A note on scaling the Linpack benchmark

Dimensional analysis yields a new scaling formula for the Linpack benchmark. The computational power r(p0,q0) on a set of processors decomposed into a (p0,q0) grid determines the computational power r(p,q) on a set of processors decomposed into a (p,q) grid by the formula r(p,q)=(p/p0)α(q/q0)βr(p0,q0). The two scaling parameters α and β measure interprocessor communication overhead required by the algorithm. A machine that scales perfectly corresponds to α=β=1; a machine that scales not at all corresponds to α=β=0. We have determined the two scaling parameters by imposing a fixed-time constraint on the problem size such that the execution time remains constant as the number of processors changes. Results for a collection of machines confirm that the formula suggested by dimensional analysis is correct. Machines with the same values for these parameters are self-similar. They scale the same way even though the details of their specific hardware and software may be quite different.