User:Lxm1117

Project Report for User:lxm1117
For Introduction to Numerical Analysis, Fall 2012.

Introduction
My final project is about error analysis of Newton-Cotes formulas. The topic is important because it helps students to understand the accuracy of using numerical integration methods with different rules. It is difficult to understand using only Wikipedia because the related Wikipedia page shows only the result formulas but not the detailed steps of how to get these formulas.

To facilitate learning of this topic I created my project page. I showed steps in details to obtain the error term of Newton-Cotes formulas, also an example illustrating the meaning of the error terms, and two exercises to develop skill at understanding the presented method.

Contribution
I created the topic page on error analysis of Newton-Cotes formulas which contains how to get general form of the error term. The method is based on the general error term of polynomial, which has already existed in Wikipedia. The method is not as complicated as expected but is easy to understand. The topic page also shows how to apply the method to numerical integration with different rules: trapezoid, simpson's 1/3 and simpson's 3/8. One numerical example has been given. I chose this particular example because it effectively demonstrates the concept of error ratio. I also added two exercises. They are both adapted from related book chapters. There has been questions on how to justify the steps involved in trapezoid composite rules, from the summation step to its next step concerning that different $$\xi_i$$ are involved in different intervals. I added a short paragraph to show that the derivation of equation (2) is based on a theorem, which is essentially analogous to the mean value theorem. The theorem is from the book by Hamming (1986) and proof is in the book.

Future Work
I decided that although adding other possible methods for comparison would be good it was too much for this project, so I just showed one method, which should be enough to clarify the topic. It would be beneficial if the topic can be expanded to related with other topics discussed in class. Also the method is derived by using Newton's polynomial, while the wikipedia one is given with Lagrange polynomial. Although these two forms are essentially the same, it would be more informative to see how the error term can be derived from the latter approach.

Conclusions
In conclusion, in this project I presented how to derive the error terms for Newton-Cotes formulas, which is missing in Wikipedia. I think this is a valuable contribution because it helps students understand the topic of numerical integration and view the concept of error term of numerical interpolation in a different context.

Quiz
{Which method cannot be used for interpolation of unequally spaced nodes? + Newton-Cotes formula - Vandermonde - Lagrange interpolation - None of those
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{ The positive real root of $$x^2-4=0$$ is { -10 _3 }.
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HW7
In wikiversity page about RungeKutta stability, I changed the expressions of those $$k$$. I changed $$k_1=f(t_n,y_n)$$ into $$k_1=hf(t_n,y_n)$$, $$k_2=f(t_n+c_2h,y_n+a_{21}hk_1)$$ into $$k_2=hf(t_n+c_2h,y_n+a_{21}k_1)$$, etc. In the following example, I changed $$y_{n+1}=y_n+\frac{h}{6}k_1+...$$ into $$y_{n+1}=y_n+\frac{1}{6}k_1+...$$.

HW8
Question 1: I am going to add contents on error analysis of Newton-Cotes formula. The wikipedia page doesn't contain enough information on how this is done. I plan to show the procedures on the order of errors for several rules of Newton-Cotes formula, including midpoint, trapezoid, simpson's and simpson's 3/8 rules, and also the corresponding composite forms. I presented this in class but failed to do well. I think it's necessary to add these information.


 * This would be useful to add and a nice project. It would be better to do a thorough analysis of a few methods rather than a shallow analysis of many methods, so you may want to limit your scope; you can decide this as you start developing the content. My recollection is that the proofs use some methods beyond what we did in class; whenever possible refer to the appropriate Wikipedia pages for background. Mjmohio (talk) 16:22, 8 November 2012 (UTC)