Interpolation Formula

Interpolation is a method of finding new values for any function using the set of values. We can determine the unknown value on a point using this formula. If linear interpolation formula is concerned then it can be used to find the new value from the two given points. If we compare it to Lagrange’s interpolation formula, the “n” set of numbers is needed. Thereafter Lagrange’s method is to be used to find the new value. Interpolation is a useful and statistical tool used to estimate values between two points. In this topic, a student will learn about the Interpolation formula and methods for applying it.

Interpolation Formula

What Is Interpolation?

Interpolation is the process for finding a particular value between two given points on a line or curve.  Here the word, ‘inter,’ means we will ‘enter,’ into the data set. This tool is not only useful in statistics but is also useful in science, business and many more real-life applications that fall within two existing data points.

So, it can be understood that interpolation formula is a method of curve fitting using linear polynomials and hence to construct new data points within the given range of a discrete set of known data points.

Linear interpolation has been used since very early time antiquity for filling the unknown values in tables. Suppose that we have a table for listing the population of some countries in 2000, 2005, 2010 and 2015. And that someone wanted to estimate the population in the year 2008.

Then linear interpolation will be is an easy method to do this. This technique of using linear interpolation for tabulation was believed to be used by Babylonian.

The basic operation of linear interpolation between two values is also useful in computer graphics.

Interpolation Formula

The interpolation formula will look like as follows:

\(y-y_1 = \frac{y_2-y_1}{x_2-x_1}\times (x-x_1)\)

in another form,

\(y= y_1 + \frac{y_2-y_1}{x_2-x_1}\times (x-x_1)\)

where two points are known as \((x_1, y_1) and (x_2,y_2)\).

The above formula is the Linear Interpolation Formula.

Linear interpolation, which is also called simply interpolation, thus is the ability to deduce a value between two values explicitly given in a table or on a line graph.

Related to the interpolation another process is extrapolation, which is the finding a corresponding value for a given set of values outside of the range listed.

Solved Examples

Q.1: Using the interpolation formula, find the value of y at x = 8. Some sets of values are given (2, 6), (5, 9)?

Solution:

The known values are,

x =8

\(x_1\)=2,

\(x_2\)=5,

\(y_1\)=6,

\(y_2\)= 9

Now we have the interpolation formula as,

y= \(y_1\) + \(\frac{y_2-y_1}{x_2-x_1}\times (x-x_1)\)

i.e. y= 6 + \(\frac{8-2}{5-2}\times (9-6)\)

y= 6+6

y = 12

Q.2: Find the value of f(2) using linear interpolation method. Given that f(-1) = 4 and f(3) =6.

Solution: Here it is obvious that given values are

X = 2

\(X_1\)= -1

\(X_2\) = 3

\(Y_1\)= 4

\(Y_2\) = 6

Thus, applying the formula,

y= \(y_1 + \frac{y_2-y_1}{x_2-x_1}\times (x-x_1)\)

Put all known values, we get

y= 4 + \(\frac{6-4}{3-(-1)}\times (2-(-1))\)

i.e. y=4 + \(\frac{2}{4} (3)\)

y = \(\frac{11}{2}\)

y= 5.5

Thus the value of f(2) is 5.5.