Use Newton\'s Method with initial guess (x_0, y_0) = (0, 0) to calculate by hand

Question

Use Newton's Method with initial guess (x_0, y_0) = (0, 0) to calculate by hand (x_1, y_1) (the first iteration) for the nonlinear system. x^2 + xy + 2x - 2y = 2 y^2 + x^2 - y = 1 Then, use the MatLab m-file for Newton's Method in two variables to calculate the root with precision 10^-4.

Explanation / Answer

clc,clear % Newton Raphson solution of two nonlinear algebraic equations % set up the iteration error1 = 1.e8; xx(1) = 0; % initial guesses xx(2) = 0.5; iter=0; maxiter=40 % begin iteration while error1>1.e-12 iter=iter+1; x = xx(1); y = xx(2); % calculate the functions f(1) = 4*x^2-20*x+0.25*y^2+8; f(2) = 0.5*x*y^2+2*x+8; % calculate the Jacobian J(1,1) = 8*x-20; J(1,2) = 0.5*y; J(2,1) = 0.5*y^2+2; J(2,2) = x*y; % solve the linear equations y = -J'; % move the solution, xx(k+1) - xx(k), to xx(k+1) xx = xx + y'; % calculate norms error1=sqrt(y(1)^2+y(2)^2); error(iter)=sqrt(f(1)^2+f(2)^2); ii(iter)=iter; if (iter > maxiter) error1 = 0; s=sprintf('****Did not converge within %3.0f iterations.****',maxiter); disp(s) end % check if error1 < 1.e-12 end x = xx(1); y = xx(2); f(1) = 4*x^2-20*x+0.25*y^2+8; f(2) = 0.5*x*y^2+2*x+8; % print results f xx iter % plot results semilogy(ii,error) xlabel('iteration number') ylabel('norm of functions') clear ii clear error

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