clear
clf
format compact
format long

x1 = .01;

np = 100;
t1 = linspace(-10*x1, 10*x1, np);
figure(1)
plot(t1, f(t1), '-')
hold on
plot(0, 0, 'or')
grid on
xlabel('t')
ylabel('f(t)')
title("Function demonstrating failure of Newton's method") 

x = newton(@f, @dfdx, x1, 20);
number_of_iterations = length(x) 

t2 = logspace(-4, 4, np);
figure(2)
semilogx(t2, f(t2), '-')
hold on
semilogx(x, f(x), 'xr')
grid on
title("Divergence of Newton's iterations") 

function y = f(x)
% F demonstrate the case of divergence for Newton's method
    alpha = 0.009;
    y = x./(alpha^2 + x.^2);
end

function y = dfdx(x)
% DFDX demonstrate the case of divergence for Newton's method
    alpha = 0.009;
    y = (alpha^2 - x^2)/(alpha^2 + x^2)^2;
end