vpasolve finds no solutions except the null solution

I have the following code to obtain the solutions of a system of 10 nonlinear equations (eqn1-eqn10) with 10 variables (x1-x10).

x = (0,0,...,0) and x = (1,1,1,1,1,0,0,0,0,0) are solutions of this system. I believe that this system has other solutions, that I want to obtain, but the code is not working and I don't understand why. In fact, with range1 the code gives the null solution as output, But with range2, it gives empty. Shouldn't the code give the solution x = (1,1,1,1,1,0,0,0,0,0)?

Other question I have is if I need to add the equations x1 + x2 + x3 + x4 + x5 = 0 and x6 + x7 + x8 + x9 + x10 = 0, the code gives an error. Can I not have more equations than variables? How do I deal with this situation?

syms x1 x2 x3 x4 x5 x6 x7 x8 x9 x10  
A1 = (x1 - x2)^2 + (x6 - x7)^2 - (x2 - x3)^2 - (x7 - x8)^2;
A2 = (x1 - x2)^2 + (x6 - x7)^2 - (x3 - x4)^2 - (x8 - x9)^2;
A3 = (x1 - x2)^2 + (x6 - x7)^2 - (x4 - x5)^2 - (x9 - x10)^2;
A4 = (x1 - x2)^2 + (x6 - x7)^2 - (x1 - x5)^2 - (x6 - x10)^2;
D1 = (x1 - x3)^2 + (x6 - x8)^2 - (x1 - x4)^2 - (x6 - x9)^2;
D2 = (x1 - x3)^2 + (x6 - x8)^2 - (x2 - x4)^2 - (x7 - x9)^2;
D3 = (x1 - x3)^2 + (x6 - x8)^2 - (x3 - x5)^2 - (x8 - x10)^2;
D4 = (x1 - x3)^2 + (x6 - x8)^2 - (x5 - x2)^2 - (x10 - x7)^2;
eqn1 = (D1 + D2 + D3 + D4) * (x1 - x3) + (4 * D1 - D2 - D3 - ... 
    D4) * (x4 - x1) + (A1 + A2 + A3 + A4) * (x1 - x2) + (4 * A4 - A1 - ... 
    A2 - A3) * (x5 - x1) == 0;
eqn2 = (4 * D2 - D1 - D3 - D4) * (x4 - x2) + (4 * D4 - D1 - D2 - ... 
    D3) * (x5 - x2) + (4 * A1 - A2 - A3 - A4) * (x3 - x2) == 0;
eqn3 = (D1 + D2 + D3 + D4) * (x3 - x1) + (4 * D3 - D1 - D2 - ... 
