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IV.21
Numerical Analysis
Lloyd N. Trefethen
1
The Need for Numerical Computation
Everyone knows that when scientists and engineers
need numerical answers to mathematical problems,
they turn to computers. Nevertheless, there is a wide-
spread misconception about this process.
The power of numbers has been extraordinary. It is
often noted that the scientiFc revolution was set in
motion when Galileo and others made it a principle
that everything must be measured. Numerical measure-
ments led to physical laws expressed mathematically,
and, in the remarkable cycle whose fruits are all around
us, Fner measurements led to reFned laws, which in
turn led to better technology and still Fner measure-
ments. The day has long since passed when an advance
in the physical sciences could be achieved, or a signiF-
cant engineering product developed, without numerical
mathematics.
Computers certainly play a part in this story, yet
there is a misunderstanding about what their role is.
Many people imagine that scientists and mathemati-
cians generate formulas, and then, by inserting num-
bers into these formulas, computers grind out the nec-
essary results. The reality is nothing like this. What
really goes on is a far more interesting process of exe-
cution of
algorithms
. In most cases the job could not
be done even in principle by formulas, for most mathe-
matical problems cannot be solved by a Fnite sequence
of elementary operations. What happens instead is
that fast algorithms quickly converge to “approximate”
answers that are accurate to three or ten digits of pre-
cision, or a hundred. ±or a scientiFc or engineering
application, such an answer may be as good as exact.
We can illustrate the complexities of exact versus
approximate solutions by an elementary example. Sup-
pose we have one polynomial of degree 4,
p(z)
=
c
0
+
c
1
z
+
c
2
z
2
+
c
3
z
3
+
c
4
z
4
,
and another of degree 5,
q(z)
=
d
0
+
d
1
z
+
d
2
z
2
+
d
3
z
3
+
d
4
z
4
+
d
5
z
5
.