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(b) Set x
0
D
1 and compute x
1
;:::;x
4
in Newton's method to solve
g.x/
D
0:
(4.183)
(c) You have observed that Newton's method converges much faster for (
4.182
)than
for (
4.183
). Use the graphical interpretation of Newton's method, see Fig.
4.5
on
page
112
, to try to understand the convergence behavior for these two problems.
(d) Consider
h.x/
D
x
6
:
Set x
0
D
1 and compute x
1
;:::;x
4
in Newton's method for solving
h.x/
D
0:
Discuss the convergence speed along the lines suggested in (c).
˘
Exercise 4.4.
Consider the system
e
y
x
D
1;
x
2
y
D
0:
(a) Show that x
D
y
D
0 solves the system.
(b) Set x
0
D
y
0
D
1=2 and compute .x
1
;y
1
/ and .x
2
;y
2
/ in Newton's method.
˘
Exercise 4.5.
Consider the ODE
u
0
D
e
u
;
u
.0/
D
0:
(4.184)
(a) Derive and implement an explicit Euler scheme for (
4.184
).
(b) Derive an implicit Euler scheme for (
4.184
). Use Newton's method to solve
the nonlinear equation arising at each time level. Implement your scheme on a
computer.
(c) Set t
D
1=100 and compute numerical solutions from t
D
0 to t
D
1,using
the programs developed in (a) and (b).
(d) Show that
u
.t /
D
ln.1
C
t/
solves (
4.184
). Use this to check the accuracy of the schemes discussed above.
(e) Consider the scheme
u
nC1
u
n
t
1
2
.e
u
n
C
e
u
nC1
/:
D
