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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 "mean-value theorem" to write
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