a multivariate quadratic function

I'm not asking for a full code, but a way that makes sense.

find a multivariate equation that solves, please no codes

%want to solve using a 2x2 matrix. 
f(0,1)=6, f(1,0)=6, f(1,1)=3, f(-1,-1)=7, f(1,2)=2. f(2,1)-=6

Answer 1

[Alan Stevens]

Here's one way. It uses an initial guess for each element of P, q and r, then combined them into a single column vector which is used by fminsearch.

f = @(x,P,q,r) x'*P*x + q'*x + r;
% initial guesses (only three elements guessed for P as it's a symmetric matrix)
P0 =[1; 0; 1];
q0 = [1; 1];
r0 = 1;

Pqr0 =[P0(:); q0(:); r0];   % initial guesses combined into a single column vector

Pqr = fminsearch(@(Pqr) Pqrfn(Pqr,f), Pqr0);  % fminsearch tries to find values of Pqr that minimise
                                              % the value returned from Pqrfn
% reconstruct P, q and r
P = [Pqr(1) Pqr(2); Pqr(2) Pqr(3)];           
q = Pqr(4:5);
r = Pqr(6);

% display results 
disp(P)
disp(q)
disp(r)

% Check results
d(1) = f([0;1],P,q,r);
d(2) = f([1;0],P,q,r);
d(3) = f([1;1],P,q,r);
d(4) = f([-1;-1],P,q,r);
d(5) = f([1;2],P,q,r);
d(6) = f([2;1],P,q,r);

disp(d)

function F = Pqrfn(Pqr,f)
        % construct P, q and r from Pqr
        P = [Pqr(1) Pqr(2); Pqr(2) Pqr(3)];
        q = Pqr(4:5);
        r = Pqr(6);
        
        % Calculate differences of function values from desired values
        d(1) = f([0;1],P,q,r) - 6;
        d(2) = f([1;0],P,q,r) - 6;
        d(3) = f([1;1],P,q,r) - 3;
        d(4) = f([-1;-1],P,q,r) - 7;
        d(5) = f([1;2],P,q,r) - 2;
        d(6) = f([2;1],P,q,r) - 6;
        % return the norm of the differences
        F = norm(d);
 
end

If you do

format long

before displaying the results, you will see that they are not exact, but are good to four decimal places.