1. ## numerical analysis question..

http://i40.tinypic.com/seyrys.jpg
we cant see what is the pattern of finding s?
what s represents?
in iteration 0 s is found by sum of an and bn
in the 1st 2nd and 3rd iteration its found by their subtraction
in the 4th its neither
??

2. (1) Numerical analysis does not really fit into this forum section.

(2)
we cant see what is the pattern of finding s?
Not sure what this means. Are you asking if you can see the pattern?

(3)
what s represents?
$s$ is a root of the function $f(x)=x^3-x$. There are three roots, denoted by $s_1$, $s_2$ and $s_3$.

(4)
n iteration 0 s is found by sum of an and bn
It is assumed that $a_n\le s\le b_n$, $f(a_n)<0$ and $f(b_n)>0$. Let $c_n=(a_n+b_n)/2$. If $f(c_n)>0$, then we set $a_{n+1}$= $a_n$ and $b_{n+1}=c_n$. Otherwise, we set $a_{n+1}=c_n$ and $b_{n+1}=b_n$. This way, $c_n=(a_n+b_n)/2$ converges to one of the roots $s$ as $n\to\infty$.

3. but an=-3 bn=2
between them we have all of our 3 solutions -1 1 0
why we choose to take -1

4. We did not choose -1. The textbook says that the process will converge, but one can't a priori say to which root. It turns out here that it converges to -1, so the error $|c_n-s|$ is calculated for $s_1=-1$.