    D4) * (x5 - x3) + (4 * A1 - A2 - A3 - A4) * (x2 - x3) + (4 * A2 - A1 - ...
    A3 - A4) * (x4 - x3) == 0;
eqn4 = (4 * D1 - D2 - D3 - D4) * (x1 - x4) + (4 * D2 - D1 - D3 - ...
    D4) * (x2 - x4) + (4 * A3 - A1 - A2 - A4) * (x5 - x4) == 0;
eqn5 = (4 * D4 - D1 - D2 - D3) * (x2 - x5) + (4 * D3 - D1 - D2 - ... 
    D4) * (x3 - x5) + (4 * A3 - A1 - A2 - A4) * (x4 - x5) + (4 * A4 - A1 - ... 
    A2 - A3) * (x1 - x5) == 0;
eqn6 = (D1 + D2 + D3 + D4) * (x6 - x8) + (4 * D1 - D2 - D3 - D4) * (x9 - x6) + ...
    (A1 + A2 + A3 + A4) * (x6 - x7) + (4 * A4 - A1 -A2 - A3) * (x10 - x6) == 0;
eqn7 = (4 * D2 - D1 - D3 - D4) * (x9 - x7) + (4 * D4 - D1 - D2 - ...
    D3) * (x10 - x7) + (4 * A1 - A2 - A3 - A4) * (x8 - x7) == 0;
eqn8 = (D1 + D2 + D3 + D4) * (x8 - x6) + (4 * D3 - D1 - D2 - ... 
    D4) * (x10 - x8) + (4 * A1 - A2 - A3 - A4) * (x7 - x8) + (4 * A2 - A1 - ... 
    A3 - A4) * (x9 - x8) == 0;
eqn9 = (4 * D1 - D2 - D3 - D4) * (x6 - x9) + (4 * D2 - D1 - D3 - ...
    D4) * (x7 - x9) + (4 * A3 - A1 - A2 - A4) * (x10 - x9) == 0;
eqn10 = (4 * D4 - D1 - D2 - D3) * (x7 - x10) + (4 * D3 - D1 - D2 - ...
    D4) * (x8 - x10) + (4 * A3 - A1 - A2 - A4) * (x9 - x10) + (4 * A4 - A1 - ...
    A2 - A3) * (x6 - x10) == 0;
equations = [eqn1 eqn2 eqn3 eqn4 eqn5 eqn6 eqn7 eqn8 eqn9 eqn10];
vars = [x1 x2 x3 x4 x5 x6 x7 x8 x9 x10];
range1 = [-Inf Inf; -Inf Inf; -Inf Inf; -Inf Inf; -Inf Inf; -Inf Inf; -Inf Inf; -Inf Inf; -Inf Inf; -Inf Inf];
range2 = [0.5 1.5; 0.5 1.5; 0.5 1.5; 0.5 1.5; 0.5 1.5; -0.5 0.5; -0.5 0.5; -0.5 0.5; -0.5 0.5; -0.5 0.5];
Y = vpasolve(equations, vars, range1);
[Y.x1 Y.x2 Y.x3 Y.x4 Y.x5 Y.x6 Y.x7 Y.x8 Y.x9 Y.x10]
Y = vpasolve(equations, vars, range2);
[Y.x1 Y.x2 Y.x3 Y.x4 Y.x5 Y.x6 Y.x7 Y.x8 Y.x9 Y.x10]

