AMATH 341 / CM 271 / CS 371
Assignment 5
Due : Wednesday March 10, 2010
Instructor: K. D. Papoulia
1. In this question you will prove the convergence order for Newton’s method. Let
x
∗
be the root of
f
(
x
) = 0 and
x
k
be the
k
th
approximation of this root by Newton’s
method.
Assume that
f
satisfies the conditions of the convergence theorem for
Newton’s method.
(a) Show that the error,
e
k
=
x
k
−
x
∗
, satisfies
e
k
+1
=
e
k
−
f
(
x
∗
+
e
k
)
f
′
(
x
∗
+
e
k
)
.
(1)
(b) Since
f
∈
C
2
, Taylor’s Theorem shows that (1) can be written
e
k
+1
=
e
k
−
f
′
(
x
∗
)
e
k
+
1
2
f
′′
(
θ
k
)
e
2
k
f
′
(
x
∗
) +
f
′′
(
ϕ
k
)
e
k
,
(2)
for some
θ
k
, ϕ
k
between
x
∗
and
x
∗
+
e
k
. Show that (2) can be written

e
k
+1

=
c
k

e
k

2
,
where
c
k
=
f
′′
(
ϕ
k
)
−
1
2
f
′′
(
θ
k
)
f
′
(
x
∗
) +
f
′′
(
ϕ
k
)
e
k
(c) Show that
lim
k
→∞
c
k
=
f
′′
(
x
∗
)
2
f
′
(
x
∗
)
.
Hint: use the fact that lim
k
→∞
x
k
=
x
∗
.
2. Consider the degree4 polynomial
f
(
x
) = (
x
−
2
.
2)(
x
−
2
.
3)(
x
−
2
.
4)(
x
−
2
.
5) con
sidered in an earlier problem set.
For each of the four roots of
f
, find a fixed point iteration in which
g
(
x
) has the
form
x
−
cf
(
x
) for some scalar
c
(may be different for each root) that converges to
that root. Experimentally determine an interval of starting values for each of the
four choices of roots that ensures convergence.
3. (a) Suppose
g
is a continuously differentiable function and
g
(
x
∗
) =
x
∗
.
Suppose
that

g
′
(
x
∗
)

>
1. Show that fixed point iteration based on
g
will never converge to
x
∗
, assuming that no iterate
x
k
happens to exactly be equal to
x
∗
.
(b) Consider the function given by
f
(
x
) =
−
exp(3
x
)
9
−
x
2
2
+
1
9
.
(3)
1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Plot the functions
f
(
x
) and
g
(
x
) =
x
−
f
(
x
) using
ezplot
and provide the resulting
plot, which has a root at
x
∗
= 0. Using fixed point iteration on
g
(
x
) with a suﬃ
ciently small tolerance, create a table of the results you get with the initial condition
x
0
= 1. Why are we unable to achieve convergence? Suggest a different
g
(
x
) and
x
0
that would give us convergence with fixed point iteration. Limit consideration
to
g
(
x
) of the form
x
−
cf
(
x
) for a scalar
c
.
4. Consider the problem of finding a threedimensional 5sided box (i.e., one side is
open) with volume 1 and with minimal surface area, where we count as surface area
only the 5 sides that are present. Write a Matlab program to solve this optimization
problem using Newton’s method for multivariate optimization. Terminate when the
norm of the gradient is less than 10
−
15
.
Terminate also if the method seems to
be diverging, e.g., one variable gets larger than 10 or smaller than 1
/
10. Try some
different starting points to see if the same solution is always obtained. The variables
of the problem are
x
= (
l
;
w
), the length and width of the box. The third linear
dimension is 1
/
(
lw
) due to the volume constraint. Try posing the objective function
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '10
 Papoulia,Katerina
 Numerical Analysis, x∗, Dominated convergence theorem, point iteration

Click to edit the document details