Why are numbers rounded in standard computer algorithms? [closed]

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Closed 11 years ago.

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I understand that this makes the algorithms faster and use less storage space, and that these would have been critical features for software to run on the hardware of previous decades, but is this still an important feature? If the calculations were done with exact rational arithmetic then there would be no rounding errors at all, which would simplify many algorithms as you would no longer have to worry about catastrophic cancellation or anything like that.

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Floating point is much faster than arbitrary-precision and symbolic packages, and 12-16 significant figures is usually plenty for demanding science/engineering applications where non-integral computations are relevant.

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The programming language ABC used rational numbers (x / y where x and y were integers) wherever possible.

Sometimes calculations would become very slow because the numerator and denominator had become very big.

So it turns out that it's a bad idea if you don't put some kind of limit on the numerator and denominator.

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In the vast majority of computations, the size of numbers required to to compute answers exactly would quickly grow beyond the point where computation would be worth the effort, and in many calculations it would grow beyond the point where exact calculation would even be possible. Consider that even running something like like a simple third-order IIR filter for a dozen iterations would require a fraction with thousands of bits in the denominator; running the algorithm for a few thousand iterations (hardly an unusual operation) could require more bits in the denominator than there exist atoms in the universe.

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Many numerical algorithms still require fixed-precision numbers in order to perform well enough. Such calculations can be implemented in hardware because the numbers fit entirely in registers, whereas arbitrary precision calculations must be implemented in software, and there is a massive performance difference between the two. Ask anybody who crunches numbers for a living whether they'd be ok with things running X amount slower, and they probably will say "no that's completely unworkable."

Also, I think you'll find that having arbitrary precision is impractical and even impossible. For example, the number of decimal places can grow fast enough that you'll want to drop some. And then you're back to square one: rounded number problems!

Finally, sometimes the numbers beyond a certain precision do not matter anyway. For example, generally the nnumber of significant digits should reflect the level of experimental uncertainty.

So, which algorithms do you have in mind?

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Traditionally integer arithmetic is easier and cheaper to implement in hardware (uses less space on the die so you can fit more units on there). Especially when you go into the DSP segment this can make a lot of difference.