[Matt J]

I believe that this system has other solutions, that I want to obtain,

Well, you can't get them all numerically. There is an infinite continuum of solutions, i.e. too many to write down. Any x of the form,

x=(a,a,a,a,a,b,b,b,b,b)

is a solution (and there may be others).

Answer 1

[Matt J]

With numerical algorithms like vpasolve, you never can tell what it will find, if anything. In any case, lsqnonlin seems to offer better performance, assuming you have the Optimizatio Toolbox. It finds a solution for range2 readily with and without the additional equations:

syms x [10,1]
A1 = (x1 - x2)^2 + (x6 - x7)^2 - (x2 - x3)^2 - (x7 - x8)^2;
A2 = (x1 - x2)^2 + (x6 - x7)^2 - (x3 - x4)^2 - (x8 - x9)^2;
A3 = (x1 - x2)^2 + (x6 - x7)^2 - (x4 - x5)^2 - (x9 - x10)^2;
A4 = (x1 - x2)^2 + (x6 - x7)^2 - (x1 - x5)^2 - (x6 - x10)^2;
D1 = (x1 - x3)^2 + (x6 - x8)^2 - (x1 - x4)^2 - (x6 - x9)^2;
D2 = (x1 - x3)^2 + (x6 - x8)^2 - (x2 - x4)^2 - (x7 - x9)^2;
D3 = (x1 - x3)^2 + (x6 - x8)^2 - (x3 - x5)^2 - (x8 - x10)^2;
D4 = (x1 - x3)^2 + (x6 - x8)^2 - (x5 - x2)^2 - (x10 - x7)^2;
eqn1 = (D1 + D2 + D3 + D4) * (x1 - x3) + (4 * D1 - D2 - D3 - ... 
    D4) * (x4 - x1) + (A1 + A2 + A3 + A4) * (x1 - x2) + (4 * A4 - A1 - ... 
    A2 - A3) * (x5 - x1) == 0;
eqn2 = (4 * D2 - D1 - D3 - D4) * (x4 - x2) + (4 * D4 - D1 - D2 - ... 
    D3) * (x5 - x2) + (4 * A1 - A2 - A3 - A4) * (x3 - x2) == 0;
eqn3 = (D1 + D2 + D3 + D4) * (x3 - x1) + (4 * D3 - D1 - D2 - ... 
    D4) * (x5 - x3) + (4 * A1 - A2 - A3 - A4) * (x2 - x3) + (4 * A2 - A1 - ...
    A3 - A4) * (x4 - x3) == 0;
eqn4 = (4 * D1 - D2 - D3 - D4) * (x1 - x4) + (4 * D2 - D1 - D3 - ...
    D4) * (x2 - x4) + (4 * A3 - A1 - A2 - A4) * (x5 - x4) == 0;
eqn5 = (4 * D4 - D1 - D2 - D3) * (x2 - x5) + (4 * D3 - D1 - D2 - ... 
    D4) * (x3 - x5) + (4 * A3 - A1 - A2 - A4) * (x4 - x5) + (4 * A4 - A1 - ... 
    A2 - A3) * (x1 - x5) == 0;
eqn6 = (D1 + D2 + D3 + D4) * (x6 - x8) + (4 * D1 - D2 - D3 - D4) * (x9 - x6) + ...
    (A1 + A2 + A3 + A4) * (x6 - x7) + (4 * A4 - A1 -A2 - A3) * (x10 - x6) == 0;
eqn7 = (4 * D2 - D1 - D3 - D4) * (x9 - x7) + (4 * D4 - D1 - D2 - ...
    D3) * (x10 - x7) + (4 * A1 - A2 - A3 - A4) * (x8 - x7) == 0;
eqn8 = (D1 + D2 + D3 + D4) * (x8 - x6) + (4 * D3 - D1 - D2 - ... 
    D4) * (x10 - x8) + (4 * A1 - A2 - A3 - A4) * (x7 - x8) + (4 * A2 - A1 - ... 
    A3 - A4) * (x9 - x8) == 0;
eqn9 = (4 * D1 - D2 - D3 - D4) * (x6 - x9) + (4 * D2 - D1 - D3 - ...
    D4) * (x7 - x9) + (4 * A3 - A1 - A2 - A4) * (x10 - x9) == 0;
eqn10 = (4 * D4 - D1 - D2 - D3) * (x7 - x10) + (4 * D3 - D1 - D2 - ...
    D4) * (x8 - x10) + (4 * A3 - A1 - A2 - A4) * (x9 - x10) + (4 * A4 - A1 - ...
    A2 - A3) * (x6 - x10) == 0;
equations = [eqn1 eqn2 eqn3 eqn4 eqn5 eqn6 eqn7 eqn8 eqn9 eqn10];


range2 = [0.5 1.5; 0.5 1.5; 0.5 1.5; 0.5 1.5; 0.5 1.5; -0.5 0.5; -0.5 0.5; -0.5 0.5; -0.5 0.5; -0.5 0.5];

equations= lhs(equations)-rhs(equations);

F=matlabFunction(equations, File="eqnFunc.m", Vars={x});

xInitial=mean(range2,2) + randn(10,1);
opts=optimoptions('lsqnonlin', 'ConstraintTol', 1e-16,'FunctionTol',1e-16,...
     'StepTol',1e-16,'OptimalityTol',1e-16);

[xnum,fval]=lsqnonlin(F,xInitial, range2(:,1), range2(:,2),opts)

Local minimum possible.

lsqnonlin stopped because the final change in the sum of squares relative to 
its initial value is less than the value of the function tolerance.

xnum = 10×1

0.9196 0.9198 0.9197 0.9191 0.9192 0.0088 0.0088 0.0088 0.0088 0.0088

<mw-icon class=""></mw-icon>

<mw-icon class=""></mw-icon>

fval = 2.2809e-18

Can I not have more equations than variables? How do I deal with this situation?

