Interpolation Formula

What is Interpolation?

In Mathematics, interpolation is used to find or estimate the value of f(x) or a function of x, from particularly known values of a function. If the values such as x₀ <....< xn and y₀ = f(x₀).....yn = f(xn) are known, and if x₀ < x < xn then then estimated values of f(x) is determined as an Interpolation. If x < x₀  or x > xn, then the approximate value of f(x) is said to be an extrapolation. In this article, we will discuss what is an interpolation, what is an interpolation formula, interpolating meaning, what is an example of interpolation


Interpolating Meaning

Interpolation is a procedure of deriving a basic function from the provided discrete data set such that the function passes through the given data points. This enables us to estimate the data points in between the given data. Generally, the interpolation is used to calculate the value of a function for an intermediate value of the independent function. In other words, interpolation is a method of estimating the unknown values that falls in between the given data points. It is used to estimate the unknown values for any geographical related data points such as noise level, rainfall, elevation, etc.


What is the Interpolation Formula?

The Linear Interpolation formula and Lagrange interpolation formula is used to determine the unknown values of a given set of data points.

The linear interpolation formula is stated as :

y = \[y_1+\frac{x-x_1}{x_2-x_1} × (y_2 - y_1)\]

Similarly, the Lagrange interpolation formula is given as:

y = \[\frac{(x-x_1)(x-x_2)....(x-x_n)}{(x_0-x_1)(x_0 -x_2)....(x_0-x_n)}y\]  + \[\frac{(x-x_0)(x-x_2)....(x-x_n)}{(x_1-x_0)(x_1-x_2)....(x_1-x_n)}y_1\] + .. + \[\frac{(x-x_1)(x-x_2)....(x-x_n)}{(x_0-x_1)(x_0-x_2)....(x_0-x_n)}y_n\]


What is an example of Interpolation?

The answer to the above question what is an example of interpolation is described below:

As we know, interpolation is used to find a value within two known values in a given sequence of values. Polynomial interpolation is considered as the feasible method of determining values between known data points. The interpolation meaning can be precisely understood through an example of interpolation given below 

Suppose, a gardener planted a sugarcane plant and measured the height of the plant while planting in the garden. The gardener used to record the height of the sugarcane plant each day.

On the fourth day, the gardener was curious to determine the height of the plant. So he looked upon the records which he maintained. The gardener record of plants heights is maintained in tabular form as shown below:

Day

Height (in mm)

1

0

3

4

5

8

7

12

9

16

11

20


If you examine the records closely then you will find that it is not much difficult to find out the height of the plant on the fourth day. Hence, the plant height on the fourth day is estimated as 6 mm. Also, we can see the sugarcane plant grows in a linear pattern.

There is a linear relationship between the numbers of the days and the height of the sugarcane plant. A linear pattern always forms a straight line and we could even determine the height of the plant by plotting the data on the graph.

But, what if the height of the sugarcane plant does not grow with a linear pattern and  forms the shape of a curve? We can easily solve this problem with the help of the above linear interpolation formula.


Interpolation Method

The different types of interpolation methods are as follows:

Linear Interpolation Method – The linear interpolation method enforces a distinct linear polynomial between each pair of data points for curves, or within the sets of three points for surfaces.

Nearest Neighbour Method – The nearest neighbor interpolation method enters the value of an interpolated point to the value of the most adjacent data point. Hence this method does not introduce any new data points.

Cubic Spline Interpolation Method – The cubic spline interpolation method places a distinct cubic polynomial between each pair of data points for curves, or between sets of three points for surfaces.

Shape-Preservation Method – The shape-preservation method is also known as Piecewise cubic Hermite interpolation (PCHIP). It preserves the monotonicity and the shape of the data. This method is used only for curves.

Thin-plate Spline Method – This method includes smooth surfaces that extrapolate well. Hence, it is used only for surfaces 

Biharmonic Interpolation Method – This method is used only for surfaces.


Solved Examples

1. Determine the values of y at x = 4 given some set of values (2,4), (6,7) by using the interpolation formula

Solution:

Given  = x₀ = 4, x₁ = 2 ; x₂ = 6 ; y₁ = 4, y₂ = 7


Using the interpolation formula, we can find the value of ay at x = 6

The interpolation formula is 

y = \[y_1+\frac{x-x_1}{x_2-x_1} × (y_2 - y_1)\]

y = 4 + (4 - 2)/(6 - 2) * (7 - 4)

y = 4 + 3/2

y = 11/2

2. Determine the values of y at x = 8 given some set of values (2,6), (5,9) by using the interpolation formula

Solution:

Given  = x₀ = 8, x₁ = 2 ; x₂ = 5 ; y₁ = 6, y₂ = 9

Using the interpolation formula, we can find the value of ay at x = 6

The interpolation formula is 

y = \[y_1+\frac{x-x_1}{x_2-x_1} × (y_2 - y_1)\]

 y = 6 + ((8 - 2)/(5 - 2) * (9 - 6))

y = 6 + 6

y = 12


Quiz Time

1. The process which is used to construct new data points within the range of discrete set of known data points is defined as 

  1. Interpolation

  2. Extrapolation

  3. Antipolation

  4. Dentipolation

2. The method of determining, beyond the original observation range, the value of the variable on the basis of its relationship with another variable is termed as 

  1. Interpolation

  2. Extrapolation

  3. Antipolation

  4. Dentipolation

3. Drawing tangent line at the end of the given data and enlarging it beyond the limit is defined as 

  1. Interpolation

  2. Extrapolation

  3. Antipolation

  1. Dentipolation

FAQ (Frequently Asked Questions)

1. Explain Linear Interpolation.

Ans. Linear interpolation is the easiest method of determining values at a position in between the given data points. The points are just joined by a simple line segment. Each segment ( joined by two data points) can be independently interpolated.  We can estimate the values on the interrelated line through the parameter mu. We can see 0 at the first point and 1 at the second point on the interpolated line. The value of mu falls outside this range results in extrapolation. This method is followed for all the other interpolation methods given above.

The linear interpolation method is used to determine the values of a security or internet rate for a point in which no data is provided.

2. What are the advantages and disadvantages of Lagrange Interpolation ?

Ans.  The advantages of Lagrange Interpolation are:

  1. The Lagrange Interpolation formula is simple and easy to remember.

  2. The application of the Lagrange Interpolation formula is not swift.

The disadvantages of Lagrange Interpolation are:

  1. There is always an option to commit errors

  2. We cannot verify the whether the functional values used in the caluclation is estimated correctly or not.