The function eps in MATLAB, which stands for epsilon, is used to determine the relative accuracy of the floating-point arithmetic of the system you're using. More specifically, eps(x) returns the distance between x and the next larger in magnitude floating-point number of the same precision as x. This value represents the smallest difference that the system can distinguish from x given its floating-point arithmetic system.
eps(2):
- The result is approximately 4.440892098500626e-016.
- This means that in the floating-point representation of numbers, the smallest value that can be added to 2 to get a distinguishably different number (according to the computer's arithmetic system) is about 4.44e-16. Any number smaller than this added to 2 would still be considered 2 by the system due to the limits of precision.
eps(20):
- The result is approximately 3.552713678800501e-015.
- Similarly, for the number 20, the smallest increment that can be recognized by the system is about 3.55e-15. Again, this is the smallest difference from 20 that the system can recognize as a different number.
eps(2000000000) (2 billion):
- The result is approximately 2.384185791015625e-007.
- For much larger numbers, like 2 billion, the smallest distinguishable increment becomes larger, in this case, about 2.38e-7. This is because floating-point precision is relative; as numbers get larger, the precision in terms of absolute difference also becomes larger.
The precision or epsilon value depends on the magnitude of the number you're looking at because of how floating-point numbers are stored in computers. Computers use a finite number of bits to represent real numbers, leading to a trade-off between the range of values that can be represented and the precision of those values.
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