You can, but of course you cannot be guaranteed of an exact solution,

equations=[equations, (x1 + x2 + x3 + x4 + x5) , (x6 + x7 + x8 + x9 + x10)];
F=matlabFunction(equations, File="eqnFunc.m", Vars={x});

[xnum,fval]=lsqnonlin(F, xInitial, range2(:,1), range2(:,2), opts)

Local minimum possible.
lsqnonlin stopped because the size of the current step is less than
the value of the step size tolerance.

xnum = 10×1

0.5000 0.5000 0.5000 0.5000 0.5000 -0.0000 0.0000 -0.0000 0.0000 -0.0000

<mw-icon class=""></mw-icon>

<mw-icon class=""></mw-icon>

fval = 6.2500

[Catarina Pina]

But this last solution obtained by lsqnonlin doesn't satisfy the equation x1 + x2 + x3 + x4 + x5 = 0. How to obtain solutions that satisfy all the equations involved?

[Walter Roberson]

x = (1,1,1,1,1,0,0,0,0,0) does not satisfy that constraint either.

[Matt J]

@Catarina Pina From above "You can, but of course you cannot be guaranteed of an exact solution,"

[Matt J]

From my comment above, it might be that all solutions of the first 10 equations are of the form,

x=(a,a,a,a,a,b,b,b,b,b)

which means the only solution that can solve all 12, would have to be with a=b=0, i.e., the null solution,

x = (0,0,...,0)

[Catarina Pina]

Yes, correct, that one is a solution without considering the constraints x1 + x2 + x3 + x4 + x5 = 0 and x6 + x7 + x8 + x9 + x10 = 0, i.e., the first output gives a solution that satisfy all the equations involved (10). But considering the additional constraints we get a solution that doesn't satisfy all the equations involved (12). In this last case, how to obtain solutions?

[Catarina Pina]

x = (0,0,...,0) is not the only solution that can solve all 12 equations, since we have, for instance, the following solution, that is not of the form x=(a,a,a,a,a,b,b,b,b,b).

x1 = 0
X2 =  (5/2 - 5^(1/2)/2)^(1/2)*(5^(1/2)/4 + 1/4)
X3 = (5/2 - 5^(1/2)/2)^(1/2)/2
X4 = -(5/2 - 5^(1/2)/2)^(1/2)/2
x6 = 1
x5 = -((5/2 - 5^(1/2)/2)^(1/2)*(5^(1/2)/4 + 1/4))-( (5/2 - 5^(1/2)/2)^(1/2)/2)-(-(5/2 - 5^(1/2)/2)^(1/2)/2)
X7 =  5^(1/2)/4 - 1/4
X8 =  - 5^(1/2)/4 - 1/4
X9 = - 5^(1/2)/4 - 1/4
x10 = -(- 5^(1/2)/4 - 1/4)-( - 5^(1/2)/4 - 1/4)-( 5^(1/2)/4 - 1/4)-1

This solution was obtained particularizing x1 = 0 and x6 = 1 and using the function solve. My question is still how to obtain solutions of the system of 12 equations using a matlab function (without assigning specific values to variables).

[Matt J]

This solution was obtained particularizing x1 = 0 and x6 = 1 and using the function solve. My question is still how to obtain solutions of the system of 12 equations using a matlab function (without assigning specific values to variables).

What makes you think a finite set of solutions exists? The code below found no less than 190 solutions, just by randomly assigning fixed values to x1 and x6 in a loop.

