What happens when numbers get very small?

Hi

My computation works with very small physical quantities. I have numbers in the order of 10^-14 which will be multiplied with numbers in about the order of 10^3 to 10^-8. Typing "eps" on the console results in something around 10^-16.

Does Matlab automatically adjust precision to handle these extremely small numbers? Do I have to worry about quantization errors when numbers get small? What about performance issues?

(Basically, I could also change the units of physical constants to bigger values, but I'd like to avoid this.)

Best regards, Michael

Answer 1

[Iain]

Matlab simply uses standard double format numbers.

This gives you about 15 significant figures of precision from 10^-300 to 10^300 -ish.

When multiplying "a * b" you're likely to get quantisation errors in the order of: eps(a*b), eps(a) * b or eps(b)*a.

You should worry about these quantisations errors when you add large numbers of small values to bigger values (like the obvious way to numerically integrate).

Answer 2

[Andreas Goser]

In most applications I am aware of, one simply uses scaling like multiplying each datapoint by 1e12 etc. In my experience very small numbers are not a problem per se - only in situations where there very different order of magnitudes in ONE matrix or there are operation with data with very different order of magnitudes.