The Function eps :: Nondouble Data Types (Mathematics)

Mathematics

The Function eps

Because there are only finitely many double-precision numbers, you cannot represent all numbers in double-precision storage. On any computer, there is a small gap between each double-precision number and the next larger double-precision number. You can determine the size of this gap, which limits the precision of your results, using the eps function. For example, to find the distance between 5 and the next larger double-precision number, enter

format long
eps(5)

ans =

    8.881784197001252e-016

This tells you that there are no double-precision numbers between 5 and 5 + eps(5). If a double-precision computation returns the answer 5, the result is only accurate to within eps(5).

The value of eps(x) depends on x: as x gets larger, so does eps(x). For example,

eps(50)

ans =

    7.105427357601002e-015

so that eps(50) is larger than eps(5).

If you enter eps with no input argument, MATLAB returns the value of eps(1), the distance from 1 to the next larger double-precision number.

Similarly, there are gaps between any two single-precision numbers. If x has type single, eps(x) returns the distance between x and the next larger single-precision number. For example,

```
x = single(5);
eps(x)
```

returns

```
ans =

  4.7684e-007
```

Note that this result is larger than eps(5). Because there are fewer single-precision numbers than double-precision numbers, the gaps between the single-precision numbers are larger than the gaps between double-precision numbers. This means that results in single-precision arithmetic are less precise than in double-precision.

For a number x of type double, eps(single(x)) gives you an upper bound for the amount that x is rounded when you convert it from double to single. For example, when you convert the double-precision number 3.14 to single, it is rounded by

double(single(3.14) - 3.14


ans =

  1.0490e-007

The amount that 3.14 is rounded is less than

eps(single(3.14))

ans =

  2.3842e-007

Single-Precision Arithmetic Example -- Writing M-Files for Different Data Types