use sostools to find if x^2 is sum of squares in matlab

To verify if a polynomial like x^2 can be written as a sum of squares (SOS) using sostools in MATLAB, we can use the sosineq function. Here is an example code:

main.m
% Define the polynomial to verify
p = [1 0 0];  % x^2

% Create a symbolic variable for x
syms x

% Define the SOS constraint using the symbolic variable
sos_constraint = sosineq(p - sdpvar(x)^2);

% Verify if the polynomial is SOS
result = soscheck(sos_constraint)

% If the result is 0, the polynomial is SOS. Otherwise, it is not.
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In this code, we firstly define the polynomial of interest using the coefficient vector p. Then, we create a symbolic variable x to use in the SOS constraint. The sosineq function creates an SOS constraint by subtracting the polynomial p from the squared SDP variable x^2. This constraint ensures that p is SOS if and only if there exists an SDP variable x that satisfies the constraint. Finally, we verify the constraint using the soscheck function, which returns 0 if the constraint is satisfied (i.e., p is SOS) or a non-zero value otherwise.