Fixed Point Method

This lecture introduces the fixed point method, where a function g is given and the goal is to find x bar such that x bar = g(x bar). The method involves iterating xn+1 = g(xn) starting from a given x0 to determine if the sequence xn converges. An illustrative example is provided with a function g having two fixed points, x1 bar and x2 bar. The convergence of xn is demonstrated by iterating from x0 to x1 bar, showing that xn tends towards x1 bar as n approaches infinity. Conversely, if x0 is greater than x2 bar, the sequence diverges towards positive infinity. The lecture concludes by hinting at the study of theorem 8.3 in the book.