*trace_deriv -= real(trace(S2.H @ S3))
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*trace_deriv *= 2
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*trace_deriv += (-s2 * trace_YtLY + s2 * trace_DtLD
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+ 2*c2 * trace_YtLD) * (c * lag + s * d_lag);
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*trace_deriv += (c*c * trace_YtLY + s*s * trace_DtLD
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+ 2*s*c * trace_YtLD) * (-s * lag + c * d_lag);
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}
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return trace;
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}
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/**************************************************************************/
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#define EIG_HISTORY_SIZE 5
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/* find generalized eigenvectors Y of (A,B) by minimizing Rayleigh quotient
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tr [ Yt A Y / (Yt B Y) ] + lag * tr [ Yt L Y ]
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where lag is a Lagrange multiplier and L is a Hermitian operator
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implementing some constraint tr [ Yt L Y ] = 0 on the eigenvectors
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(if L is not NULL).
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Constraints that commute with A and B (and L) are specified via the
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"constraint" argument, which gives the projection operator for
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the constraint(s).
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*/
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void eigensolver_lagrange(evectmatrix Y, real *eigenvals,
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evectoperator A, void *Adata,
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evectoperator B, void *Bdata,
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evectpreconditioner K, void *Kdata,
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evectconstraint constraint, void *constraint_data,
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evectoperator L, void *Ldata, real *lag,
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evectmatrix Work[], int nWork,
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real tolerance, int *num_iterations,
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int flags)
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{
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real g_lag = 0, d_lag = 0, prev_g_lag = 0;
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short usingConjugateGradient = 0, use_polak_ribiere = 0,
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use_linmin = 1;
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real E, prev_E = 0.0;
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real d_scale = 1.0;
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real traceGtX, prev_traceGtX = 0.0;
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real theta, prev_theta = 0.5;
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int i, iteration = 0, num_emergency_restarts = 0;
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mpiglue_clock_t prev_feedback_time;
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real linmin_improvement = 0;
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G = Work[0];
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X = Work[1];
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BY = Y;
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D = X;
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BD = D; /* storage for B*D (re-use B*Y storage) */
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prev_G = G;
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restartY:
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eigenvals *= 0.0
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convergence_history = [10000.0] * n
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constraint(Y, constraint_data)
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do {
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real y_norm, gamma_numerator = 0;
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YtBY = Y.H @ Y
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y_norm = sqrt(real(trace(YtBY)) / Y.p);
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Y /= y_norm
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YtBY /= y_norm*y_norm
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U = YtBY
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if (!sqmatrix_invert(U, 1, S2)) /* non-independent Y columns */
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/* emergency restart with random Y */
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...
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/* If trace(1/YtBY) gets big, it means that the columns
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of Y are becoming nearly parallel. This sometimes happens,
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especially in the targeted eigensolver, because the
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preconditioner pushes all the columns towards the ground
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state. If it gets too big, it seems to be a good idea
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to re-orthogonalize, resetting conjugate-gradient, as
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otherwise we start to encounter numerical problems. */
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if (flags & EIGS_REORTHOGONALIZE) {
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traceU = real(trace(U))
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if (traceU > EIGS_TRACE_U_THRESHOLD * U.p) {
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Y = Y @ sqrtm(U).H /* orthonormalize Y */
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prev_traceGtX = 0.0;
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YtBY = Y.H @ Y
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y_norm = sqrt(real(trace(YtBY)) / Y.p)
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Y /= y_norm
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YtBY /= y_norm * y_norm
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U = YtBY
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/* G = AYU; note that U is Hermitian: */
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G = A @ Y @ U
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YtAYU = Y.H @ G
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E = real(trace(YtAYU))
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convergence_history[iteration % EIG_HISTORY_SIZE] =
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200.0 * fabs(E - prev_E) / (fabs(E) + fabs(prev_E));
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if (iteration > 0 &&
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fabs(E - prev_E) < tolerance * 0.5 * (E + prev_E + 1e-7))
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break; /* convergence! hooray! */
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/* Compute gradient of functional: G = (1 - BY U Yt) A Y U */
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G += -Y @ (U @ YtAYU)
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/* set X = precondition(G): */
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X = K @ G
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//TIME_OP(time_KZ, K(G, X, Kdata, Y, NULL, YtBY));
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/* We have to apply the constraint here, in case it doesn't
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commute with the preconditioner. */
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constraint(X, constraint_data);
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d_scale = 1.0;
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/* Minimize the trace along Y + lambda*D: */
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if (!use_linmin) {
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real dE, E2, d2E, t, d_norm;
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/* Here, we do an approximate line minimization along D
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by evaluating dE (the derivative) at the current point,
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and the trace E2 at a second point, and then approximating
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the second derivative d2E by finite differences. Then,
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we use one step of Newton's method on the derivative.
