 
Summary: PROPAGATION OF ERROR
Suppose we are evaluating a function f(x) in the ma
chine. Then the result is generally not f(x), but rather
an approximate of it which we denote by ef(x). Now
suppose that we have a number xA xT . We want
to calculate f(xT ), but instead we evaluate ef(xA).
What can we say about the error in this latter com
puted quantity?
f(xT ) ef(xA) = [f(xT )  f(xA)]+
h
f(xA)  ef(xA)
i
The quantity f(xA)  ef(xA) is the "noise" in the
evaluation of f(xA) in the computer, and we will
return later to some discussion of it. The quantity
f(xT )  f(xA) is called the propagated error; and it
is the error that results from using perfect arithmetic
in the evaluation of the function.
If the function f(x) is differentiable, then we can use
the "meanvalue theorem" to write
