Homework #4
1.
(a) Starting with the initial guess of 1, find three additional approximations to the
root of
f
(
x
) =
x
2

2 using Newton’s method.
(b) What is the absolute error of the final approximation?
2.
(a) Consider
f
(
x
) =
(
√
x,
x
≥
0

√

x,
x <
0
.
Starting with the initial guess of
x
0
=
a >
0, find three additional approximations
to the root of
f
(
x
) using Newton’s method.
(b) Give a graphical description of how Newton’s method is arriving at the approxi
mations.
(c) Does Newton’s method converge to the exact root at 0 for
x
0
sufficiently close
to 0?
Why does this not violate the theorem on the convergence of Newton’s
method?
3.
(a) Starting with the initial guess of
x
0
= 1, find three additional approximations to
the root of
f
(
x
) = (
x

2)
2
using Newton’s method.
(b) What is the absolute error of the initial guess and each of the three approxima
tions? Note the exact root is
x
*
= 2.
(c) What is

x
*

x
k
+1

/

x
*

x
k

for
k
= 0
,
1
,
2? From this, what do you guess to be
the order of convergence and asymptotic error constant in this case?
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '08
 staff
 Numerical Analysis, Addition, Approximation, initial guess, Rootfinding algorithm

Click to edit the document details