HP 15c - Manual (Page 157)

Appendix: Accuracy of Numerical Calculations 157
It all seems like much ado about very little. After a blizzard of formulas and examples, we
conclude that the error caused by p ≠ π is negligible for engineering purposes, so we need not
have bothered to know about it. That is the burden that conscientious error analysts must
bear; if they merely took for granted that small errors are negligible, they might be wrong.
Backward Error Analysis
Until the late 1950's, most computer experts inclined to paranoia in their assessments of the
damage done to numerical computations by rounding errors. To justify their paranoia, they
could cite published error analyses like the one from which a famous scientist concluded that
matrices as large as 40 × 40 were almost certainly impossible to invert numerically in the
face of roundoff. However, by the mid 1960's matrices as large as 100×100 were being
inverted routinely, and nowadays equations with hundreds of thousands of unknowns are
being solved during geodetic calculations worldwide. How can we reconcile these
accomplishments with the fact that that famous scientist's mathematical analysis was quite
correct?
We understand better now than then why different formulas to calculate the same result
might differ utterly in their degradation by rounding errors. For instance, we understand why
the normal equations belonging to certain least-squares problems can be solved only in
arithmetic carrying extravagantly high precision; this is what that famous scientist actually
proved. We also know new procedures (one is presented on page 118) that can solve the
same least-squares problems without carrying much more precision than suffices to represent
the data. The new and better numerical procedures are not obvious, and might never have
been found but for new and better techniques of error analysis by which we have learned to
distinguish formulas that are hypersensitive to rounding errors from formulas that aren't. One
of the new (in 1957) techniques is now called "backward error analysis," and you have
already seen it in action twice: first, it explained why the procedure that calculates λ(x) is
accurate enough to dispel the inaccuracy in example 2; next, it explained why the calculator's
Æ functions very nearly satisfy the same identities as are satisfied by trig functions even
for huge radian arguments x at which Æ(x) and trig(x) can be very different. The
following paragraphs explain backward error analysis itself in general terms.
Consider some system F intended to transform an input x into an output y = f(x). For
instance, F could be a signal amplifier, a filter, a transducer, a control system, a refinery, a
country's economy, a computer program, or a calculator. The input and output need not be
numbers; they could be sets of numbers or matrices or anything else quantitative. Were the
input x to be contaminated by noise Δx,
then in consequence the output y + Δy = f(x + Δx) would generally be contaminated by noise
Δy = f(x + Δx) − f(x).