Fast and scalable parallel algorithms for matrix chain product and matrix powers on distributed memory systems

Given N matrices A₁, A₂,…, A_N of size N×N, the matrix chain product problem is to compute A₁×A₂×···A_N . Given an N×N matrix A, the matrix powers problem is to calculate the first N powers of A, i.e., A, A²A³,…, A^N. We consider distributed memory systems (DMS) with p processors that can support one-to-one communications in O(T(p)) time. Assume that the time complexity of the best known sequential algorithm for matrix multiplication is O(N^α), where α<2.3755. Let p be arbitrarily chosen in the range 1⩽p⩽N^α+1/log N. We show that the two problems can be solved on a p-processor DMS in T _chain(N,p)=O(N^α+1/p+T(p)(N ^2(1+1/α)/p²/ ^α(log p/N)^1-2α/+log(p log N/N^α) log N)) and T_power(N,p)=0(N^α+1/p+T(p)(N ^2(1+1/α)/p²/ ^α(log p/log N) ^1-2α/+(log N)²)) times, respectively. We also give instantiation of the above results in distributed memory parallel computers and DMS with hypercubic networks, and show that our parallel algorithms are either fully scalable or highly scalable