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This has the advantage of requiring two fewer O(np^2)
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matrix multiplications compared to the exact linmin. */
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d_norm = sqrt(real(trace(D.H @ D)) / Y.p);
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/* dE = 2 * tr Gt D. (Use prev_G instead of G so that
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it works even when we are using Polak-Ribiere.) */
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dE = 2.0 * SCALAR_RE(evectmatrix_traceXtY(prev_G, D)) / d_norm;
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/* shift Y by prev_theta along D, in the downhill direction: */
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t = dE < 0 ? -fabs(prev_theta) : fabs(prev_theta);
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Y += t/d_norm * D
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U = inv(Y.H @ Y)
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E2 = real(trace((Y.H @ A @ Y).H @ U))
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/* Get finite-difference approximation for the 2nd derivative
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of the trace. Equivalently, fit to a quadratic of the
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form: E(theta) = E + dE theta + 1/2 d2E theta^2 */
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d2E = (E2 - E - dE * t) / (0.5 * t * t);
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theta = -dE/d2E;
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/* If the 2nd derivative is negative, or a big shift
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in the trace is predicted (compared to the previous
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iteration), then this approximate line minimization is
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probably not very good; switch back to the exact
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line minimization. Hopefully, we won't have to
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abort like this very often, as it wastes operations. */
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if (d2E < 0 || -0.5*dE*theta > 20.0 * fabs(E-prev_E)) {
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if (flags & EIGS_FORCE_APPROX_LINMIN) {
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} else {
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use_linmin = 1;
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evectmatrix_aXpbY(1.0, Y, -t / d_norm, D);
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prev_theta = atan(prev_theta); /* convert to angle */
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/* don't do this again: */
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flags |= EIGS_FORCE_EXACT_LINMIN;
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}
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}
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else {
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/* Shift Y by theta, hopefully minimizing the trace: */
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evectmatrix_aXpbY(1.0, Y, (theta - t) / d_norm, D);
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}
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}
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if (use_linmin) {
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d_scale = sqrt(real(trace(D.H @ D)) / Y.p);
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D /= d_scale
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AD = A @ D
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DtD = D.H @ D
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DtAD = D.H @ AD
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YtD = Y.H @ D
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YtAD = Y.H @ AD
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symYtD = (YtD + YtD.H) / 2
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symYtAD = (YtAD + YtAD.H) / 2
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U_sYtD = U @ symYtD.H
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dE = 2.0 * (real(trace(U.H @ symYtAD)) - real(trace(YtAYU.H @ U_sYtD)))
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S2 = DtD - 4 * symYtD @ U_sYtD
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d2E = 2.0 * (real(trace(U.H @ DtAD)) -
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real(trace(YtAYU.H @ U @ S2)) -
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4.0 * real(trace(U.H @ symYtAD @ U_sYtD)))
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d_lag = lag = trace_YtLY = trace_DtLD = trace_YtLD = 0
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/* this is just Newton-Raphson to find a root of
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the first derivative: */
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theta = -dE/d2E;
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if d2E < 0 or abs(theta) >= K_PI:
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||||
theta = -abs(prev_theta) * numpy.sign(dE)
|
||||
|
||||
/* Set S1 to YtAYU * YtBY = YtAY for use in linmin.
|
||||
(tfd.YtAY == S1). */
|
||||
YtAY = YtAYU @ YtBY.H
|
||||
|
||||
theta = linmin(&new_E, &new_dE, theta, E, dE,
|
||||
0.1, min(tolerance, 1e-6), 1e-14,
|
||||
0, -numpy.sign(dE) * K_PI,
|
||||
trace_func, &tfd,
|
||||
flags & EIGS_VERBOSE)
|
||||
linmin_improvement = abs(E - new_E) * 2.0/abs(E + new_E);
|
||||
|
||||
/* Shift Y to new location minimizing the trace along D: */
|
||||
Y *= cos(theta)
|
||||
Y += D * sin(theta)
|
||||
}
|
||||
|
||||
/* In exact arithmetic, we don't need to do this, but in practice
|
||||
it is probably a good idea to keep errors from adding up and
|
||||
eventually violating the constraints. */
|
||||
constraint(Y, constraint_data);
|
||||
|
||||
prev_traceGtX = traceGtX;
|
||||
prev_theta = theta;
|
||||
prev_E = E;
|
||||
|
||||
/* Finally, we use the times for the various operations to
|
||||
help us pick an algorithm for the next iteration: */
|
||||
{
|
||||
real t_exact, t_approx;
|
||||
t_exact = EXACT_LINMIN_TIME(time_AZ, time_KZ, time_ZtW,
|
||||
time_ZS, time_ZtZ, time_linmin);
|
||||
t_approx = APPROX_LINMIN_TIME(time_AZ, time_KZ, time_ZtW,
|
||||
time_ZS, time_ZtZ);
|
||||
|
||||
/* Sum the times over the processors so that all the
|
||||
processors compare the same, average times. */
|
||||
mpi_allreduce_1(&t_exact,
|
||||
real, SCALAR_MPI_TYPE, MPI_SUM, mpb_comm);
|
||||
mpi_allreduce_1(&t_approx,
|
||||
real, SCALAR_MPI_TYPE, MPI_SUM, mpb_comm);
|
||||
|
||||
if (!(flags & EIGS_FORCE_EXACT_LINMIN) &&
|
||||
linmin_improvement > 0 &&
|
||||
linmin_improvement <= APPROX_LINMIN_IMPROVEMENT_THRESHOLD &&
|
||||
t_exact > t_approx * APPROX_LINMIN_SLOWDOWN_GUESS) {
|
||||
use_linmin = 0;
|
||||
}
|
||||
else if (!(flags & EIGS_FORCE_APPROX_LINMIN)) {
|
||||
use_linmin = 1;
|
||||
prev_theta = atan(prev_theta); /* convert to angle */
|
||||
}
|
||||
}
|
||||
} while (++iteration < EIGENSOLVER_MAX_ITERATIONS);
|
||||
|
||||
evectmatrix_XtX(U, Y, S2);
|
||||
CHECK(sqmatrix_invert(U, 1, S2), "singular YtBY at end");
|
||||
eigensolver_get_eigenvals_aux(Y, eigenvals, A, Adata,
|
||||
X, G, U, S1, S2);
|
||||
|
||||
}
|
||||
|
||||
Loading…
Reference in New Issue