Variational algorithms for linear algebra

Fast Algorithms for Linear Algebra Modulo N

Many linear algebra problems over the ring ZZN of integers modulo N can be solved by transforming via elementary row operations an n ⇥ m input matrix A to Howell form H. The nonzero rows of H give a canonical set of generators for the submodule of (ZZN ) m generated by the rows of A. In this paper we present an algorithm to recover H together with an invertible transformation matrix P which sat...

Parallel algorithms in linear algebra

This paper provides an introduction to algorithms for fundamental linear algebra problems on various parallel computer architectures, with the emphasis on distributed-memory MIMD machines. To illustrate the basic concepts and key issues, we consider the problem of parallel solution of a nonsingular linear system by Gaussian elimination with partial pivoting. This problem has come to be regarded...

Performance of CAP-Specified Linear Algebra Algorithms

The traditional approach to the parallelization of linear algebra algorithms such as matrix multiplication and LU factorization calls for static allocation of matrix blocks to processing elements (PEs). Such algorithms suffer from two drawbacks : they are very sensitive to load imbalances between PEs and they make it difficult to take advantage of pipelining opportunities. This paper describes ...

Numerical Linear Algebra Algorithms and Software

Randomized algorithms in numerical linear algebra

This survey provides an introduction to the use of randomization in the design of fast algorithms for numerical linear algebra. These algorithms typically examine only a subset of the input to solve basic problems approximately, including matrix multiplication, regression and low-rank approximation. The survey describes the key ideas and gives complete proofs of the main results in the field. A...