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\begin{document}
\section{Handout}
\begin{center}
{\bf Mathematics 423, Spring Semester 1998}\\ January 12, 1998
\end{center}

\bigskip

\begin{tabbing}
Text: \=Elements of \=Modern Algebra, by Gilbert and Gilbert\kill {\em Instructor}: \>\> Andrew Sommese\\
\>\>	231 CCMB (On Juniper, just south of the main library)\\
\>\>e-mail: sommese.1@nd.edu\ \ \ \	Phone: 631-6498\\
{\em Text}: \>Numerical Analysis (sixth ed.), by Burden and Faires\\ \end{tabbing}
{\em Office Hours}: Open Door: I am in my office almost all of every weekday, and encourage you to
visit any time. If you just come to my office you will probably find me, but if you set up a time with me before hand, then you can be sure that I will be there. \\

\noindent{\em Examinations, homework, and grades:\ } 
There will be two one-hour departmental examinations worth 100 points and a two-hour final
examination worth 150 points.	The final exam will cover all
the material of the course with emphasis on the material covered after the second exam.



A student who misses an examination will receive no points for that exam unless he or she has written permission from the {\em Vice President for Student
Affairs}. (Travel plans are not considered to be a sufficient excuse for taking an exam
on a different date.)



Homework will be assigned regularly, and is an integral part of the course. I ask students to form groups of three (and one or two groups of size four depending on the number of students in the class modulo 3) to do all the assignments. If there are people who would like to be in the same group, please let me know by the end of class on Friday, January 16. I will hand out the lists of members of the class by groups on Monday, January 19. If the class enrollment changes I might have to add members
to a few groups.
Typically I will give assignments throughout the week and collect them the following Monday.
I strongly encourage you to see me if there is anything connected with the course or the mathematics in the course that you are unclear on or would like to know more about.	You are allowed and encouraged
to use your notes and C programs, any numerical analysis or C books, and any library books while doing the homework. 



Both examinations and the homework are conducted under the honor code. People within a group are graded together on the homework assignments, and are expected to work together.
People in different groups are encouraged to discuss the mathematics, but should not
discuss how to do the week's assignment before it is handed in! 



Homework will be worth 100 points.
Thus the total number of possible points for the semester is 450. The numerical break points for letter grades (A, A-, B+,\ldots) will be based only on the
test scores and the homework.





\medskip


\begin{tabbing}
Exam 1:\ \ \=8:00 A.M.; Tuesday, Nov.\ 29 1994\ \ \ \ \ \ \=(in class)\kill
Exam 1: \> Wednesday, February 25 in class \\ Exam 2: \> Wednesday, April 8 in class.\\ Final: \> Thursday, May 7, 1998: 8:00-10:00 AM.\\ \end{tabbing}
{\em The most recent version of this handout plus other useful materials can be found in}\ \ \
{\verb+/afs/nd.edu/coursesp.98/math/math423.01+}. 

\newpage
\section{Homework Assigned}


\begin{tabbing}
August 31 (Wednesday )\=\ pagepage \ \ \ \ \=16, 17, 23, 27, 29, 37\kill \ \ \ \ {\em Due Date} \>{\em Page Number}\ \>{\em Problems Assigned}\\ \smallskip
1st week\>26\>1a,c;3a,c;5e,h;6e,h;7e,h;8e,h;27\\ January 28 (Wed.)\>53\>1 (use a calculator); 7c,d; 13\\ extended to Feb. 2\>63\>1; 2; 12; 17 \\
\>75\>1 (use a calculator); 6a,f; 8i:a,f; 13b,c\\ \>86\>4; 6\\
February 2 (Mon.)\>119-123\>1a,c; 2a,c; 3a,b; 5; 7a,b; 14; 16\\ February 9 (Mon.)\>132-134\>1; 2; 4; 13\\ \>141-142\>1a,c; 2a,c; 7\\
February 16 (Mon.)\>197\>1ab; 3ab; 5ab\\ \>205--206\>1a,b; 2a,b; 3ab; 7\\
\>221--222\>1a; 2a; 3ab; 7\\
\>Exercise 1 of the Bernoulli polynomial handout\\ February 23 (Mon.)\>177-178\>1 (forward diff.\ only); 3a,b\\ \>186-187\>9; 10; 11\\
\>213\>1h; 2h\\
March 2 (Mon.)\>259\>4\\
\>267\>1a\\
\ \ \ for Mar.\ 16 (Mon.) Read to pg.\ 292\\ March 18 (Wed.)\>258\>1c,d; \\
\>267\>1c; 2c; 3b,c; 4c\\
\>274\>1a,d; 2a; 3b\\
March 23 (Mon.)\>285\>11a; 15--use maple\\ \>293\>6 (use rkf45; maple makes this easy)\\ \>304\>1a; 2a;\ (only two-step methods, only $t\le 0.4$)\\ March 30 (Mon.)\>328\> 2\\
(write as a system of first order equations, do not solve)\\ \>340\>4b,c\\
\>483-485\>5acd(use logs in d); 7; 10\\
\>496\>11; 12a\\
\>506\>1a\\
April 6 (Mon.)\>597\> 2, 4, 5a, 9a\\
\>604\>1d, 3a; \ Also do the Problems 1, 2 below.\\ April 20 (Mon.) \>631\> 3b\\
\>638\> 3a,c\\
\>644\> 3a,d\\
\> 547\> 3\\
April 27 (Mon.) \>651\> 3c,d\\
\>154\> 1; 3a,d; 11\\
\end{tabbing}

\bigskip

\noindent {\bf Problem 1.} Let $V_4$ denote the degree $\le 4$ polynomials. Compute the degree $\le 4$ polynomial $p(x)$ such that $\displaystyle
||p(x)-T_6(x)||=\min_{q\in V_4}||q(x)-T_6(x)||. $
where $T_6(x)$ is the sixth Chebychev polynomial and $\displaystyle ||f||=\sqrt{<f,f>}$, where for two functions $f,g$ on $[-1,1]$ $$
<f,g> =\int_{-1}^1\frac{f(x)g(x)}{\sqrt{1-x^2}}\mbox{\rm d}x. $$

\noindent {\bf Problem 2.} Let $V_3$ denote the degree $\le 3$ polynomials. For two functions $f,g$ on $[0,1]$ let
$$
<f,g> =\int_{0}^1 f(x)g(x)\mbox{\rm d}x. $$
Find a basis of $V_3$ that contains $1$ and is orthogonal relative to $<f,g>$.




\end{document}
