Week by week links to resources and summary of course progress.

• Week 1, Aug 26, 28
  • Reading: Chapters 1-3 of MCA.
  • Basic polynomial arithmetic, similarity of polynomials and integers, countable and uncountable sets.
  • Algebraic categories, primarily ring, CR1, ID, UFD, PID, ED, Field, and PIR.
  • Handout 1 (pdf)
• Week 2, Sept 2, 4
  • Reading: Chapter 4 of MCA.
  • Extended Euclidean Algorithm for Greatest Common Divisor
  • Basic modular arithmetic using EEA for inversion.
  • Karatsuba's divide and conquer polynomial multiplication, division. (MCA chapter 8.1)
  • Ring and Field classes in LinBox.
  • Homework set #1: hw1.pdf.
• Week 3, Sept 9, 11
  • Finite field representations. Non prime fields, field extensions, by polynomials modulo an irreducible.
  • (non prime) field representation by Zech logs with respect to a primitive element.
  • Fraction inducing and fraction free Gaussian Elimination formatrices over a PID. Reference: Sylvester's Identity and Multistep Integer-Preserving Gaussian Elimination by Erwin H. Bareiss.
  • Maple script LU.map
  • Linear time reductions among basic matrix problems. notes
• Week 4, Sept 16, 18
  • Select your presentation topic from among these.
  • Wrap up of basic matrix problem complexity relationships.
  • Square free factorization of polynomials over finite fields.
  • Thursday: Novocin on polynomial factorization.
• Week 5, Sept 23, 25
  • Chakraborty on Distinct Degree factorization.
  • Square free factorization wrapup
  • Wiedemann's black box minimal polynomial algorithm
  • If time permits: a menagerie of Lin Alg problems.
• Week 6, Sept 30, Oct 2
  • Tue: Novocin on LLL lattice reduction (short vector in lattice), Examples of use, cloud.sagemath.com worksheets
  • Thursday:
• Week 7, Oct 7, 9
  • Tue: Stachnik on real root isolation
  • minpoly via Wiedemann's blackbox algorithm
  • Applications of minpoly.
  • Thu: Cesarz on Fast Fourier Transform
• Week 8, Oct 14, 16
  • Tue: Lu on the Toeplitz preconditioner for Wiedemann's algorithm
  • Handout: Wiedemann's paper
  • Thu: Kolagunda on Strassen's matrix multiplication algorithm.
  • Runtime of Wiedemann's algorithm is O(n^1+α)
  • Handout on project topics.
  • Gilbert project idea for PDE soling with symbolic component, based on papers Invertpars2011a_rev-0902.pdf and symbolic-cortical-cancellous-version.pdf.
• Week 9, Oct 21, 23
  • Tue: Lambert on 4 Russians matrix multiplication algorithm.
  • Handout on similar matrices
  • Cost analysis of Wiedemann's algorithm: O(n^1+α).
  • Thu: Ulnicy on fast multipoint evaluation and interpolation.
  • Thu: Project choice due
  • Correctness analysis of Wiedemann's algorithm, the case of a diagonalizable matrix..
• Week 10, Oct 28, 30
  • Tue: Baldwin on OpenCL
  • Correctness of Wiedemann's algorithm, the cases of nonlinear irreducibles, Jordan blocks, generalized Jordan blocks.
• Week 11, Nov 6
  • Application of Wiedemann's alg to determinant, Notes.
  • Wikipedia on Schwartz/Zippel lemma.
• Week 12, Nov 11, 13
  • Application of Wiedemann's alg to rank and singular system solving.
  • Butterfly preconditioners.
  • Further consideration of diagonal preconditioners.
• Week 13, Nov 18, 20
  • Integer polynomial factorization
  • Block Wiedemann applications, notes
  • Project status report
• Week 14, Nov 25, Dec 2
  • Block Wiedemann, the algorithm
  • Integer and polynomial Smith Normal Form
• Exam week, Dec 11, 10:30 to 12:30
  • Project reports conference