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decompose_tropical_complex_curve.m
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decompose_tropical_complex_curve.m
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%% [produced_rays, ray_multiplicities, aux_data] = decompose_tropical_complex_curve(varargin)
%
% a function which decomposes a complex curve's tropicalization.
%
% the input for this function is a bertini input file, whos name is input
% by default.
%
% the main two output variables are produced_rays, and
% ray_multiplicities. all other data is output via the aux_data
% struct, and includes cauchy paths, puiseux series coefficient,
% and a number of other things
%
% options are passed to this function by string-valued option names, as
% detailed below.
%
% command-line options:
%
% * 'input' - a string specifying the name of the Bertini input file to use
% * 'options' - a struct, the fieldnames of which are bertini options, and the
% * values of which are the values of those options
% * 'purge' - boolean. true means delete previously existing temporary
% directories prior to starting, false is leave them in place. false is
% default.
% * 'defaultslicevalue' - positive float, the default value for slicing near
% coordinate axes. the slice value used is the lesser of (this value, and
% the nearest absolute value of critical points). the default for this
% setting was arbitrarily chosen to be 0.1.
% * 'numsamplepoints' - even integer, determines the number of samples
% taken for monodromy around the coordinate axes. default is 8. larger
% problems may require a higher number.
% * 'puiseuxthreshold' - computed puiseux series coefficients less than this
% tolerance are considered zero.
% * 'intersectionthreshold' - coordinates of intersection points less than this
% tolerance are considered zero.
% * 'intersectionuniquethreshold' - intersection points are unique if
% separated by this distance or more.
% * 'cauchyuniquethreshold' - points on cauchy loops are unique if separated
% by more than this distance.
%
%
% copyright 2015, 2016 Daniel Brake
% University of Notre Dame
% Applied and Computational Mathematics and Statistics
% danielthebrake@gmail.com
%
% 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/>.
%% The main function
function [produced_rays, ray_multiplicities, aux_data] = decompose_tropical_complex_curve(varargin)
%%%%%%%%%%%%%%%%%%%%%
%
% preliminary setup
%
%%%%%%%%%%%%%%%%%%%%%
%% parse the command line and read the input file
[curve_system_name, cauchy_walk_point_uniqueness_threshold, intersection_point_uniqueness_threshold, intersection_zero_coord_threshold, puiseux_zero_coeff_threshold,min_slice_value,default_slice_value,remove_prev_dir,num_sample_points,user_config] = parse_varargin(varargin{:});
b_input = bertini_input(curve_system_name);
if isfield(b_input.config,'tracktype')
b_input.config = rmfield(b_input.config,'tracktype');
end
f = fieldnames(user_config);
for ii = 1:length(f)
b_input.config.( lower(f{ii}) ) = user_config.(f{ii});
end
system_variables = b_input.variable_group;
num_vars = numel(b_input.variable_group);
%% get the numerical irreducible decomposition of the system
if isempty(dir('witness_data'))
error('cannot continue without bertini file ''witness_data''. please generate it.');
end
w_data = witness_data('parse','dehom');
selections = w_data.unidimensionsal_select(1); % choose only from one-dimensional components
w = w_data.construct_agglomeration(selections);
aux_data.initial_start_points = w.points;
aux_data.random_line_coefficients = w.linears;
component_degree = size(aux_data.initial_start_points,2); %the degree of the complex component
starting_directory = move_into_working_directory(remove_prev_dir);
%%%%%%%%%%%%%%
%
% now starts the actual work
%
%%%%%%%%%%%%%%%%%%
%% get only the unique intersection points.
[raw_intersection_points] = compute_intersection_points(b_input, aux_data, 'complex');
%this is here, and not in the compute_intersection_points function, so that
%the intersections points can be thresholded differently if the setting
%changes on a subsequent call.
