Euler

function euler

  % y' = (-y+x*sin(x))/x; y(1) = 0.3011687, x = [1, 2]

  x = [1, 2];
  y0 = 0.3011687;
  h1 = 0.1;
  h2 = 0.01;

  printf('\n  I \n\n h1 = 0.1 \n y = %i', euler_method(x, y0, h1));
  printf('\n\n h2 = 0.01 \n y = %i\n\n\n', euler_method(x, y0, h2));

  printf('  II \n\n h1 = 0.1 \n y = %i', euler_midpoint(x, y0, h1));
  printf('\n\n h2 = 0.01 \n y = %i\n\n\n', euler_midpoint(x, y0, h2));


end



function y = euler_method(x, y_i, h)

  for x = x(1):h:x(2)

    y = y_i + h*f(x, y_i);
    y_i = y;

  endfor

endfunction


function y = euler_midpoint(x, y_i, h)

  for x = x(1):h:x(2)-h

    y = y_i + h/2*( f(x, y_i) + f(x+h, y_i + h*f(x, y_i)) );
    y_i = y;

  endfor

endfunction


function f = f(x, y)

  f = (-y + x*sin(x)) / x;

endfunction