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HW#5
EGM6341
From textbook by Atkinson
pp185194
#6,
The total error e(x) is the sum of the truncation error TE(x) and the roundoff error R(x)
)
(
)
(
)
(
x
R
x
TE
x
+
=
ε
The truncation error is given by
)
)(
)(
(
"
2
1
)
(
1
0
x
x
x
x
f
x
TE


=
ξ
by for some
ξ
between x
0
and x
i
.
Note:
max 
4
/

)
)(
(

2
1
0
h
x
x
x
x
=


,
h=


0
1
x
x

, and max
)
(
"
f
 = max sin
ξ
 = 1.
The maximum roundoff error is
Max R(x) = 5
x
10
7
Thus we need
)
(
)
(
)
(
x
R
x
TE
x
+
=
<
8
/
2
h
+ 5
x
10
7
< 10
6
which gives
h < 2
x
10
3
.
#11
Let z = e
x
.
Then the polynomial becomes
)
(
)
(
0
0
z
p
z
c
e
c
x
p
n
j
j
n
j
jx
j
n
j
n
=
∑
=
∑
=
=
=
…(the rest of the proof follows naturally).
#18.
f
x
D
x
D
2
x
D
3
x
D
4
x
D
5
x
Linear
Quad
Cubic
4th
order
5th
order
0.22389
1
2
1.74569
0.2840
7
0.7793
0.7648
1.672
2.39084
2.4014
4
2.4046
5
2.40482
5
2.404825
6
0.16660
7
2.1
1.77794
0.4152
8
0.9487
1.2202
2.787
2.39622
2.4038
5
2.4048
2
2.40482
6
2.404825
5
0.11036
2
2.2
1.82407
0.5709
6
1.2110
1.9545
4.946
2.40131
2.4048
1
2.4048
3
2.40482
5
2.404825
7
0.05554
2.3
1.88565
0.7632
1
1.6159
3.2050
9.336
2.40473
2.4048
3
2.4048
2
2.40482
6
2.404825
2
0.00250
8
2.4
1.96497
1.0093
9
2.2505
5.4511
18.826
2.40493
2.4048
1
2.4048
3
2.40482
2
0.04838
2.5
2.06521
1.3356
2
3.2728
9.7211
2.40008
2.4063
3
2.4041
5
2.40535
0.0968
2.6
2.19085
1.7828
6
4.9830
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2.7
2.34815
2.4182
4
0.18504
2.8
2.54612
0.22431
2.9
D
k
x= kth
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This note was uploaded on 04/23/2009 for the course EGM 6341 taught by Professor Mei during the Spring '09 term at University of Florida.
 Spring '09
 MEI

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