Let yk denote f(x), where k = 0, 1, 2, 3, 4. (i) Find the value L (23) of the Lagrange polynomial at x = 23 and the relative error in the approximation f(23)~ L(23). Here and below, all calculations are to be carried out in the FPA5. (ii) Show your work by filling in the standard table for the method showing the process of evaluation of the Lagrange polynomial at the given point (entries in each row of the table are to be separated by single spaces): Ik Yk Lk (x) Yk Lk (x) L(23) = and the relative error in question is given by RE(f(23) L(23)) =
Let yk denote f(x), where k = 0, 1, 2, 3, 4. (i) Find the value L (23) of the Lagrange polynomial at x = 23 and the relative error in the approximation f(23)~ L(23). Here and below, all calculations are to be carried out in the FPA5. (ii) Show your work by filling in the standard table for the method showing the process of evaluation of the Lagrange polynomial at the given point (entries in each row of the table are to be separated by single spaces): Ik Yk Lk (x) Yk Lk (x) L(23) = and the relative error in question is given by RE(f(23) L(23)) =
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
Transcribed Image Text:Let yk denote f(x), where k = 0, 1, 2, 3, 4.
(i) Find the value L (23) of the Lagrange polynomial at x = 23 and the relative error in the approximation f(23)~ L(23). Here
and below, all calculations are to be carried out in the FPA5.
(ii) Show your work by filling in the standard table for the method showing the process of evaluation of the Lagrange polynomial at
the given point (entries in each row of the table are to be separated by single spaces):
Ik
Yk
Lk (x)
Yk Lk (x)
L(23) =
and the relative error in question is given by
RE(f(23) L(23)) =
Transcribed Image Text:(Lagrange Polynomials). The following table represents the historical data on the consumer price index (CPI) in Germany from 1977
to 2021, where t is time since 1977 (measured in years) and f(t) is the corresponding value of the CPI:
t
f(t)
t
f(t)
t
f(t)
t
f(t)
t
f(t)
Let L(x) be the fourth Lagrange polynomial for the function f(x) with the nodes
xk
0
46.30
9
62.96
18
80.48
27
91.05
36
105.69
Yk
1
2
3
4
5
47.56 49.48
52.18 55.49
58.40
11
12
13
14
10
63.12
63.93
65.70
67.48
70.21
19
20
21
22
23
81.65
83.23 83.99
84.48
85.70
28
29
30
31
32
92.46
93.92
96.07 98.60 98.91 100.00 102.08
38
37
106.65 107.20
39
40
41
107.73 109.35 111.25
42
43
112.85 113.43
Lk (x)
Let y denote f(x), where k = 0, 1, 2, 3, 4.
(i) Find the value L(23) of the Lagrange polynomial at x = 23 and the relative error in the approximation f(23)~ L(23). Here
and below, all calculations are to be carried out in the FPA5.
6
7
8
60.32 61.77
63.05
15
16
17
73.76
77.06
79.13
24
25
26
87.40
88.64 89.56
33
34
35
104.13
44
116.99
(ii) Show your work by filling in the standard table for the method showing the process of evaluation of the Lagrange polynomial at
the given point (entries in each row of the table are to be separated by single spaces):
xo = 2, x₁ = 12, x2 = 22, x3 = 33, x4 = 42.
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