Department of Mathematics
University of Houston
Numerical Analysis I
Dr. Ronald H.W. Hoppe
Numerical Mathematics I
(8. Homework Assignment)
Exercise 27
(
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 28
(
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
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 Spring '12
 Staff
 Numerical Analysis, Newton’s method, newton iteration

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