Update 02_newton.md

02_newton.md

@@ -12,11 +12,27 @@ Iterative techniques for solving $f(x) = 0$ for $x$.
 
 *Bisection*: start with an interval $[a, b]$ bracketing the root.
 Evaluate the midpoint.  Replace one end, maintaining a root bracket.
-Linear convergence.  Slow but **robust**.
+Linear convergence.  Slow but **robust**. Error bound is defined by
+$|x_{n}-{x}^*|<(b-a)/{2}^n$.
+
+*Fixed-point iteration*: A number $p$ is a **fixed point** for a given 
+function g if $g(p)=p$. so given a root-finding problem $f(p)=0$,we can
+define function $g$ with a fixed point at $p$ in a number of ways, for 
+example, as $g(x)=x-f(x)$ or as $g(x)=x+3f(x)$. Conversey, if the function
+$g$ has a fixed point at $p$, then the function defined by $f(x)=x-g(x)$
+has a zero at $p$.
 
 *Newton's Method*: $x_{k+1} = x_k - f(x_k) / f'(x_k)$.  Faster,
 quadratic convergence (number of correct decimals places doubles each
-iteration).
+iteration).when $|p_{n}-p_{n-1}|<e$ or $|f(p_{n})|<e$, we can stop the
+the iterations.
+
+*Secant method*: Newton's method is an extremly powerful technique, but it
+has a major weekness:the need to know the value of the derivative of $f$ at
+each approximation. Frequently, $f(x)'$ is far more arithmetic operation to
+calculate than $f(x)$. We introduce the **secant method**. 
+
+  $$ p_{n}=p_{n-1}-f(p_{n-1})(p_{n-1}-p_{n-2})/(f(p_{n-1})-f(p_{n-2})) $$.
 
 Downsides of Newton's Method: need derivative info, and additional
 smoothness.  Convergence usually not guaranteed unless "sufficiently