How to interpretate numbers below de MATLAB precision

How to interpretate numbers below 2.2204e-16?

Answer 1

[John D'Errico]

No problem.

A = 1e-250
A =
     1e-250

MATLAB has no problem with small numbers. Well, there is such a thing as too small.

Perhaps your question REALLY is how to work in higher precision than a double. For that you need to use either the symbolic toolbox, of my own HPF toolbox. Either will suffice, although they will be slower than working with doubles.

[Fidel Souza]

Thank you for your answer. I'm calculating a determinant of a matrix and the result is too small. I'm seeing that for values below 1e-324 the MATLAB returns zero. These HPF toolbox it's yours or it's a MATLAB toolbox?

[per isakson]

Search the File Exchange for the HPF Toolbox

[James Tursa]

If you are seeing numbers below 1e-324 I would seriously examine what problem you are solving and how you are going about doing it. Numbers that small are often a red flag that something in your method is wrong.

Answer 2

[Roger Stafford]

You may have a misunderstanding of double precision floating point numbers, Fidel. The number 2.2204e-16 is the value of the least bit for a number that lies between 1 and 2, which is to say it is equal to 2^(-52). However, for smaller numbers, the least bit becomes correspondingly smaller. For example, if a number lies between, say, 1/32 and 1/16, the least bit will have a value of 2^(-57) = 6.9389e-18, representing a more precise value in absolute terms. The basic idea is that double precision (matlab's 'double') numbers have 53 bits to use for their significand (mantissa), so the ratio between the least bit value and the value of the number is always between 2^(-52) and 2^(-53), or roughly sixteen decimal places.

It is only when double numbers are smaller than 2^(-1022) where they become 'denormalized' that they begin to lose the sixteen decimal place precision, and these are exceedingly small numbers.

[Fidel Souza]

Thank you Roger. I undertood.