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MA-693/893
Computer Algebra 2 Spring 2022 Reading Course |
Syllabus | People | Maple | Projects | Homeworks | Reading | Grading | Academics |
Current Announcements
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Peoples' home pages:
Erich Kaltofen,
Maple programs for the course (Maple hints).
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Course Outline* | |||||
Lecture | Topic(s) | Notes | Book(s) | ||
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1. Jan 11 |
Administrative meeting.
Strassen's matrix multiplication
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GG §12.1
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2. Jan 16 |
Algebraic complexity;
analysis of Strassen's scheme
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3. Jan 18 |
Winograd's scheme
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4. Jan 22 |
LUP factorization < matrix multiplication
(Bunch-Hopcroft algorithm)
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5. Jan 25 |
Outline of black box linear algebra
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6. Jan 29 |
Hensel lifting
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GG §15.4
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7. Feb 1 |
Dixon's algorithm
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Numer. Math., vol. 40, nr. 1 (1982)
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8. Feb 5 |
P-adic numbers;
recap of black box linear algebra
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GG §9.6
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9. Feb 8 |
the Berlekamp-Massey algorithm
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Section 2.2 in
JSC vol. 36 (2003)
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GG §12.3
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10. Feb 12 |
Proof of Wiedemann algorithm via the
Schwartz/Zippel lemma
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Section 3 in
Math. Comput. vol. 64 (1995)
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GG §6.9, Lemma 6.44
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11. Feb 15 |
Wiedemann algorithm for singular matrices;
the use of preconditioners
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See
talk at Fq6
Further reading LAA vol. 343-344 (2002) |
GG §12.4
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12. Feb 19 |
Gauss's lemma;
GCD of several multivariate polynomials
via linear combinations
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Theorem 6.2 in
J. ACM, vol. 35 (1988)
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GG §6.2
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13. Feb 22 |
GCD of several multivariate polynomials
via linear combinations continued
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14. Feb 27 |
Integration of rational functions
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My Lect. Notes
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GG §22
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15. Mar 1 |
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Week of Mar 5-9 | Spring Break, no class | ||||
16. Mar 12 |
Lattice basis reduction (LLL)
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GG §12.1-12.3
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17. Mar 15 |
LLL continued
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18. Mar 19 |
Integration of rational functions continued;
proof that log(x) is irrational;
Rothstein-Trager theorem
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Wed, Mar 21, 5pm | Last day to drop the course |
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19. Mar 22 |
GGH public key crypto-system;
knapsack based crypto
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Section 4 in
JSC vol. 29, pp. 891-919 (2000)
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20. Mar 26 |
Differential fields;
quotient fields;
algebraic function fields;
proof that log(x) is transcendental
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My Lect. Notes
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21. Mar 29 |
Elementary Liouville extensions;
structure theorem (not proved)
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22. Apr 2 |
Liouville's theorem (not proved);
impossibilities of closed form solution of the integral
of certain transcendental functions
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Tue, Apr 3, 5pm
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Papers for class presentation must be declared at 5pm | ||||
23. Apr 5 |
Semi-algebraic sets;
the principle of quantifier elimination
on examples
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Section 2 in
JSC vol. 29, pp. 891-919 (2000)
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24. Apr 9 |
Proof of Sturm's theorem;
Cauchy root bound
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GG Exercise 4.32
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Tue, Apr 10
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Approvals of papers for presentation by me are posted | ||||
25. Apr 12 |
Cauchy principle of argument;
Cauchy index of a real rational function
on an interval
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Handout: chapters in Marden's and Gantmacher's books
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26. Apr 16 |
Routh-Hurwitz algorithm for
complex root isolation
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27. Apr 19 |
Seidenberg's method
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Handout: Seidenberg's paper;
Section 6 in
AAECC vol. 1, nr. 2, pp. 135-148 (1990)
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Sat, Apr 21 | East Coast Computer Algebra Day, Washington College | ||||
28. Apr 23 |
Paper presentations begin;
recap of Fall-Spring courses
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29. Apr 26 |
Extra topic: FFT
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GG §8.2
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??? | Presentations | ||||
Friday, May 6, 4:59pm | Spring 2022 grades due |
On-line information: All information on courses that I teach (except individual grades) is now accessible via html browsers, which includes this syllabus. Click on my courses' page of my resume. You can also find information on courses that I have taught in the past.
There will be six homework assignments of approximately equal weight. At the end of the course, each student will give a 30 minute presentation and write a 3-5 page discussion of the ideas, or write a program implementing the algorithms in what you have read. I expect that you attend my office hours. If you need assistance in any way, please let me know (see also the University's policy).
©2022 Erich Kaltofen. Permission to use provided that copyright notice is not removed.