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Derive Newton's method for solving the nonlinear equation arising at each time
level. Implement the scheme and discuss its accuracy and compare the results
with those obtained by the implicit and explicit Euler schemes.
(f) Redo (e) using the scheme
u
nC1
u
n
t
D
e
2
.
u
nC1
C
u
n
/
:
˘
Exercise 4.6.
Suppose you have a number c and you want to find its reciprocal 1=c.
Can you compute that number only by applying multiplication? In order to do this
we define
1
x
f.x/
D
c
and seek a solution of
f.x/
D
0:
(4.185)
(a) Show that Newton's method for (
4.185
) can be written in the form
x
kC1
D
.2
cx
k
/x
k
:
(4.186)
(b) Set c
D
4, x
0
D
0:2 and compute x
1
;:::;x
4
by (
4.186
). Comment the accuracy
of the scheme.
˘
4.8
Project: Convergence of Newton's Method
There are basically three things you need to know about the convergence of Newton's
method:
(a) Usually, the method converges
8
very fast, typically in just a few iterations.
(b) Sometimes the convergence is slow.
(c) From time to time, the method does not converge at all.
It is possible to derive sharp results on how fast and in which cases Newton's
method converges, see e.g. Stoer and Bulirsch [27]. These results, however, tend to
8
This is not completely precise. We should write something like “the sequence of approximations
generated by Newton's method usually converges very fast”. The method does not converge, how-
ever; it is the numbers generated by the method that, hopefully, converge. But that is really a whole
lot of words, so we will sacrifice precision for clarity and readability.
