Significant figures discrepancy in complex number operations

Attached is my m-file where I take 2 complex numbers X & Y and multiply them by a decaying exponenential, F, & then sum the results. I executed the operation in 2 separate ways: 1) X*F+Y*F; 2) (X+Y)*F;

The results do not concur past 10^-16. You can see this by my workspace variable "alert". Is this due to the precision of significant figures within MATLAB? The 2 approaches should yield the identical results if I am correct.

One might ask why be so concerned past 16 significant figures? What concerns me is that some unforeseen error in computing phase information might crop up. Is there some trick I could implement to avert any unforeseen errors?

Answer 1

[Walter Roberson]

IEEE 754 Double Precision uses (effectively) 53 bits of mantissa, able to reflect relative changes down to 2^(-53) . If you are using values in the approximate range of 1.0, then 2^(-53) of that is 1.1102230246251565404236316680908203125 * 10^(-16) which is not quite as good as 10^(-16) . In the area of 1.0, a single bit difference translates to about 1E-16 change in output.

It is a truism in floating point calculation that the same calculation performed two different ways that are "algebraically equivalent" can give you different results due to differences in rounding. http://matlab.wikia.com/wiki/FAQ#Why_is_0.3_-_0.2_-_0.1_.28or_similar.29_not_equal_to_zero.3F

Also, I seem to recall that some of the hardware instructions are permitted to have an accuracy less than 1 ULP (Unit In The Last Place); the details of the standard would have to be checked for that kind of detail.

If you need higher precision you will need to use a higher precision package such as the Symbolic Toolbox