Lectures: MC 320, Monday, 9:30 am - 11:30 am.

Office Hours: MC 415, Tuesday and Wednesday, 9:30 am - 11 am.

Phone: 519 661 2111 ext 86994 (e-mail contact preferred!)

- Basic algorithms on power series: multiplication, inversion, composition.
- Further algorithms: Newton iteration, Padé and Hermite-Padé approximants.
- Various aspects of the notion of "solving" linear differential equations and recurrences.
- Discovering and proving identities.
- Extension to multivariate settings and non-commutative Gröbner bases.

- Lecture 1 Introduction, polynomial multiplication, rational generating series
- Lecture 2 Newton iteration: inverse, exponential, first-order differential equations
- Lecture 3 D-finiteness and P-recursiveness
- Lecture 4 hypergeometric summation
- Lecture 5 Definite hypergeometric summation
- Lecture 6 Operator algebras

- March 23: slides are online for lecture 6.
- March 16: slides are online for lecture 5.
- March 9: slides are online for lecture 4.
- March 3: slides are online for lecture 3.
- Feb. 18: more papers for the oral presentation are online.
- Feb. 10: the first papers for the oral presentation are online.
- Feb. 10: updated slides for lecture 2.
- Jan. 28: slides are online for lectures 1 and 2.
- There is
**no class**on January 7th. The first class is on**Monday, January 14th.**

*Modern Computer Algebra*, by Joachim von zur Gathen and Jürgen Gerhard, Cambridge University Press, 1999.*A=B*, by Marko Petkovsek, Herbert Wilf and Doron Zeilberger, available here.

- D. J. Bernstein, Scaled remainder trees.
- V. Shoup, Efficient computation of minimal polynomials in algebraic extensions of finite fields (proc. ISSAC'99).
- R. P. Brent and H. T. Kung, Fast algorithms for manipulating formal power series (Journal of the ACM , 25(4):581-595, 1978).
- C. Umans, Fast polynomial factorization, modular composition, and multipoint evaluation of multivariate polynomials in small characteristic (proc. STOC'08).
- A. Bostan, F. Chyzak, F. Ollivier, B. Salvy, E. Schost and A. Sedoglavic, Fast computation of power series solutions of systems of differential equations (proc. SODA'07).
- S. Gerhold and M. Kauers, A procedure for proving special function inequalities involving a discrete parameter (proc. ISSAC'05).
- G. Almkvist and D. Zeilberger, The method of differentiating under the integral sign. (Journal of Symbolic Computation, 10: 571-591, 1990).
- H. Wilf and D. Zeilberger Rational functions certify combinatorial identities, (J. Amer. Math. Soc. 3, 147-158, 1990).
- H. Cheng, G. Hanrot, E. Thomé, E. Zima, P. Zimmermann Time- and space-efficient evaluation of some hypergeometric constants (proc. ISSAC'07).
- F. Chyzak An extension of Zeilberger's fast algorithm to general holonomic functions Discrete Mathematics 217(1-3): 115-134 (2000)