syms x [10,1]
A1 = (x1 - x2)^2 + (x6 - x7)^2 - (x2 - x3)^2 - (x7 - x8)^2;
A2 = (x1 - x2)^2 + (x6 - x7)^2 - (x3 - x4)^2 - (x8 - x9)^2;
A3 = (x1 - x2)^2 + (x6 - x7)^2 - (x4 - x5)^2 - (x9 - x10)^2;
A4 = (x1 - x2)^2 + (x6 - x7)^2 - (x1 - x5)^2 - (x6 - x10)^2;
D1 = (x1 - x3)^2 + (x6 - x8)^2 - (x1 - x4)^2 - (x6 - x9)^2;
D2 = (x1 - x3)^2 + (x6 - x8)^2 - (x2 - x4)^2 - (x7 - x9)^2;
D3 = (x1 - x3)^2 + (x6 - x8)^2 - (x3 - x5)^2 - (x8 - x10)^2;
D4 = (x1 - x3)^2 + (x6 - x8)^2 - (x5 - x2)^2 - (x10 - x7)^2;
eqn1 = (D1 + D2 + D3 + D4) * (x1 - x3) + (4 * D1 - D2 - D3 - ... 
    D4) * (x4 - x1) + (A1 + A2 + A3 + A4) * (x1 - x2) + (4 * A4 - A1 - ... 
    A2 - A3) * (x5 - x1) == 0;
eqn2 = (4 * D2 - D1 - D3 - D4) * (x4 - x2) + (4 * D4 - D1 - D2 - ... 
    D3) * (x5 - x2) + (4 * A1 - A2 - A3 - A4) * (x3 - x2) == 0;
eqn3 = (D1 + D2 + D3 + D4) * (x3 - x1) + (4 * D3 - D1 - D2 - ... 
    D4) * (x5 - x3) + (4 * A1 - A2 - A3 - A4) * (x2 - x3) + (4 * A2 - A1 - ...
    A3 - A4) * (x4 - x3) == 0;
eqn4 = (4 * D1 - D2 - D3 - D4) * (x1 - x4) + (4 * D2 - D1 - D3 - ...
    D4) * (x2 - x4) + (4 * A3 - A1 - A2 - A4) * (x5 - x4) == 0;
eqn5 = (4 * D4 - D1 - D2 - D3) * (x2 - x5) + (4 * D3 - D1 - D2 - ... 
    D4) * (x3 - x5) + (4 * A3 - A1 - A2 - A4) * (x4 - x5) + (4 * A4 - A1 - ... 
    A2 - A3) * (x1 - x5) == 0;
eqn6 = (D1 + D2 + D3 + D4) * (x6 - x8) + (4 * D1 - D2 - D3 - D4) * (x9 - x6) + ...
    (A1 + A2 + A3 + A4) * (x6 - x7) + (4 * A4 - A1 -A2 - A3) * (x10 - x6) == 0;
eqn7 = (4 * D2 - D1 - D3 - D4) * (x9 - x7) + (4 * D4 - D1 - D2 - ...
    D3) * (x10 - x7) + (4 * A1 - A2 - A3 - A4) * (x8 - x7) == 0;
eqn8 = (D1 + D2 + D3 + D4) * (x8 - x6) + (4 * D3 - D1 - D2 - ... 
    D4) * (x10 - x8) + (4 * A1 - A2 - A3 - A4) * (x7 - x8) + (4 * A2 - A1 - ... 
    A3 - A4) * (x9 - x8) == 0;
eqn9 = (4 * D1 - D2 - D3 - D4) * (x6 - x9) + (4 * D2 - D1 - D3 - ...
    D4) * (x7 - x9) + (4 * A3 - A1 - A2 - A4) * (x10 - x9) == 0;
eqn10 = (4 * D4 - D1 - D2 - D3) * (x7 - x10) + (4 * D3 - D1 - D2 - ...
    D4) * (x8 - x10) + (4 * A3 - A1 - A2 - A4) * (x9 - x10) + (4 * A4 - A1 - ...
    A2 - A3) * (x6 - x10) == 0;
eqn11 = x1 + x2 + x3 + x4 + x5==0;
eqn12 = x6 + x7 + x8 + x9 + x10==0;
equations = [eqn1 eqn2 eqn3 eqn4 eqn5 eqn6 eqn7 eqn8 eqn9 eqn10 eqn11 eqn12];


equations= lhs(equations)-rhs(equations);
F=matlabFunction(equations, File="eqnFunc.m", Vars={x});

opts=optimoptions('lsqnonlin', 'ConstraintTol', 1e-16,'FunctionTol',1e-16,...
     'StepTol',1e-16,'OptimalityTol',1e-16,'Display','off');


