Element wise multiplication .* rounding problem

Hello,

My name is Robin and I am currently using Matlab to compute some calculations.

But I have an issue. Let's take :

M = 564;
P = sin(5);
disp((M.*10.*P) - (M.*(10.*P)));

The result should be 0. Instead I get :

ans = 9.0949e-13

I understand the gap is very small, but I think Matlab rounding algorithm should finally provide 0.

Can you help me ?

Answer 1

[John D'Errico]

Nope. It does not. Welcome to the wild, whacky wonderful world of floating point numbers and arithmetic. A place where computations are only an approximation to mathematics, a model thereof. Where things like the associative law, the distributtive laws of arithmetic are only approximately true. A place where you need to learn to use tolerances on results, where you need to never trust the least significant bits of a result, at least not until you know, absolutely, because you fully understand the computations, what is happening. And even then, tolerances are still a good safety net, a good idea.

MATLAB does not automatically round everything that you do, expecting that a small number found really should be exactly zero. If it did, it would then introduce potential inaccuracies, even beyond those created by the use of floating point numbers themselves.

And if it did, do you think a chemist would be happy to see the reciprocal of Avogadro's number always be shown as exactly zero?

Answer 2

[madhan ravi]

Second John D’Erricos answer as a workaround if you have symbolic math toolbox:

P=sym(sin(5)); % just alter your line to this

Answer 3

[Robin L.]

Thank you John D'Errico and madhan ravi !

So I consider that no formula is better than the other, between :

(M.*10).*P

and

M.*(10.*P)

Because each result is an approximation, and the gap between both values is finally unsignificant for me.

[John D'Errico]

Please don't add an answer just ot make a comment.

But, yes, the associative law has problems in floating point arithmetic. Both expressions are valid, yet they can yield different results in the least significant bits.