Primeri uporabe newton.m in krivulja.m / Examples of use of newton.m and krivulja.m

To use the function newton.m, we first need to define the function \(\mathbf{F}\) and its Jacobi matrix \(J\mathbf{F}\). Inline functions in Octave can be defined as:

octave:1> F = @(X) [X(1)^2 - X(2)^2 - 1; X(1) + X(2) - X(1)*X(2) - 1]
octave:2> JF = @(X) [2*X(1), -2*X(2); 1 - X(2), 1 - X(1)]

The above defines the function \(\mathbf{F}\) (and corresponding Jacobi matrix \(J\mathbf{F}\)) for the example nonlinear system in the 1st exercise. We can now run Newton's iteration:

octave:3> newton(F, JF, [2; 1])

and check that we have actually obtained a solution:

octave:4> F(ans)

For the 2nd exercise (and function krivulja.m) we need to define \(f\) and \(\mathrm{grad}\, f\) (as inline functions). Let's do this for an ellipse \(x^2 + \frac{y^2}{4} = 1\) first:

octave:5> f = @(x, y) x^2 + y^2/4 - 1
octave:6> gradf = @(x, y) [2*x; y/2]

Then call:

octave:7> K = krivulja(f, gradf, [1; 0], 0.1, 20);
octave:8> plot(K(1, :), K(2, :))

octave:9> K = krivulja(f, gradf, [1; 0], 0.1, 100);

octave:10> length(K)

octave:11> f = @(x, y) x^4 + y^2 - x^2*y - 1
octave:12> gradf = @(x, y) [4*x^3 - 2*x*y; 2*y-x^2]
octave:13> K = krivulja(f, gradf, [1; 0], 0.05, 200);
octave:14> plot(K(1, :), K(2, :))