X1=(-100:100);
X6=X1(randperm(end));
for i=1:200
    
 lb=-inf(10,1); ub=-lb;
 lb(1)=X1(i); ub(1)=lb(1);
 lb(6)=X6(i); ub(6)=lb(6);

 xInitial=zeros(10,1);
 [xnum,fval(i)]=lsqnonlin(F, xInitial , lb,ub, opts);
 
end

numSolutions = sum(abs(fval)<=1e-10)

numSolutions = 191

[Catarina Pina]

I realize there are an infinite number of solutions, I just wanted to find some of them. With this code I can do what I need, thanks!

[Matt J]

You are quite welcome, but if this answer addresses your question, please Accept-click it.

Answer 2

[Paul]

This code exceeds the run time limit here, and is taking forever running on my computer, so don't know what it produces.

If solve produced a solution when specifying two of the variables, then try using it with ReturnConditions = true to get a parameterization of all solutions.

Might also want to assume all of the xi are real if that is, in fact, the case. Might narrow down the solution space.

syms x1 x2 x3 x4 x5 x6 x7 x8 x9 x10  
A1 = (x1 - x2)^2 + (x6 - x7)^2 - (x2 - x3)^2 - (x7 - x8)^2;
A2 = (x1 - x2)^2 + (x6 - x7)^2 - (x3 - x4)^2 - (x8 - x9)^2;
A3 = (x1 - x2)^2 + (x6 - x7)^2 - (x4 - x5)^2 - (x9 - x10)^2;
A4 = (x1 - x2)^2 + (x6 - x7)^2 - (x1 - x5)^2 - (x6 - x10)^2;
D1 = (x1 - x3)^2 + (x6 - x8)^2 - (x1 - x4)^2 - (x6 - x9)^2;
D2 = (x1 - x3)^2 + (x6 - x8)^2 - (x2 - x4)^2 - (x7 - x9)^2;
D3 = (x1 - x3)^2 + (x6 - x8)^2 - (x3 - x5)^2 - (x8 - x10)^2;
D4 = (x1 - x3)^2 + (x6 - x8)^2 - (x5 - x2)^2 - (x10 - x7)^2;
eqn1 = (D1 + D2 + D3 + D4) * (x1 - x3) + (4 * D1 - D2 - D3 - ... 
    D4) * (x4 - x1) + (A1 + A2 + A3 + A4) * (x1 - x2) + (4 * A4 - A1 - ... 
    A2 - A3) * (x5 - x1) == 0;
eqn2 = (4 * D2 - D1 - D3 - D4) * (x4 - x2) + (4 * D4 - D1 - D2 - ... 
    D3) * (x5 - x2) + (4 * A1 - A2 - A3 - A4) * (x3 - x2) == 0;
eqn3 = (D1 + D2 + D3 + D4) * (x3 - x1) + (4 * D3 - D1 - D2 - ... 
    D4) * (x5 - x3) + (4 * A1 - A2 - A3 - A4) * (x2 - x3) + (4 * A2 - A1 - ...
    A3 - A4) * (x4 - x3) == 0;
eqn4 = (4 * D1 - D2 - D3 - D4) * (x1 - x4) + (4 * D2 - D1 - D3 - ...
    D4) * (x2 - x4) + (4 * A3 - A1 - A2 - A4) * (x5 - x4) == 0;
eqn5 = (4 * D4 - D1 - D2 - D3) * (x2 - x5) + (4 * D3 - D1 - D2 - ... 
    D4) * (x3 - x5) + (4 * A3 - A1 - A2 - A4) * (x4 - x5) + (4 * A4 - A1 - ... 
    A2 - A3) * (x1 - x5) == 0;
eqn6 = (D1 + D2 + D3 + D4) * (x6 - x8) + (4 * D1 - D2 - D3 - D4) * (x9 - x6) + ...
    (A1 + A2 + A3 + A4) * (x6 - x7) + (4 * A4 - A1 -A2 - A3) * (x10 - x6) == 0;
eqn7 = (4 * D2 - D1 - D3 - D4) * (x9 - x7) + (4 * D4 - D1 - D2 - ...
    D3) * (x10 - x7) + (4 * A1 - A2 - A3 - A4) * (x8 - x7) == 0;
eqn8 = (D1 + D2 + D3 + D4) * (x8 - x6) + (4 * D3 - D1 - D2 - ... 
    D4) * (x10 - x8) + (4 * A1 - A2 - A3 - A4) * (x7 - x8) + (4 * A2 - A1 - ... 
    A3 - A4) * (x9 - x8) == 0;
eqn9 = (4 * D1 - D2 - D3 - D4) * (x6 - x9) + (4 * D2 - D1 - D3 - ...
    D4) * (x7 - x9) + (4 * A3 - A1 - A2 - A4) * (x10 - x9) == 0;
eqn10 = (4 * D4 - D1 - D2 - D3) * (x7 - x10) + (4 * D3 - D1 - D2 - ...
    D4) * (x8 - x10) + (4 * A3 - A1 - A2 - A4) * (x9 - x10) + (4 * A4 - A1 - ...
    A2 - A3) * (x6 - x10) == 0;
equations = [eqn1 eqn2 eqn3 eqn4 eqn5 eqn6 eqn7 eqn8 eqn9 eqn10];
vars = [x1 x2 x3 x4 x5 x6 x7 x8 x9 x10];

