Interval Arithmetic, Part I
Show that under the assumption of small percentage tolerances there is a single formula for the approximate percentage tolerance of the product of two intervals in terms of the tolerances of the factors. You may simplify the problem by assuming that all numbers are positive.
Suppose we have two intervals, spanning intervals which are completely positive.
Let the first interval be centered at "x", with width "a".
Let the second interval be centered at "y", with width "b".
(define i1 (make-center-width 'x 'a))
(define i2 (make-center-width 'y 'b))The intervals look like the following:
Since all numbers are positive, the product of the two intervals is given by:
Let's try to factor the product "xy" out of the last three terms:
We can define the relative percentage errors as follows:
It is given that these percentages are small, i.e., much less than 1 and close to 0.
The product can be rewritten as :
Since the percentages are small, we can neglect p1*p2, which is second order.
We are left with (the approximate relation):
