Progress Report—11/13 宗慶. Problem Statement Find kernels of large and sparse linear systems over GF(2)

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Presentation transcript:

Progress Report—11/13 宗慶

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

Well-Known Solutions Block Lanczos Algorithm Block Wiedemann Algorithm

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.

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

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

Block Wiedemann Algorithm

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

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

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