Really big problems in scientific computing are solved by coupling several computers together in a network. These computers have to exchange information during the computation. The overall computing...

Really big problems in scientific computing are solved by coupling several computers together in a network. These computers have to exchange information during the computation. The overall computing time thus depends on how fast each computer does its job and how fast they are able to communicate with each other. In this exercise, we will discuss a model of the communication speed. Let us assume that we want to exchange vectors of real numbers (stored in double precision) between a computer A and a computer B. The vector y has n entries. Let T = T_N
be the time needed to send the vector y from A to B. Actually, we send y first from A to B and then back to A. The cost of the “one-way” communication can be found by dividing the “round-trip” time by a factor 2. In this way we only need to do time measurement on one computer, say A. In the table below we have listed values of n and the communication time Tn.

(a)    Construct a linear least squares model of the form

(b) Try to explain why T (O)=

 (c) In computer terms, ˛ is referred to as the latency of the communication network, and 1/βcan be regarded as the practical bandwidth, which should be somewhat lower than the peak-performance bandwidth. Normally, bandwidth is measured in gigabits (109) per second. Use the fact that one double-precision number has 64 bits and report the bandwidth of the network that has been used for producing the measurements given in Table 5.6.

Table 5.6.