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cauchy_walk_complex.m
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cauchy_walk_complex.m
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% walk around a pole in complex time. the pole is where one variable
% takes on value 0. the purpose of the walk is to discover which other
% points lie on the same curve around the pole.
%
%
%
% copyright 2016 Daniel Brake
% University of Notre Dame
% Applied and Computational Mathematics and Statistics
%
% Bertini (TM) is a registered trademark.
%
% This file is part of Bertini_tropical.
%
% Bertini_tropical 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.
%
% Bertini_tropical 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/>.
function [walked_paths, unvisited_starting_points, unvisited_starting_connections] = cauchy_walk_complex(starting_points, starting_connections, center_points, slice_value, component_degree, b_input_copyme, variable_index, num_sample_points, unique_point_threshold)
thresh = unique_point_threshold;
b_input = copy(b_input_copyme);
b_input.config.UserHomotopy = 2;
system_variables = b_input.variable_group;
num_vars = numel(system_variables);
b_input.declare_symbols({'walk_around'},'function');
b_input.declare_and_define('eps',slice_value,'constant');
num_paths = 0;
walked_paths = repmat(cauchy_path(),1,size(starting_points,2));
unvisited_starting_connections = starting_connections;
unvisited_starting_points = starting_points; % take a copy
while size(unvisited_starting_points,2)>0 %we delete points from unvisited_starting_points as this loop proceeds.
num_paths = num_paths+1;
walked_paths(num_paths) = cauchy_path(); % initialize to new class object.
walked_path = walked_paths(num_paths); %using handle class here, so that modification of one is propagated to the other.
walked_path.path_variable = variable_index; %should this be the name or symbol rather than index?
walked_path.set_start_point(unvisited_starting_points(:,1));
walked_path.center_point = center_points(:,unvisited_starting_connections(1));
walked_path.set_radius(abs(slice_value));
%walk around t = eps to t = eps in complex time
current_point = unvisited_starting_points(:,1);
periodic_point = current_point;
%initialize some data containers
visited_starting_points = [];
walked_path.add_connecting_point(current_point);
unvisited_starting_connections(1) = [];
unvisited_starting_points(:,1) = []; %delete the point from the list
cycle_number = 0;
iteration = 0;
%enter the loop for walking around t=0 in complex time
while iteration<component_degree
iteration = iteration+1; % increment this counter
for jj = 1:num_sample_points
startname = sprintf('start%i_%i',iteration,jj);
inputname = sprintf('input_walk_%i_%i',iteration,jj);
b_input.define_symbol('walk_around',...
sprintf('%s - eps*exp( I*(s*2*Pi*%i/%i + (1-s)*2*Pi*%i/%i) )',system_variables{variable_index}, jj-1,num_sample_points,jj,num_sample_points));
write_generic_solns(current_point,startname);
write_bertini_input_file(system_variables, b_input.functions,'filename',inputname,'options',b_input.config,'constants',b_input.constant,'subfunctions',b_input.subfunction);%
bertini('filename',inputname,'startname',startname,'stifle','rerunonfail',10); %,'mpi',min(4,size(current_point,2))
next_point = get_generic_solns('nonsingular_solutions',num_vars);
movefile('nonsingular_solutions',['nonsingular_solutions_' num2str(iteration) '_' num2str(jj)])
if isempty(next_point)
next_point = get_generic_solns('finite_solutions',num_vars);
if ~isempty(next_point)
warning('empty point in cauchy path...read from finite_solutions instead of nonsingular_solutions.consider changing condnumthreshold.');
else
error('path failure in cauchy_path tracking. consult a Bertini expert.');
end
end
walked_path.add_point_to_path(next_point);
current_point = next_point;
end %re: for jj
cycle_number = cycle_number+1;
%now we check to see if we came back to where we started. if so,
%then done with this particular loop, and will either be done, or
%start a new loop on the first unvisited starting point if it
%exists.
if ~isempty(find_same_point(periodic_point, current_point, unique_point_threshold))
break;
end
%got here, so not back at the beginning of the loop.
visited_index = find_same_point(unvisited_starting_points, current_point, unique_point_threshold);
%are we at a previously encountered point? if not, we skip the
%below things.
if isempty(visited_index)
continue
end
%well, we're here, so did not return to the start of the path, and
%are at one of the points on the same line as the starting point.
if length(visited_index)>1
thresh = unique_point_threshold;
while length(visited_index)>1
thresh = thresh/2;
visited_index = find_same_point(unvisited_starting_points, current_point, thresh);
if isempty(visited_index)
thresh = thresh*sqrt(5);
end
end
warning('had to shrink unique_point_threshold to %1.4e\nto get a unique visited starting point on monodromy loop\n',thresh);
end
walked_path.add_connecting_point(unvisited_starting_points(:,visited_index));
visited_starting_points = [visited_starting_points unvisited_starting_points(:,visited_index)];
unvisited_starting_points = unvisited_starting_points(:,1:end~=visited_index); %delete the visited points.
unvisited_starting_connections = unvisited_starting_connections(1:end~=visited_index);
% unvisited_starting_points
% visited_starting_points
% max(abs(periodic_point-current_point))
%
end %re: while and(max(...
%need to do better error catching here for challenging decompositions.
if isempty(find_same_point(periodic_point, current_point, thresh))
warning('exceeded maximum theoretical upper bound on cycle number\ndid not return to start in %i cycles',component_degree)
pause
end
walked_path.cycle_number = cycle_number;
end %re: while size(unvisited_starting_points...
walked_paths = walked_paths(1:num_paths);
end %re: function cauchy_walk