Granularity edit

Floating point numbers, because they are comprised of only a certain number of bits, have a granularity. They therefore cannot express an infinite amount of fractional values. This means that there is a "Largest Possible Value", ε, that satisfies the following equation:


This value is called the machine epsilon of the floating point system.

Epsilon (ε) edit

When a real number is rounded to the nearest floating point number, the machine epsilon forms an upper bound on the relative error. This fact makes the machine epsilon extremely useful in determining the convergence tolerances of many iterative numerical algorithms.

Determining Epsilon edit

The machine epsilon[1] can be computed according to the formula:


So, for IEEE 754 single precision we have


and for IEEE 754 double precision we have


When   is not known, but   is known to be 2, the machine epsilon can be found by starting with a trial value of, say, 0.5 and successively dividing the value by 2 until   is true.

Granularity Effects edit

An effect of this granularity is that some basic algebraic properties don't strictly hold. For instance, if we have three floating-point values, x, y, and z, we can show that:


When the floating-point numbers are used in iterative calculations, round-off and granularity errors can result in large errors.

References edit

  1. Machine epsilon in wikipedia