How to solve N nonlinear equations using fsolve?

I'm trying to solve a system on N nonlinear equations. For , for example, I can simply input 5 equations by hand that has 5 variables in total. Here's an example just to show what I use;

%%%%% Example %%%%%
f=@(x) [x(1)^2-x(2);
             x(2)^2-x(3);
             x(3)^2-x(4);
             x(4)^2-x(5);
             x(5)^2-x(1)];
X=ones(1,N-3);
S=fsolve(f,X);
%%%%% Example %%%%%

What I really want to do is to solve N nonlinear equations that has N variables in total. In this case there are 2 issues I need to solve.

1-) How can I input N variables?

2-) How can I input N functions?

Is there any loop I can use?

Thanks in advance.

[Walter Roberson]

do you have the symbolic toolbox? if so then create a vector of equations and matlabFunction that

Answer 1

[Walter Roberson]

If you do not have the symbolic toolbox, you will need to enter the equations as text. strjoin() the equations with ';' and put '@(x)[' ']' around that, and str2func() to create the function handle.

[Berk Özdemir]

Thanks for the reply, but I think I made a mistake by using the word "input". What I want to do is to create a system of nonlinear equations which should be written in script such as;

x(1)^2-x(2)

x(2)^2-x(3)

x(3)^2-x(4)

...

x(N-1)^2-x(N)

x(N)^2-x(1)

which depends to N. As you know, for large values of N, it is impossible to write them by hand, so I'm looking for a loop if it is possible do so.

EDIT : I managed to create the system of equations by using matlabFunction. This time, I got "not enough input" error even though my initial vector has the same size as the number of variables when I try to solve it by using fsolve.

[Walter Roberson]

Make sure you use the 'vars' option of matlabFunction()

As you know, for large values of N, it is impossible to write them by hand

You can generate the equations symbolically if you have suitable patterns, and then use matlabFunction. Or you can generate the equations as text, and join the text together using

str2func(['@(x)[', strjoin(CellOfText', ';'), ']'])

You could also,

funs = arrayfun(@(idx) @(x) x(idx).^2 - x(idx+1), 1:N-1, 'uniform', 0);
obj = @(x) arrayfun(@(idx) funs{idx}(x), 1:length(funs));
fsolve(obj, x0)

However, I would point out that if you are restricting yourself to real numbers, then the solution to the above pattern is that x(N) must be non-negative, and that each x(N-1) = sqrt(x(N)) in series, except that x(1) can be +/- sqrt(x(2)). You can generate the entire sequence as x(N).^((1/2).^(N-1:-1:0))

[Berk Özdemir]

Dear Walter,

Thank you for your response, using vars option of matlabFunction worked, but that led me another problem.

I create a vector of symbolic variables such as;

A=sym('x', [1;N-3]);

where N is entered by the user. I checked for ,

f=matlabFunction(F,'vars',{[A(1) A(2) A(3) A(4) A(5) A(6) A(7)});

S=fsolve(f,X0);

and it worked. But what if A has 1000 variables?

EDIT : I simply changed

f=matlabFunction(F,'vars',{[A(1) A(2) A(3) A(4) A(5) A(6) A(7)});

to

f=matlabFunction(F,'vars',{[A]});

and it worked.

That helped a lot, thank you.

[Walter Roberson]

You could also just use

f=matlabFunction(F,'vars',{A});