filter function not linear in numerator coeffcients

Why is the built-in "filter" function not linear in the numerator coefficients?

b = [1.000000000000000
 0.156960680127524
-0.008820403825717];
a = [1.000000000000000
-1.829126729842689
 0.834081514068486];
x = [1; zeros(250,1)];
scale = 0.941;
result1 = filter(scale*b,a,x);
result2 = scale*filter(b,a,x);
result1-result2

My intuition is that it should make no difference whether one scales the impulse b (as in result1) or the response (as in result2). Numerically there is, however, if only tiny.

Background: I am computing the MA coefficients for an ARMA process and want to test different impulses.

Any hints greatly appreciated!

Answer 1

[Jan]

This difference between the solutions is tiny:

max(abs(result1 - result2))
>> 1.1546e-14

This is an expected effect for computations with numbers stored in floating point format with limited precision. See: Why is 0.3-0.2 not equal 0.1.

Search in the net for "IEE754", which defines the standard method of the type double. You will find many explanations of the expected effects.