1. Progress Report—11/13 宗慶

2. Problem Statement Find kernels of large and sparse linear systems over GF(2)

3. Well-Known Solutions Block Lanczos Algorithm Block Wiedemann Algorithm

4. Published in 1994  Solving Homogeneous Linear Equations Over GF(2) via Block Wiedmemann Algorithm by Don Coppersmith Don Coppersmith proposed a block version of the Wiedemann algorithm which take advantage of the ability to perform simultaneous operations on block of vectors.

5. Wiedemann algorithm Based on the fact that when a square matrix is repeatedly applied to a vector, the resulting vector sequence is linear recursive.

6. Advantages of Blocking Parallel implementation Faster sequential running time Better probability of success [Villard 1997]

7. Block Wiedemann Algorithm

8. Three-tier Parallelism on Cell blades SIMD on SPE Heterogeneous Multi-cores per node MPI between nodes

9. Available Resources Implementation of block Lanczos algorithm  openmp LinBox  A C++ template library for exact, high- performance linear algebra computation with dense, sparse, and structured matrices over the integers and over finite fields. Many papers….. Cell Blades  Connected via Ethernet ITRI plan to purchase Infiniband

10. Challenges awaits me Symbolic computation  Map algebra to computer Data structures  How to represent data efficiently Many similar block Wiedemann algorithms