[intersection_points,~] = unique_up_to_threshold(raw_intersection_points, intersection_point_uniqueness_threshold); % ~ replaced indices
%set up a bunch of empty variables which grow during the
%variable-by-variable loop which follows.
raw_corresponding_coefficients = {};
raw_corresponding_variables = {};
raw_rays = [];
considered_points = [];
aux_data.paths = [];
% compute the critical points with respect to the current time variable
[aux_data.generic_critical_start_points, detjac_input] = compute_critical_points_start_points(b_input,aux_data.initial_start_points,aux_data.random_line_coefficients);
%cache a copy into isect for use and modification throughout the next loop.
isect = intersection_points;
slice_values = zeros(num_vars,1);
%% we do a variable-by-variable study, computing the rays for each intersection point.
for ii= 1:num_vars
%% get the indices of those intersection points such that variable ii is 0
intersect_this_variable = find(abs(isect(ii,:))<intersection_zero_coord_threshold);
% put the found points into a matrix for use in this iteration of the
% loop. then delete from the isect variable, so that we don't study the
% intersection points more than once.
local_isect = isect(:,intersect_this_variable);
isect(:,intersect_this_variable) = []; %delete the used points. don't worry, we have a cached copy up top.
% if there are no intersection points for this variable, then there's
% nothing to do. continue.
if isempty(intersect_this_variable)
fprintf('\n\n\nno unstudied intersection points for variable %s\n\n\n\n',system_variables{ii});
slice_values(ii) = nan;
continue;
end
%% compute the critical points with respect to the current time variable
aux_data.finite_critical_points.(system_variables{ii}) = compute_specific_critical_points(detjac_input,ii,aux_data.generic_critical_start_points);
% by default slice at default (arbitrary, but up to user to override) place.
slice_value = default_slice_value;
if ~isempty(aux_data.finite_critical_points.(system_variables{ii}))
%the following syntax is likely broken on older versions of matlab.
crit_point_values = aux_data.finite_critical_points.(system_variables{ii})(ii,:); % project onto the kth variable
%threshold away critical values at 0.
crit_point_values = crit_point_values( abs(crit_point_values)>intersection_zero_coord_threshold );
nonzero_intersection_coordinates = intersection_points(ii,:);
nonzero_intersection_coordinates = nonzero_intersection_coordinates(abs(nonzero_intersection_coordinates)>intersection_zero_coord_threshold);
if ~isempty(crit_point_values)
slice_value = max(min([default_slice_value abs(crit_point_values)/2 abs(nonzero_intersection_coordinates)/2]),min_slice_value);
end
end
slice_values(ii) = slice_value;
fprintf('slicing value for t=%s is %1.4e\n',system_variables{ii},slice_value)
%% slice the curve near the intersection points, and then do monodromy
%loops around the intersection points to determine the number of unique
%paths leading to the point.
%
%this call builds the paths, and stores them. to the paths are
%attached a bunch of data -- the cycle number, the center point, the previously known
%points which were visited, and the path itself.
[tmp_paths,rejected_path_intersection_points,slice_points, connections] = slice_and_monodromy_complex(slice_value, local_isect, aux_data.initial_start_points, aux_data.random_line_coefficients, component_degree, b_input, ii, num_sample_points, cauchy_walk_point_uniqueness_threshold);
if size(unique_up_to_threshold([considered_points rejected_path_intersection_points],intersection_point_uniqueness_threshold),2) ~= size(considered_points,2)
warning('have encountered intersection points which were not previously studied. the results of this decomposition are almost certainly incorrect. these points have been stored.');
aux_data.rejected_path_intersection_points.(system_variables{ii}) = rejected_path_intersection_points;
end
%add the paths which were computed.
aux_data.paths = [aux_data.paths tmp_paths];
aux_data.connections.(system_variables{ii}) = connections;
aux_data.studied_intersection_points.(system_variables{ii}) = local_isect;
aux_data.slice_points.(system_variables{ii}) = slice_points;
%preallocate
numerators = zeros(num_vars,1);
%% compute the rays using cauchy integrals
% for each path walked during slice and separate.
for zz = 1:length(tmp_paths)
computed_coefficients = zeros(num_vars,1);
%grab the current center point.
curr_center_point = tmp_paths(zz).center_point;
%now to compute the cauchy integrals, for the puiseux coefficients.
for vv=1:num_vars
if 0 %abs(curr_center_point(vv))>intersection_zero_coord_threshold
%a non-zero coordinate in the center point of a loop
%indicates that the puiseux series has a constant term.
