Interpolation Formula

The Interpolation formula is the method to find the new values of any function using the set of values that are already available and is done by interpolation. The interpolation formula uses interpolation, which is the process involving finding a value between two points on the curve of a function. The interpolation formula is explained below along with a few solved examples in the following section.

What is the Interpolation Formula?

The interpolation or linear interpolation formula finds the point on a straight line between two points, so if the two known points are known to us with the coordinates (\(x_{0},y_{0} \) )and \( (x_{1},y_{1})\), then on linear interpolation, a point on the straight line between these points is the result. Linear interpolation formula for polynomials of the first order can be given as,

 \(f(x) = f(x_0) + (x − x_0) \dfrac{f(x_0) − f(x_1)}{x_0 − x_1}\)

Examples Using the Interpolation Formula

Example 1: Using the interpolation formula, find the value of y at x = 0, if given some set of values are (-2, 5), (1, 7)?

Solution:

Given the known values are,

\(x\)= 0 ; \(x_0\) = -2 ; \(x_1\) = 1 ; \(y_0\) = 5 ; \(y_1\) = 7 ;

Using the interpolation formula,

 \(f(x) = f(x_0) + (x − x_0) \dfrac{f(x_0) − f(x_1)}{x_0 − x_1}\)
y = 5 + (0 + 2)×(5 - 7)/(-2 - 1)
y = 5 + 4/3
y = 19/3 = 6.33...

Therefore, the value of y at (x = 0) = 6.333.

Example 2: Using interpolation formula find y(10) from the following table:

Solution:

Given the known values are,

x = 10; \(x_0\) = 5 ; \(x_1\) = 6  ; \(y_0\) = 12 ; \(y_1\) = 13;

Using the interpolation formula,

 \(f(x) = f(x_0) + (x − x_0) \dfrac{f(x_0) − f(x_1)}{x_0 − x_1}\)
y = 12 + (10 - 5)×(13 - 12)/(5 - 6)
y = 12+ 5
y = 17

Therefore, the value of y at (x = 10) is 17.

Example 3: Using the interpolation formula, find the value of y at x = 1, if given some set of values are (-4, 5), (2, 6)?

Solution:

Given the known values are,

\(x\)= 1 ; \(x_0\) = -4 ; \(x_1\) = 2 ; \(y_0\) = 5 ; \(y_1\) = 6 ;

Using the interpolation formula,

 \(f(x) = f(x_0) + (x − x_0) \dfrac{f(x_0) − f(x_1)}{x_0 − x_1}\)
y = 5 + (1 +4)×(5 - 6)/(-4 - 1)
y = 5 + 5×1/5
y = 6

Therefore, the value of y at (x = 0) = 6