For numerical calculations, the computer is limited by two major constraints: the amount of memory available and the speed of the processor. Let’s examine how these constraints affect our ability to perform Gauss elimination on an old Mac. The Mac SE/30s was equipped with 2 MB (megabytes) of Random Access Memory (RAM).

A fairly large part of it is occupied by the operating system and the QuickBASIC application. Let’s assume that 1 MB is available for our use.

(a) What is the largest system of equations Ax = b that we could solve in double precision on the Mac according to our memory limitation? Assume that we only need to fit the matrix A (n² numbers) and the vector b (n numbers) into memory.

1 MB = 1 megabyte = 1024 KB

1 KB = 1 kilobyte = 1024 bytes

1 byte = 8 bits

1 bit holds one binary piece of information (0 or 1)

1 double precision variable requires 64 bits

(b) Repeat this calculation for the Cray-2 supercomputer, which has 512 megawords (MW) of memory. Assume that 500 MW is available for our use.

1 MW = 1 megaword = 1024 × 1024 words

1 word = 64 bits

(Notice that, on the Cray, the default variable size is double precision.)

(c) What is the largest system of equations Ax = b that we could reasonably solve on a Mac SE/30 if computer speed is our only limitation?

Assume that we will be using Gauss elimination and that “reasonable” means in less than one hour.

A rough estimate of the speed of a compiled program on a Mac SE/30 is 0.01

MFLOPS (for double precision).

1 MFLOPS = 1 million floating-point operations per second.

Gauss elimination takes approximately n³/3 operations.

(d) Repeat this calculation for the Cray-2 supercomputer, if its nominal speed is assumed to be 100 MFLOPS.

(e) Based on your answers to (a) through (d), what is the limiting factor for Gauss elimination on a Mac? On a Cray-2?