Parallelized solution of banded linear systems with an introduction to p-adic computation

Abstract

We present an approach that supports a parallelized solution of banded linear systems without communication between processors. We do this by adding unknowns to the system equal to the number of superdiagonals q. We then perform r forward substitution processes in parallel (where r is the number of nonzero terms in the right-hand side vector), and superimpose the resulting solution vectors. This leads to the determination of the extra unknowns, and by extension, to the overall solution. However, some systems exhibit exponential growth behavior during the forward substitution process, which prevents the approach from working. We present several modifications to address this, extending the approach (in a modified form) to be used for general systems. We also extend it to block banded systems. Numerical results for well-behaved test systems show a speedup of 20-80 over conventional solvers using only 8 processors. Theoretical estimates assuming q processors demonstrated a speedup of a factor exceeding 300 for 105 unknowns when q D 2000; for 109 unknowns, the speedup exceeds a factor of 104 when q D 45;000. We also introduce some fundamentals of p-adic computation and modular arithmetic as the basis of the development and implementation of a fully parallel p-adic linear solver, which allows error-free computation over the rational numbers, and is better suited to control coefficient growth.