MATLAB: Matlab floating point precision.

matlab floating point precision

My question pertains to matlab precision. I encountered this problem I have a long time series of 549354 record. and I am looking at values the end point and a point somewhere in the beginning.
First point Tcum(64996) and the second point is Tcum(549354)
Q1 = Tcum(64996) which is actually = 0.048229777756042
Q2 = Tcum(549354) which is actually = 1.060933335844769e+05
X2 = Q2-Q1;
Check = Q2-X2;
Now I expected Check to be identically equal to Q1; these are the values I get.
Check =  0.048229777748929
Q1    =  0.048229777756042
They are clearly different. Is this behaviour normal and is a result of dealing with largely varying magnitude in numbers? Or am i doing something wrong here?

Best Answer

Never trust the least significant bits of a floating point number, at least unless you know enough about the extent that you can trust them. But in general, don't.
A double precision floating point number carries roughly 16 digits, actually 52 binary bits of precision.
What happens when you add two numbers that are of different magnitude?
X0 = 1/3
X0 =
         0.333333333333333
So X0 is 1/3. Not exactly because we cannot represent the fraction 1/3 exactly in binary. But that is not really pertinent here.
sprintf('%0.55f',X0)
ans =
    '0.3333333333333333148296162562473909929394721984863281250'
Suppose we add and subtract delta?
delta = 100000.1
delta =
                  100000.1
X1 = X0 + delta - delta
X1 =
       0.333333333328483
What happened? We added a number that was more than 1e5 times as large as X0. Stored in a double, that means we lose the bottom 5 digits of the sum. Subtracting delta off again, does not help, because we already lost that information. Once information is lost, it cannot be magically regained.
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