modify newton's method from tanglent line to 1st order and 2nd order
tangent curve.


ove 1 = 1 + 2^{-1074}

given the number of bits to store exponent and fraction, 
how to derive machine epsilon and etc.

% \item
% modify newton's method from tanglent line to 1st order and 2nd order
% tangent curve.

\item 
Use any method to
find a solution of $ (1+x)^{1/3}-(1-x)^{1/3} = 10^{-10} $ to 10 correct
digits.

\item 
Find $p_20$ to 10 correct digits if
$p_0 = 1$, $p_1 = 0.9$, $p_n = 4 p_{n-1} + 2.79 p_{n-2}$.

% \item
% Find order of convergence numerically
% from the data, identify rate of convergence, or \alpha

% \item
% nested 

% \item 

\item 
find convergent fixed point iteration

\item
proof of fixed point iteration

\item
modified q2p5.txt

\iem
Aitken's method to \sin{1\over n} and find limit

\item

t or f
method and error estimate of Lagrange interpolation
$sin x$

\item
Give a cubicly convergent method to solve for $e^x-1$ = 0.
Give the formula and prove that it is cubically convergent. 
If you cannot do it, do the same for
a quadratiacally convergent method for partial credit.


Find $\alpha$ and $\lambda$ (the constants
in the definition of order of convergence)
for your method if it is applied to
solve the equation $x$ instead. 
You can do it analytically or numerically.
Both will get full credit, id correct.
If cannot do it, do the same for
a quadratiacally convergent method for partial credit
(no matter whether you did cubically or quadratically convergent
method in (a) ).