Solution of nonlinear equation

Hello everyone!

When solving the following equations, I defined variable parameters by syms, but it implied that wasn't correct. I don't know how to do, could you help me solve it? Thank you very much!

Best regards

clc; clear 
%Solve nonlinear equations 
syms a(-6) a(-5) a(-4) a(-3) a(-2) a(-1) a(0) a(1) a(2) a(3) a(4) a(5) a(6) a(7)
[Sa(-6),Sa(-5),Sa(-4),Sa(-3),Sa(-2),Sa(-1),Sa(-0),Sa(1),Sa(2),Sa(3),Sa(4),Sa(5),Sa(6),Sa(7)] ...
 =solve(e(-3)==0,e(-2)==0,e(-1)==0,e(0)==0,e(1)==0,e(2)==0,a(-6)==a(7),a(-5)==a(6), ...
 a(-4)==a(5),a(-3)==a(4),a(-2)==a(3),a(-1)==a(2),a(0)==a(1))


function y=di(x)
%  Define dirichlet function
y=0.*(x~=0)+1.*(x==0);
end


function F=e(n)
F=9*symsum(a(s),s,-6,7)*symsum(a(t),t,-6,7)*di(9*n+2+3*+s)-di(n);
end

Answer 1

[Torsten]

format long
a0 = rand(1,7);
%a0 = [5.3916,0.0342,1.789,0.619,-0.0104,0.2155,0.0575];
options = optimset('MaxFunEvals',1000000,'MaxIter',1000000,'TolFun',1e-14,'TolX',1e-14);
%a = fsolve(@fun,a0,options)
a = lsqnonlin(@fun,a0,[],[],options)

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.

a = 1×7

0.197493709597352 0.281302916777374 0.140651449715022 -0.140651455082789 0.000000000021439 0.140651453041311 -0.140651455260134

norm(fun(a))

ans = 
     3.040264027305917e-09

function res = fun(a)
  A = [fliplr(a),a];
  res = zeros(1,7);
  for n = -3:3
    sum_j = 0.0;
    for j = -6:7
      aj = A(j+7);
      sum_k = 0.0;
      for k = -6:7
        ak = A(k+7);
        deltak = double(9*n+2+3*k+j==0);
        sum_k = sum_k + ak*deltak; 
      end
      sum_j = sum_j + aj*sum_k;
    end
    res(n+4) = 9*sum_j - double(n==0);
  end
end

[lei]

Hi Torsten,

Thank you very much! It does help for me. And I have some questions.

1) As the result is different after running, whether it implies that there are several solutions about the nonlinear equations.

2) Is there some appoach that give the relation between variable parameters in the above nonlinear equation?

Thank you again.

Best regards

[Torsten]

As the result is different after running, whether it implies that there are several solutions about the nonlinear equations.

The code gives different "solutions" depending on the initial guess for the variables. I cannot decide whether they are really roots of the system.

Is there some appoach that give the relation between variable parameters in the above nonlinear equation?

I don't know what you mean.

[lei]

Because the difference of "solutions", I think maybe there is a connection between the "solutions". Now I think I misunderstand it. Thank you a lot. And have a nice weekend!

[Torsten]

Because the difference of "solutions", I think maybe there is a connection between the "solutions".

Maybe. But you have a complicated system of 7 equations quadratic in the unknowns. I think it will be impossible to analyze its solutions on a theoretical basis.

[Torsten]

Here is a solution of your system:

syms sum_j sum_k
a = sym('a',[1 7]);
res = sym('res',[1 7]);
A = [fliplr(a),a];
for n = -3:3
    sum_j = 0;
    for j = -6:7
        aj = A(j+7);
        sum_k = 0;
        for k = -6:7
            ak = A(k+7);
            deltak = (9*n+2+3*k+j==0);
            sum_k = sum_k + ak*deltak; 
        end
        sum_j = sum_j + aj*sum_k;
    end
    res(n+4) = 9*sum_j - (n==0);
end
res

res = 

[a1,a2,a3,a4,a5,a6,a7,parameters,conditions] = solve(res==0,'ReturnConditions',1)

a1 = 

a2 = 

a3 = 

a4 = 

a5 = 

a6 = 

a7 = 

parameters = 

z

conditions = 

[lei]

Hi Torsten,

I am confused that whether it implies that the equations only two exact solutions?

[Torsten]

I am confused that whether it implies that the equations only two exact solutions?

Seems the solution is parametrized by one parameter (see above).

[Torsten]

@lei comment moved here:

Torsten, thank you very much! The result is what I want. Thanks again and have a nice day!