Dorothy Goettler
NAE – HW 8
Problem 8.4.1
The Pade approximations of degree two for the given function are determined by first finding the
Maclaurin series expansion for the function. This is combined with the appropriate polynomial series of constants p and q as illustrated in the textbook and the resulting expression in expanded. Finally, the resulting linear system of equation is solved using partial pivoting and the values of the coefficients of p and q are output. m

2 , n

0 : 1

2 x

2 x
2 m

1 , n

1 : ( 1
 x ) /( 1
 x ) m

0 , n

2 : ( 1

2 x

2 x
2
)

1 i
1
2
3 x i
0.2
0.4
0.6 f(x i
)
1.4918
2.2255
3.3201 r
2,0
(x i
)
1.48
2.12
2.92 r
1,1
(x i
)
1.50
2.33
4.0 r
0,2
(x i
)
1.4706
1.9231
1.9231
4
5
0.8
1.0
4.9530
7.3891
3.88
5.0
9.0
-
1.4706
1.0
Problem 8.4.9
The Chebyshev rational approximations of degree two for the given function are determined using a method similar to that used for the Pade approximation above. The only difference is that each x k term in the Pade approximation is replaced by the k th degree Chebyshev polynomial T k
(x) . As seen in the table below, r
T1,1
(x i
) yields the best approximation to f(x) for x = 0.25, 0.5, and 1.0. r
T2,0
(x) = (1.266066 T
0
(x) – 1.130318 T
1
(x) + 0.2714953 T
2
(x)) / T
0
(x) r
T1,1
(x) = (0.9945705 T
0
(x) – 0.4569046 T
1
(x)) / (T
0
(x)) + 0.48038745 T
1
(x)) r
T0,2
(x) = 0.7940220 T
0
(x) / (T
0
(x)) + 0.8778575 T
1
(x) + 0.1774266 T
2
(x)) i
1
2
3 x i
0.25
0.50
1.0 f(x i
)
0.77880078
0.60653066
0.36787944 r
T2,0
(x i
)
0.74592811
0.56515935
0.40724330 r
T1,1
(x i
)
0.78595377
0.61774075
0.36319269 r
T0,2
(x i
)
0.74610974
0.58807059
0.38633199
Problem 8.6.3d
Algorithm 8.3 was used to compute the trigonometric interpolating polynomial of degree four for the given function. This method consists of organizing the problem so that the number of data points being used can be easily factored, particularly into factors of two.
S
4
(x) = -0.1526819 + 0.04754278 cos x + 0.6862114 cos 2x – 1.216913 cos 3x + 1.176143 cos 4x –
0.8179387 sin x + 0.1802450 sin 2x + 0.2753402 sin 3x