MATLAB Examples

# A point on each connected component

We wish to find the decomposition of the real solutions of the sextic plane curve:

$x_2^2 + x_1^2(x_1-1)^3(x_1-2).$

We begin by solving the Fritz John homotopy (see Section 14.4):

polysyms x1 x2 l0 l1 y1 = 0.5; y2 = 0.8; eps = 1.1 + 0.7i; a0 = -1.3; a1 = 1.7; F = x2^2 + x1^2*(x1-1)^3*(x1-2) - eps; G1 = l0*(x1-y1) + l1*x1*(x1-1)^2*(6*x1^2-13*x1+4); G2 = l0*(x2-y2) + 2*l1*x2; patch = a0*l0 + a1*l1 - 1; poly_system = BertiniLab('variable_group',{[x1 x2],[l0 l1]}, ... 'function_def',[F; G1; G2; patch],'config',struct('SecurityLevel',1)); poly_system = solve(poly_system); results = poly_system.solve_summary; istart = strfind(results,'NOTE: nonsingular'); iend = strfind(results,'The following files'); disp(results(istart:iend-1)) 
NOTE: nonsingular vs singular is based on condition number and identical endpoints | Number of real solns | Number of non-real solns | Total ------------------------------------------------------------------------------------------ Non-singular | 0 | 16 | 16 Singular | 0 | 0 | 0 ------------------------------------------------------------------------------------------ Total | 0 | 16 | 16 Finite Multiplicity Summary Multiplicity | Number of real solns | Number of non-real solns ------------------------------------------------------------------------------------------ 1 | 0 | 16 Infinite Solution Summary NOTE: nonsingular vs singular is based on condition number and identical endpoints | Number of real solns | Number of non-real solns | Total ------------------------------------------------------------------------------------------ Non-singular | 0 | 0 | 0 Singular | 0 | 1 | 1 ------------------------------------------------------------------------------------------ Total | 0 | 1 | 1 Infinite Multiplicity Summary Multiplicity | Number of real solns | Number of non-real solns ------------------------------------------------------------------------------------------ 20 | 0 | 1 ------------------------------------------------------------------------------------------ 

There should be 16 finite solutions. We use these as the starting point for a user homotopy:

polysyms t T F = x2^2 + x1^2*(x1-1)^3*(x1-2) - t*eps; poly_system.function_def = [F; G1; G2; patch]; poly_system.config.UserHomotopy = 2; poly_system.pathvariable = T; poly_system.parameter = [t T]; poly_system.starting_points = poly_system.read_solutions('finite_solutions'); poly_system = solve(poly_system); results = poly_system.solve_summary; istart = strfind(results,'NOTE: nonsingular'); iend = strfind(results,'The following files'); disp(results(istart:iend-1)) 
NOTE: nonsingular vs singular is based on condition number and identical endpoints | Number of real solns | Number of non-real solns | Total ------------------------------------------------------------------------------------------ Non-singular | 1 | 10 | 11 Singular | 2 | 0 | 2 ------------------------------------------------------------------------------------------ Total | 3 | 10 | 13 Finite Multiplicity Summary Multiplicity | Number of real solns | Number of non-real solns ------------------------------------------------------------------------------------------ 1 | 1 | 10 2 | 1 | 0 3 | 1 | 0 ------------------------------------------------------------------------------------------ 

There are three real roots, one each with multiplicity 1, 2 and 3:

sols = poly_system.match_solutions('real_finite_solutions'); real(double([sols.x])) 
ans = 0.0000 0.0000 1.0000 1.0000 1.0000 1.8796 0.0000 0.0000 0.0000 0.0000 0.0000 -0.5380