How to use fsolve to solve all the equations?

%For the following equations want to solve all of its solutions, using solve to solve the time is too long, but using fsolve to solve I wrote the following program, but feel that the solution is not quite right, given too many results

function [F] = modulation_equation(X)
a=X(1);
c=X(2);
F(1)= 0.1096622461e0 * a ^ 6 - 0.4223705634e-1 * a ^ 8 + 0.4066962405e-2 * a ^ 10 - 0.1613995833e-2 * c ^ 2 * a ^ 4 + 0.5938649525e-5 * c ^ 4 * a ^ 2 - 0.3163847188e7 * c ^ 2 * a ^ 8 + 0.8689234851e5 * c ^ 4 * a ^ 6 + 0.9705023202e3 * c ^ 6 * a ^ 4 - 0.8469105945e0 * c ^ 2 * a ^ 6 + 0.1203014830e-1 * c ^ 4 * a ^ 4 - 0.4506937871e-4 * c ^ 6 * a ^ 2 + 0.8083313632e1 * c ^ 8 * a ^ 2;
F(2)= -0.6970477491e12 * a ^ 12 + 0.1914345842e11 * c ^ 2 * a ^ 10 + 0.2138367102e9 * c ^ 4 * a ^ 8 + 0.1815447799e7 * c ^ 6 * a ^ 6 - 0.5120907418e3 * c ^ 8 * a ^ 4 + 0.1885489314e1 * a ^ 2 * c ^ 10 - 0.8368566306e4 * c ^ 2 * a ^ 8 - 0.4108305749e5 * c ^ 4 * a ^ 6 + 0.6065237395e3 * c ^ 6 * a ^ 4 + 0.1220234699e5 * c ^ 2 * a ^ 6 - 0.1795926849e3 * c ^ 4 * a ^ 4 + 0.6608059271e0 * c ^ 6 * a ^ 2 - 0.2232435902e1 * c ^ 8 * a ^ 2;
end
clear
x_range = -1:0.01:1;    % 定义 a 的初值范围
y_range = -1:0.05:1;  % 定义 c 的初值范围
sol = [];  % 用于存储唯一解的矩阵
options = optimoptions('fsolve', 'Display', 'off');  % 设置 fsolve 参数，不显示输出
% 遍历区间中的初值组合
for j = x_range
    for i = y_range
        x0 = [j, i];  % 当前初始点
        try
            % 尝试使用 fsolve 求解
            [X_sol, fval, exitflag] = fsolve(@modulation_equation, x0, options);
            if exitflag > 0  % 仅当成功求解时处理
                % 检查新解是否已存在于 sol 中（判断相似解）
                if isempty(sol) || all(vecnorm(sol - X_sol, 2, 2) > 1e-2)
                    sol = [sol; X_sol];  % 添加唯一解 (a, c) 作为一行
                end
            end
        catch
            % 如果 fsolve 失败，跳过该点
            continue
        end
    end
end
sol = unique(sol, 'rows', 'stable');
sol1= sol(sol(:, 1) ~= 0 & sol(:, 2) ~= 0, :);
sol2=sqrt(sol1(:, 1).^2 + sol1(:, 2).^2);
sol3=sort(unique(sol2,'rows', 'stable'))

[Alex Sha]

some thing wrong? the exact solution is: a=0, c=0

Answer 1

[Naga]

To improve fsolve results, adjust the initial guess range (x_range and y_range) and step size to focus on promising regions. Modify tolerance settings using optimoptions to filter closely spaced solutions. After obtaining solutions, refine them with an appropriate distance threshold, adjusting from 1e-2 as needed for distinctness. Please go through the updated code for your reference:

function [F] = modulation_equation(X)
    a = X(1);
    c = X(2);
    F(1) = 0.1096622461 * a^6 - 0.04223705634 * a^8 + 0.004066962405 * a^10 ...
           - 0.001613995833 * c^2 * a^4 + 0.00005938649525 * c^4 * a^2 ...
           - 31638471.88 * c^2 * a^8 + 868923.4851 * c^4 * a^6 ...
           + 970.5023202 * c^6 * a^4 - 0.8469105945 * c^2 * a^6 ...
           + 0.0120301483 * c^4 * a^4 - 0.00004506937871 * c^6 * a^2 ...
           + 80.83313632 * c^8 * a^2;
    F(2) = -0.6970477491e12 * a^12 + 0.1914345842e11 * c^2 * a^10 ...
           + 0.2138367102e9 * c^4 * a^8 + 0.1815447799e7 * c^6 * a^6 ...
           - 5120.907418 * c^8 * a^4 + 1.885489314 * a^2 * c^10 ...
           - 83685.66306 * c^2 * a^8 - 410830.5749 * c^4 * a^6 ...
           + 6065.237395 * c^6 * a^4 + 122023.4699 * c^2 * a^6 ...
           - 1795.926849 * c^4 * a^4 + 0.6608059271 * c^6 * a^2 ...
           - 2.232435902 * c^8 * a^2;
end

clear
x_range = -1:0.05:1;    % Adjusted step size for a
y_range = -1:0.1:1;     % Adjusted step size for c
sol = [];  % To store unique solutions
options = optimoptions('fsolve', 'Display', 'off', 'TolFun', 1e-6, 'TolX', 1e-6);

% Iterate over initial guesses
for j = x_range
    for i = y_range
        x0 = [j, i];
        try
            [X_sol, ~, exitflag] = fsolve(@modulation_equation, x0, options);
            if exitflag > 0
                if isempty(sol) || all(vecnorm(sol - X_sol, 2, 2) > 1e-2)
                    sol = [sol; X_sol];
                end
            end
        catch
            continue
        end
    end
end

% Ensure unique solutions
sol = unique(sol, 'rows', 'stable');
sol1 = sol(sol(:, 1) ~= 0 & sol(:, 2) ~= 0, :);
sol2 = sqrt(sol1(:, 1).^2 + sol1(:, 2).^2);
sol3 = sort(unique(sol2));

Answer 2

[Matt J]

given too many results

Perhaps you should use uniquetol, rather than unique.

[fayu]

Thank you for your suggestion, but using uniquetol command still can not solve

[Matt J]

We don't know what that means. You haven't shown your attempt to use uniquetol() and the outcome.