Analysis of Parallel Algorithms for Matrix Chain Product and Matrix Powers on Distributed Memory Systems

Given N matrices A
₁
, A
_2,
...,A
_N
of size NtimesN, the matrix chain product problem is to compute A
₁
timesA
₂
times...timesA
_N
. Given an NtimesN matrix A, the matrix powers problem is to calculate the first N powers of A, that is, A, A
²
, A
³
,..., A
^N
. We solve the two problems on distributed memory systems (DMSs) with p processors that can support one-to-one communications in T(p) time. Assume that the fastest sequential matrix multiplication algorithm has time complexity O(N
^alpha
), where the currently best value of a is less than 2.3755. Let p be arbitrarily chosen in the range 1lesplesN
^alpha+1
/(log N)
²
. We show that the two problems can be solved by a DMS with p processors in T
_chain
(N,p)=O((N
^alpha+1
/p)+T(p))((N
^2(2+1/alpha
/p
^2/alpha
)(log+p/N)
^1-2/alpha
+log+((p log N)/N
^alpha
)) and T
_power
(N,p)=O(N
^alpha+1
/p+T(p)((N
^2(1+1/alpha)
/p
^2/alpha
)(log+p/2 log N)
^1-2/alpha
+(log N)
²
))) times, respectively, where the function log+ is defined as follows: log+ x=log x if xges1 and log+ x=1 if 0