Y10  = solve(equations,vars,'ReturnConditions',true)

eqn11 = x1 + x2 + x3 + x4 + x5 == 0;
eqn12 = x6 + x7 + x8 + x9 + x10 == 0;
Y12 = solve([equations,eqn11,eqn12],vars,'ReturnConditions',true)

Answer 3

[Alex Sha]

For:

range2 = [0.5 1.5; 0.5 1.5; 0.5 1.5; 0.5 1.5; 0.5 1.5; -0.5 0.5; -0.5 0.5; -0.5 0.5; -0.5 0.5; -0.5 0.5];

result-1:

x1: 1.08082083137654
x2: 1.08080951998694
x3: 1.08081167383851
x4: 1.0808049613845
x5: 1.0807961790416
x6: 0.139381627943366
x7: 0.139378007206469
x8: 0.139387284204371
x9: 0.13939708413668
x10: 0.139379799060729

feval:
4.18856104328548E-14
-3.48053189694011E-15
1.9850994632769E-15
-1.31438771899855E-14
-3.90425935929704E-14
-9.46492326396661E-15
7.3761149429896E-16
-6.93623770817858E-15
3.28876546733261E-14
-1.31813703670225E-14

result-2:

x1: 1.02958234918508
x2: 1.02959012135357
x3: 1.0295940618282
x4: 1.02959069634112
x5: 1.02958674833031
x6: -0.0307085239327012
x7: -0.0306995429173505
x8: -0.0306983806107563
x9: -0.0307242623557914
x10: -0.0307004373970983

feval:
8.54065517352478E-15
5.87827032304209E-16
-6.61360139369969E-15
9.51411238197795E-15
5.9308056815423E-16
2.18297090864782E-14
4.55445556176149E-14
4.51489361938118E-14
-8.39926540201438E-14
2.78159953330793E-14

result-3:

x1: 1.11722991939913
x2: 1.11722093473141
x3: 1.11721577030959
x4: 1.11723361423116
x5: 1.11722092000214
x6: 0.350956747942537
x7: 0.350948443506383
x8: 0.350952713845785
x9: 0.350953745554248
x10: 0.350945892821742

feval:
8.43258754382745E-15
-9.218608646947E-15
-1.5533394955795E-14
6.34621251709236E-15
-3.51572661481046E-15
2.14134991818329E-16
-4.90905368161458E-16
-7.00918017460192E-15
4.90030773194566E-15
1.47945732346218E-16

...

There are infinite results.