User:YangOu

Homework 7
http://en.wikiversity.org/wiki/Rootfinding_for_nonlinear_equations#Convergence I have done some changes to the definition of the convergence which is not accurate and listed the definition of the superlinear and sublinear convergence.

After that I made the solution of the given example look clear. I just applied the definition of the convergence to find the approximation of the root if I have known the rate of convergence, the order of convergence, the initial error and the tolerance. When the error becomes less than the tolerance, then we can stop the iteration and draw the conclusion that the value at that step is the approximation of the root.

Things I learned

 * 1) I learned how to edit my user page.
 * 2) I got what the Wikipedia is. It is a multilingual,web-based,free-content encyclopedia project based on an openly editable model.
 * 3) I can read a lot of articles written by other users of Wikipedia.
 * 4) I have known something about the floating point.
 * 5) I can share some information I have acquire with other people on Wikipedia.

Math formula:
ex= $$\sum_{n=0}^\infty \frac{x^n}{n!}$$

$$f(x) \,\!$$ $$= \sum_{n=0}^\infty a_n x^n $$ $$= a_0+a_1x+a_2x^2+\cdots$$

$$x={-b\pm\sqrt{b^2-4ac} \over 2a}$$

Talk page edit
Talk:Bisection method

Things learned after the previous homework
Last time I mentioned the disadvantage of bisection method and I provided some description about how to do that, but that proposal may be not good enough because it is not easy for me to ensure the monotonicity of the function in the given interval and to find the relative maximums and minimums of the function.

At the end of my proposal, I talked about combining Newton's method and bisection method to solve this problem.I have found a way to use these two methods together.We can use Newton iterate instead of bisection midpoint whenever the newton iterate is within the given interval and we should discard the newton iterate if it is outside the interval.We should also give up the newton iterate if it is very close to the end point of the interval. If the bisection iterate is better than the newton iterate, so we will perform one more step of bisection method.After that we will try newton iterate again and keep going this process.

There is still problem here by using the hybrid method. we can not ensure that the method can converge and it is not slow. So next time I should find some ways to make sure the hybrid method converge and converge fast. The hybrid method is given by a_<b - \frac{f(b)}{f'(b)}<c