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2
ax3 + bx3 + cx3 + dx3 + e = y3 next page (5) close
exit The augmented matrix for system 5 is 3
x 4 x1
1
3
4 x2 x2 4
3 x3 x3 x4 x3
4
4
3
4
x5 x5 2
x1
2
x2
2
x3
2
x4
2
x5 x1
x1
x3
x4
x5 1
1
1
1
1 y1 y2 y3 . y4 y5 The
Interpolating
(6) Polynomial If x1 x2 x = x3 x 4
x5 y1 y2 y = y3 , y 4
y5 and title page note that the columns of the augmented matrix (6) are x.ˆ4, x.ˆ3, x.ˆ2, x, ones(size(x)),
and y. The coeﬃcient matrix of system 5,
4 3
2
x2 x2 x2 x1 1
4 3
2 x3 x3 x3 x3 1 (7) x4 x3 x2 x 1 ,
4 4
4
4
3
4
2
x5 x5 x5 x5 1 previous page is called a Vandermonde matrix. The Vandermonde matrix is important in a number of applications, but it is particularly useful when computing the interpolating polynomial. back For our next example, we will ﬁnd a fourth degree interpolating polynomial that passes
through the points (−2, 26), (−1, −2), (1, −4), (2, −2), and (4, 128). First, substitute each data
point into the equation for a general fourth degree polynomial, y = ax 4 + bx 3 + cx 2 + dx + e. contents next page print doc
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exit a(−2)4 + b(−2)3 + c(−2)2 + d(−2) + e = 26 The a(−1)4 + b(−1)3 + c(−1)2 + d(−1) + e = −2
a(1)4 + b(1)3 + c(1)2 + d(1) + e = −4 (8) a(2)4 + b(2)3 + c(2)2 + d(2) + e = −2
4 3 Interpolating
Polynomial 2 a(4) + b(4) + c(4) + d(4) + e = 128
The augmented matrix for system 8 is (−2)4 (−2)3 (−1)4 (−1)3 14
13 24
23 44
43 (−2)2
(−1)2
12
22
42 −2
−1
1
2
4 Note that the coeﬃcient matrix of system 5 is (−2)4 (−2)3 (−2)2 (−1)4 (−1)3 (−1)2 14
13
12 24
23
22 44
43
42 1 26 1 −2 1 −4 . 1 −2 1 128 (9) title page
−2
−1
1
2
4 1 1 1. 1 1 contents
(10) Note that the coeﬃcient matrix (10) is a Vandermonde matrix, where x1
−2 x2 −1 x = x3 = 1 . x 2 4 x5
4
Matlab’s matrix building capability makes it especially easy to create the augmented matrix (9).
First, enter the data points in two column vectors. previous page
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exit >> x=[2 1 1 2 4]’
x=
2
1
1
2
4
>> y=[26 2 4 2 128]’
y=
26
2
4
2...
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This document was uploaded on 02/14/2014.
 Summer '12

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