Approaching Optimality for Solving SDD Linear Systems

Ioannis Koutis, Gary L. Miller, Richard Peng
2014 SIAM journal on computing (Print)  
We present an algorithm that on input of an n-vertex m-edge weighted graph G and a value k, produces an incremental sparsifierĜ with n−1+m/k edges, such that the condition number of G withĜ is bounded above byÕ(k log 2 n), with probability 1 − p. The algorithm runs in timẽ O((m log n + n log 2 n) log(1/p)). As a result, we obtain an algorithm that on input of an n × n symmetric diagonally dominant matrix A with m non-zero entries and a vector b, computes a vector x satisfying ||x − A + b||A <
more » ... A + b||A, in expected timẽ O(m log 2 n log(1/ )). The solver is based on repeated applications of the incremental sparsifier that produces a chain of graphs which is then used as input to a recursive preconditioned Chebyshev iteration.
doi:10.1137/110845914 fatcat:onrjx7drjzcjnbfmvtahmjvziy