Solutions for Homework 9 Foundations of Computational
Math 1 Fall 2010
Problem 9.1
Let
f
(
x
) :
R
→
R
be given by
f
(
x
) =
x
4
−
5
x
2
+ 4
and consider applying Newton’s method for optimization. Here Newton’s method refers to
the basic form where the step size is 1 and nothing is done to alter the Hessian to guarantee
positive definiteness. Note that
f
(
x
) is a scalar function of a scalar argument and has the
form
5
4
3
2
1
0
1
2
3
4
5
8
6
4
2
0
2
4
6
8
(i) What are the values of
x
that are local minimizers or local maximizers of
f
(
x
).
Justify your answers.
(ii) Find the value
β >
0 such that
f
(
x
) has negative curvature for
−
β < x < β
, and
positive curvature outside the interval, i.e., for
x <
−
β
or
x > β
.
(iii) What happens to the Newton step at
x
=
β
?
(iv) Determine
μ
(
x
) :
R
→
R
such that the step of Newton’s method applied to
f
(
x
)
can be written as
x
k
+1
=
μ
(
x
k
)
x
k
.
(v) Find the value of
α
∈
R
such that
β > α >
0 and Newton’s method cycles
and does not converge when
x
0
=
α
or
x
0
=
−
α
.
That is,
−
α
=
μ
(
α
)
α
and
α
=
−
μ
(
−
α
)
α
.
(vi) Show that if
−
α < x < α
then

μ
(
x
)

<
1
1
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(vii) Show that if
−
α < x
0
< α
is the initial point for Newton’s method then there is
a constant 0
< γ <
1 (possibly dependent on
x
0
but independent of
k
) such that

x
k
+1

< γ

x
k

and therefore
x
k
→
0.
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 Spring '11
 Gallivan
 Multivariable Calculus, General Relativity, xk, Newton’s method

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