Steffensen's method: Difference between revisions

The method we will use is ''implicitization''. In this method, one first rewrites one of the curves as an [[wp:Implicit_function|implicit equation]] in <math>x</math> and <math>y</math>. For this we will use the parabola that is convex upwards: it has implicit equation <math>5x^2 + y - 5 = 0</math>. Then what one does is plug the parametric equations of the ''other'' curve into the implicit equation. This gives <math>5(x(t))^2 + y(t) - 5 = 0</math>, where <math>t</math> is the independent parameter for the curve that is convex ''leftwards''. After expansion, this will be a degree-4 equation in <math>t</math>. Find its four roots and you have found the points of intersection of the two curves.