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The estimate (
4.216
) states that when the iterates stay in a region where f
0
is non-
zero and f
00
is bounded- (
4.210
)and(
4.211
) - then the convergence is quadratic.
We have now concluded the discussion of quadratic convergence; we claimed ini-
tially that the convergence is usually quadratic, and you can now understand roughly
when this is the case.
Next, we will discuss slow convergence. We see from (
4.215
)thatf
0
.x
/
D
0 is
a problem. This was already discussed in Exercise
4.3
above. But we also note from
(
4.215
)thatiff
00
.x
/ is zero, we can expect faster convergence. In order to study
these two effects, we define two functions
f.x/
D
cosh.x/
1;
(4.217)
g.x/
D
sinh.x/;
(4.218)
and we want to use Newton's method to solve
f.x/
D
0
(4.219)
and
g.x/
D
0:
(4.220)
Since
cosh.0/
D
1
(4.221)
and
sinh.0/
D
0;
(4.222)
we have x
D
0 for both (
4.219
)and(
4.220
). Let us also recall that
d
dx
cosh.x/
D
sinh.x/
(4.223)
and
d
dx
sinh.x/
D
cosh.x/:
(4.224)
(q) Sketch the functions f.x/ and g.x/ for x
2
I
D
Œ
1; 1.
(r) Set x
0
D
1 and use the graphical illustration of Newton's method, see page
112
,
to explain that you expect fast convergence for (
4.218
) and slow convergence
for (
4.217
).
(s) Verify that
