Inverse Interpolation
Given: a set of values of x and y = ƒ(x), we are interested to find the value of x for a certain value of y. this is termed as inverse interpolation.
Thus for four arguments a0, a1, a2 and a3 the value of x for given value of ƒ(x) is given by the formula:
x = [ƒ(x) - ƒ(a1)] [ƒ(x) - ƒ(a2)] [ƒ(a0) - ƒ(a3)]/[ƒ(a0) - ƒ(a1)] [ƒ(a0) - ƒ(a2)] [ƒ(a0) - ƒ(a3)]× (a0) + [ƒ(x) - ƒ(a0)] [ƒ(x) - ƒ(a2)] [ƒ(x) - ƒ(a3)]/ [ƒ(a1) - ƒ(a0)][ƒ(a1) - ƒ(a2)] [ƒ(a1) - ƒ(a3)] × (a1) + [ƒ(x) - ƒ(a0)] [ƒ(x) - ƒ(a2)] [ƒ(x) - ƒ(a3)]/ [ƒ(a2) - ƒ(a0)][ƒ(a2) - ƒ(a2)] [ƒ(a2) - ƒ(a3)] × (a2) + [ƒ(x) - ƒ(a0)] [ƒ(x) - ƒ(a1)] [ƒ(x) - ƒ(a2)]/ [ƒ(a3) - ƒ(a0)][ƒ(a3) - ƒ(a2)] [ƒ(a3) - ƒ(a3)] × (a3)
Illustration: you are given the following information:
|
x
|
5
|
6
|
9
|
11
|
|
ƒ(x)
|
12
|
13
|
14
|
-16
|
Find the value of x when ƒ(x) = 15.
Solution: in the usual notation of Lagrange's formula:
|
x
|
a0 = 5
|
a1 = 6
|
a2 = 9
|
a3 = 11
|
|
y = ƒ(x)
|
12
|
13
|
13
|
13
|
We have to calculate x when ƒ(x) = 15. Using inverse interpolation formula:
x = (15 - 13) (15 - 14) (15 - 16)/(12 - 13) (12 - 14) (12 - 16) × 5 + (15 - 12) (15 - 14) (15 - 16)/(13 - 12) (13 - 14) (13 - 16) × 6 + (15 - 12) ( 15 - 13) (15 - 16)/(14 - 12) (14 - 13) (14 - 16) × 9 + (15 - 12) ( 15- 13) (15 - 14)/(16 - 12) (16 - 13) (16 - 14) × 11
= (2) × (1) × (-1)/(-1) (-2) (-4) × 5 + (3) (1) (-1)/(1) (-1) (-3) × 6 + (3) (2) (-1)/ (2) (1) (-2) × 9 + (3) (2) (1)/ (4) (3) (2) × 11
= 1.25 - 6 + 13.5 + 2.75
= 11.5