MMAE 350
D. Rempfer
Problem Set VI
Page 1 of 3
Due Date: November 06, 2006
1.
You are given the following set of (
x, y
) pairs of values, describing (errorprone) measurements
of some unknown function
f
(
x
):
x
i
1
0.8
0.6
0.4
0.2
0
0.2
0.4
0.6
0.8
1.0
y
i
1.0
1.8
2.5
3.0
2.8
3.1
3.1
2.5
2.4
2.0
0.8
Find a leastsquares approximation of the function
f
(
x
) that is of the form
f
(
x
)
≈
a
0
log
(
1
+
x
2
)
+
a
1
exp
(

x
2
)
+
a
2
x
2
.
(1)
2.
Calculate the coe
ffi
cients of the Fourier expansion
˜
f
(
t
)
=
a
0
+
5
j
=
1
a
j
cos(2
πj t
)
+
5
i
=
j
b
j
sin(2
πj t
)
for the following two functions:
a.
f
(
t
)
=
1
2

1
2
t,
x
∈
[0
,
1)
b.
f
(
t
)
=
1
2
t

1
2
t
2
,
x
∈
[0
,
1]
For each of the above, plot both the periodic extension of
f
and
˜
f
in the interval [0
,
2]. Note
that the integrals that give these coe
ffi
cients can be solved in closed form, so you do not have to
integrate numerically.
3.
You are given the task to derive an integration formula of NewtonCotes type that integrates a
function that is given at the three nonequidistant points
x
0
,
x
1
=
x
0
+
2
h
, and
x
2
=
x
1
+
3
h
.
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 '05
 rempher
 Numerical Analysis, D. Rempfer

Click to edit the document details