%this is k=0, and in this case, the puiseux coefficient
%is the center point coordinate.
k = 0;
computed_coeff = curr_center_point(vv);
else
k = 0; %initialize
computed_coeff = cauchy_integral(tmp_paths(zz).path(vv,:), tmp_paths(zz).radius, tmp_paths(zz).cycle_number, k);
largest_coeff = [computed_coeff k];
while and(abs(computed_coeff)<puiseux_zero_coeff_threshold, k<=component_degree)
k = k+1;
computed_coeff = cauchy_integral(tmp_paths(zz).path(vv,:), tmp_paths(zz).radius, tmp_paths(zz).cycle_number, k);
if abs(computed_coeff) > abs(largest_coeff(1))
largest_coeff = [computed_coeff k];
end
end
if k> component_degree
warning('computed valuation for variable %i exceeds maximum theoretical valuation. this is certainly an error. largest puiseux coefficient was %e, for term %i press any key to continue.',vv,largest_coeff(1),largest_coeff(2));
pause
end
end % re: if abs()
computed_coefficients(vv) = computed_coeff;
numerators(vv) = k;
end
raw_rays(1:num_vars,end+1) = -numerators;
raw_corresponding_variables{end+1} = ii;
raw_corresponding_coefficients{end+1} = computed_coefficients;
end
considered_points = [considered_points local_isect];
end
if size(considered_points,2) ~= size(intersection_points,2)
display('the number of considered points does not match the total number of intersection points!');
pause;
end
cd(starting_directory);
%% compute the number of times each unique ray appears in raw_rays.
produced_rays = unique_up_to_threshold(raw_rays, 1e-4);
ray_multiplicities = zeros(1,size(produced_rays,2));
corresponding_variables = cell(1,size(produced_rays,2));
corresponding_coefficients = cell(1,size(produced_rays,2));
corresponding_paths = cell(1,size(produced_rays,2));
corresponding_centers = cell(1,size(produced_rays,2));
for ii = 1:size(produced_rays,2)
%first, copy the current unique ray, and repmat it. then subtract
%from raw_rays, and count the number of coordinates of that
%difference which were 0.
a = sum(abs(repmat(produced_rays(:,ii) , [1 size(raw_rays,2)]) - raw_rays) < 1e-4,1);
% this gets those indices of raw_rays (columns of) which were
% exactly the current (ii) produced_ray.
% todo: replace this call with an all() call
ind = find(a==num_vars);
for jj = 1:length(ind)
corresponding_variables{ii} = [corresponding_variables{ii} raw_corresponding_variables{ind(jj)}];
corresponding_coefficients{ii} = [corresponding_coefficients{ii} raw_corresponding_coefficients(ind(jj))];
corresponding_centers{ii} = [corresponding_centers{ii} aux_data.paths(ind(jj)).center_point];
corresponding_paths{ii} = ind; % which paths led to the computation of the iith produced_ray.
end
g = gcd_all(produced_rays(:,ii)); %reduce
produced_rays(:,ii) = produced_rays(:,ii)/g;
ray_multiplicities(ii) = g * sum( a==num_vars );
end
if ~isempty(aux_data.paths)
centers = [aux_data.paths(:).center_point];
for ii = 1:size(intersection_points,2)
if isempty(find_same_point(centers,intersection_points(:,ii), intersection_point_uniqueness_threshold,1))
warning('intersection point %i was not the center of any path',ii);
end
end
end
aux_data.raw_intersection_points = raw_intersection_points;
aux_data.slice_values = slice_values;
aux_data.raw_rays = raw_rays;
aux_data.intersection_points = intersection_points;
aux_data.corresponding_variables = corresponding_variables;
aux_data.corresponding_coefficients = corresponding_coefficients;
aux_data.corresponding_paths = corresponding_paths;
aux_data.corresponding_centers = corresponding_centers;
end
%% change directories into a working directory
function starting_directory = move_into_working_directory(remove_prev_dir)
starting_directory = pwd;
[~, folder_name, ~] = fileparts(starting_directory);
% set up and and move into a temporary directory, so we keep the directory
% relatively clean. otherwise there's an explosion of files.
