Department of Mathematics
University of Houston
Numerical Analysis I
Dr. Ronald H.W. Hoppe
Numerical Mathematics I
(6th Homework Assignment)
Exercise 21
(
Regula falsi of higher order
)
Assume that
f
∈
C
1
([
a,c
))
,
0
< a < c,
has a simple zero
x
*
∈
[
a,c
] and that
f
(
a
)
f
(
c
)
<
0. Let
b
such that
a < b < c
and consider the quadratic polynomial
p
∈
P
2
(lR) interpolating
f
in
a,b,c
, i.e.
p
(
z
) =
f
(
z
)
,z
∈ {
a,b,c
}
.
(i) Show that
p
has exactly one zero in [
a,c
].
(ii) Denote by
REAL zero(
a,b,c,f
)
the function which computes the zero of
p
for given
a,b,c
. Using this function,
construct an iterative algorithm which provides an approximation of
x
*
up to
machine accuracy eps.
Exercise 22
(
Newton’s method for the computation of the
m
th root
)
(i) For
m
∈
lN, the positive
m
th root
x
*
>
0 of
a >
0 is the solution of the
equation
f
(
x
) = 0 where
f
(
x
) :=
x
m

a .
Assume
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 '12
 Staff
 Numerical Analysis, Newton’s method, Rootfinding algorithm

Click to edit the document details