Exercises

1. An application running on processors consumes random numbers per second per processor. For how long can the application execute before it exhausts all available numbers, assuming that the leapfrog method is applied to the linear congruential generator of Equation 10.3 on a 32-bit machine? A 64-bit machine?

2. A Monte Carlo simulation must perform independent trials. How many random numbers can be employed in each trial without duplication, assuming that the modified leapfrog method is applied to the linear congruential generator of Equation 10.1 on a 32-bit machine? A 64 bit machine?

3. Monte Carlo integration techniques are sometimes to compute integrals of high dimension. They approximate the r-dimensional integral

   

   of a function f as

   

   where each is an r -vector of random values. Design and implement a parallel algorithm for this method, and use it to compute the one-dimensional integral

   

   Measure and account for the performance of the parallel program as a function of N and P.