dirname = sprintf('%s_complex_tropical_decomposition',folder_name);
if remove_prev_dir
dirlist = dir([dirname '*']);
for ii = 1:length(dirlist)
rmdir(dirlist(ii).name,'s')
end
end
if ~isdir(dirname)
mkdir(dirname);
end
cd(dirname);
end
%% set program parameters to defaults, and parse the command line options
function [curve_system_name, cauchy_walk_point_uniqueness_threshold, intersection_point_uniqueness_threshold, intersection_zero_coord_threshold, puiseux_zero_coeff_threshold,min_slice_value,default_slice_value,remove_prev_dir,num_sample_points,user_config] = parse_varargin(varargin)
curve_system_name = 'input';
intersection_point_uniqueness_threshold = 1e-8;
cauchy_walk_point_uniqueness_threshold = 1e-8;
intersection_zero_coord_threshold = 1e-8;
puiseux_zero_coeff_threshold = 1e-10;
min_slice_value = 1e-5;
default_slice_value = 0.1;
remove_prev_dir = false;
num_sample_points = 16; %this number MUST be even
user_config = struct;
option_counter = 0;
%% parse out the options from varargin
while option_counter < length(varargin)
option_counter = option_counter+1;
curr_opt_name = varargin{option_counter};
switch curr_opt_name
case 'input'
option_counter = option_counter+1;
curve_system_name = varargin{option_counter};
case 'options'
option_counter = option_counter+1;
user_config = varargin{option_counter};
if isfield(user_config,'tracktype')
error('do not specify a track type');
end
case 'purge'
remove_prev_dir = true;
case 'defaultslicevalue'
option_counter = option_counter+1;
curve_system_name = varargin{option_counter};
case 'numsamplepoints'
option_counter = option_counter+1;
num_sample_points = varargin{option_counter};
if mod(num_sample_points,2)~=0
error('the number of sample points for monodromy MUST be even');
end
case 'puiseuxthreshold'
option_counter = option_counter+1;
puiseux_zero_coeff_threshold = varargin{option_counter};
if ~isnumeric( puiseux_zero_coeff_threshold )
error('threshold for puiseux coefficients must be a number');
end
if puiseux_zero_coeff_threshold < 0
error('zero-threshold for puiseux coefficients must be greater than 0.')
end
case 'intersectionthreshold'
option_counter = option_counter+1;
intersection_zero_coord_threshold = varargin{option_counter};
if ~isnumeric( intersection_zero_coord_threshold )
error('threshold for intersection coordinates must be a number');
end
if intersection_zero_coord_threshold < 0
error('zero-threshold for intersection coordinates must be greater than 0.')
end
case 'intersectionuniquethreshold'
option_counter = option_counter+1;
intersection_point_uniqueness_threshold = varargin{option_counter};
if ~isnumeric( intersection_point_uniqueness_threshold )
error('uniqueness threshold for intersection points must be a number');
end
if intersection_point_uniqueness_threshold < 0
error('uniqueness threshold for intersection points must be greater than 0.')
end
case 'cauchyuniquethreshold'
option_counter = option_counter+1;
cauchy_walk_point_uniqueness_threshold = varargin{option_counter};
if ~isnumeric( cauchy_walk_point_uniqueness_threshold )
error('uniqueness threshold for points on cauchy loop must be a number');
end
if cauchy_walk_point_uniqueness_threshold < 0
error('uniqueness threshold for points on cauchy loop must be greater than 0.')
end
case 'minslicevalue'
option_counter = option_counter+1;
min_slice_value = varargin{option_counter};
if ~isnumeric( min_slice_value )
error('minimum slice value must be a number');
end
if min_slice_value < 0
error('minimum slice value must be greater than 0.')
end
otherwise
error('bad option name %s',curr_opt_name);
end
end
end