CISC 822 : Algebraic Algorithms

Approximate Lecture sequence

1. Preface
   • Some history of CA.
   • History of saclib. See sachis.tex or sachis.ps
   • Overview of quantifier elimination. See qeov.tex or qeov.ps
   • Overview of symbolic linear algebra.
   • Assignment: read Knuth section 4.3.1, pp 265-284.
2. Integers 1
   
   • The saclib system, esp lists (see saclists.tex or saclists.ps ), and integer representation (see intrep.tex or intrep.ps ).
   • Integer addition, subtraction, sign, abs., bit length. See intadd.tex or intadd.ps
   • Integer multiplication, the classical algorithm.
3. Integers 2
   
   • Integer multiplication - Karatsuba algorithm. See intmult.tex or intmult.ps
   • Integer division - quotient/remainder and exact division. See intdiv.tex or intdiv.ps
   • Integer exponentiation and square root.
4. Integers 3
   
   • Integer greatest common divisors (gcd). Euclidean alg., incl. analysis.
5. Integers 4
   
   • Extended Euclidean algorithm, Lehmer and Lehmer-like.
6. Integers 5
   
   • Rational number arithmetic. See rnarith.tex or rnarith.ps
   • Prime number generation. See primegen.tex or primegen.ps
   • Integer factorization.
7. Integer assignment: ex1.tex or ex1.ps
   Modification of problem 6: ex1b.tex or ex1b.ps
8. Polynomials 1
   
   • Polynomial representation,
   • input and output, See polyio.tex or polyio.ps
   • addition and multiplication.
9. Polynomials 2
   Polynomial division -
   • quotient/remainder (trial division),
   • exact division,
   • pseudo-quotient/pseudo-remainder.
10. Polynomials 3
    
    • Polynomial substitution and reordering variables.
    • Polynomial evaluation.
11. Polynomials 4
    
    • Polynomial multiplication via FFT algorithm. fft.tex
    • Integer multiplication via FFT.
12. Polynomials 5
    
    • Modular arithmetic. See modarith.tex or modarith.ps
    • Modular Polynomial arithmetic.
    • Evaluation and interpolation. See polyinterp.tex or polyinterp.ps
    • Chinese remainder theorem/algorithm. See Chirem.tex or Chirem.ps
13. Polynomial GCDs and Resultants 1
    
    • Euclidean algorithm over a field.
    • Content and primitive part
    • primitive polynomial remainder sequence algorithm.
14. Polynomial GCDs and Resultants 2
    
    • Fundamental theorem of polynomial remainder sequences
    • Subresultant polynomial remainder sequence.
15. Polynomial GCDs and Resultants 3
    
    • Modular gcd algorithm
      [ full analysis not completed. ]
16. Polynomial GCDs and Resultants 4
    
    • Modular resultant algorithm.
17. Polynomial GCDs and Resultants 5
    
    • Discriminants, analysis. See discriminant.tex or discriminant.ps
18. Polynomial GCDs and Resultants 6
    
    Squarefree factorization squarefree.tex
19. Midterm Exam.
20. Polynomial factorization 1
    
    • Berlekamp's algorithm
21. Polynomial factorization 2
    
    • Distinct degree factorization
22. Polynomial factorization 3
    
    • Hensel algorithms.
23. Polynomial factorization 4
    
    • Factor coefficient bounds.
24. Polynomial factorization 5
    
    • Primality testing.
    • Modular factor testing.
25. Unspecified lecture
    
    • a basic algebra lecture on groups, rings, fields. See rings.tex or rings.ps [ could have profitably been a little earlier in course. ]
    • homomorphisms,
    • direct sum/product, quotient ring, localization,
26. Unspecified lecture
    
    • A basic linear algebra lecture on determinants. (Prep for Sylvester matrix, Resultant) Equivalent formulations, multi-linearity, Determinant algorithms. Elimination(s), incl. fraction free, Modular, [ not done as separate lecture ]
27. Unspecified lecture
    
    • Catch up for slippage.
28. Groebner bases 1
    
    • Terminology: Monomial, Head term, Admissible term orderings,
    • Defs: Groebner basis, canonical basis.
    • Buchberger's algorithm, incl partial correctness
29. Groebner bases 2
    
    • Buchberger's algorithm - termination, analysis.
    • Heuristic improvements, parallelism
    • Applications of Groebner bases: Ideal membership, canonical form in quotient ring.
    • GBasis reading?

• (3 weeks) Real root isolation and refinement, interval arithmetic. Reading: ?
• (2 weeks) Computation with algebraic numbers. Reading: ?
• (5 weeks) Cylindrical algebraic decomposition algorithm and quantifier elimination. Reading: ?
• (4 weeks) Matrix topics (fraction free elimination, Krylov space methods, matrix normal forms - Smith, Hermite, Frobenius, Jordan). Reading: notes