Reference Textbook: A Friendly Introduction to Numerical Analysis- Brian Bradie.
Rate of convergence
Let $\{x_n\}$ be a sequence that converges to $x$. If there exists a sequence $\{y_n\}$ that converges to $0$,a positive constant $\lambda>0$ (independent of $n$) and a Natural number $N$ such that
$\forall n\geq N \implies |x_n-x|<\lambda |y_n|$
Then we say $\{x_n\}$ be a sequence that converges to $x$ with a rate of convergence $O(y_n).$
- What is the rate of convergence of Newton-Raphson Method(If the sequence $x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)} converges)$ ? Is it possible to find the rate of convergence to Newton-Raphson method? This was the question came to my mind when the author misses the rate of convergence of Newton Raphson method and Secant Method.
2.What is the rate of convergence of Secant Method?
My attempt
- Let for the initial guess $x_0$, $x_n$ converges to $x.$
$ |x_n-x|=|x_{n-1}-\frac{f(x_{n-1})}{f'(x_{n-1})}-x|=|\frac{(x_{n-1}-x)f'(x_{n-1})-f(x_{n-1})}{f'(x_{n-1})}|$
I am not able to find $\lambda, N $ and $\{y_n\}$.
- For Secant Method,
$ |x_n-x|=|x_{n-1}-f(x_{n-1})\cdot \frac{(x_{n-1}-x_{n-2})}{f(x_{n-1})-f(x_{n-2})}-x|=|(x_{n-1}-x)-f(x_{n-1})\cdot \frac{(x_{n-1}-x_{n-2})}{f(x_{n-1})-f(x_{n-2})}|=| \frac{(x_{n-1}-x)\cdot (f(x_{n-1})-f(x_{n-2}))-f(x_{n-1})\cdot (x_{n-1}-x_{n-2})}{f(x_{n-1})-f(x_{n-2})}|$
I am not able to reduce further. I am not able to find $\lambda, N $ and $\{y_n\}$.