South Dakota School of Mines and Technology
Department of Computer and Mathematical Sciences
Math 373
Interpolation
Interpolation_04
1. Below are some regularly-spaced data varying with the independent variable x.
x
1
f (x)
-14
2
9
3
128
4
463
5
1182
6
2501
Double Click Me to Open in Excel
a) What is the order (highest power of x) of the polynomial from which the data was
generated?
b) Find the equation that fits the data.
c) Interpolate for f(2.2) using a
i) first order (linear) approximation
ii) second order approximation
iii) third order approximation
2
Below are some regularly-spaced data varying with the independent variables x and y. Suggest a
method by which a value might be accurately interpolated at any (x, y) such as (723, 2.34).
y
x
1
2
3
4
5
6
7
8
100
0.39
3.68
9.44
17.77
28.76
42.46
58.92
78.19
100.30 125.29
9
10
300
-0.70
2.58
8.34
16.68
27.66
41.36
57.82
77.09
99.20
124.19
500
-1.21
2.07
7.83
16.16
27.15
40.85
57.31
76.58
98.69
123.68
700
-1.55
1.74
7.49
15.83
26.81
40.51
56.97
76.24
98.35
123.34
900
-1.80
1.48
7.24
15.58
26.56
40.26
56.72
75.99
98.10
123.09
1100
1300
-2.00
-2.17
1.28
1.12
7.04
6.87
15.38
15.21
26.36
26.20
40.06
39.89
56.52
56.36
75.79
75.62
97.90
97.73
122.89
122.72 .
3. Using the following data:
x
f(x)
1.00
2.00
1.10
2.92
1.20
2.68
1.30
2.27
1.40
1.71
1.50
1.10
1.60 1.70 1.80 1.90 2.00
-0.83 -2.73 -3.63 -5.43 -5.00
estimate the value of f(1.43) using Newton’s Difference Interpolation to the order of
a) 0
b) 1 (Tip: simply add the current-order term to the above, lower-order term.)
c) 2
d) 3
For your convenience:
This is a transposed array Horizontal to Vertical).
x
f(x)
x
1.00
1
2
1.1
2.92
1.2
2.68
1.3
2.27
f(x)
2.00
Double Click while in Word ® to open in Excel ®
1.10
2.92
1.20
2.68
1.4
1.71
4. Use the Lagrange Interpolation Equation for a quadratic order interpolation of problem #3
and compare it to the Newton Difference Equation result obtained above.