Linear interpolation is the simplest method which is used for estimating a channel from the vector of the given channel’s estimates. It is very helpful in data prediction, data forecasting, market research, mathematical and scientific applications. This article will elaborate and provide knowledge on this concept with Linear Interpolation Formula and suitable examples. Let us learn it!
Introduction to Linear Interpolation
Interpolation is a method which is used for estimating the value of a function between any two known values. There are some relationships and through the help of experiments on a range of values to predict other values. Interpolation is helpful to estimate the function of the un-tabulated points. Interpolation is helpful in estimation of any desired value at some specific known coordinate point. It is a method of curve fitting by using linear polynomials to plot new data points which lie in the range of a discrete set of known data points.
Linear interpolation is being used from a very ancient time for calculation of the unknown values in tables. Suppose if we have a table for listing the population of some countries in 2000, 2005, 2010 and 2015. And that we wanted to estimate the population in the year 2007.
Then linear interpolation will be the easiest way or method to do this. It is believed that linear interpolation for tabulation was used by Babylonian.
The basic operation of linear interpolation between two values is also helpful in computer graphics.
The Formula of Linear Interpolation
Its simplest formula is provided below:
y = y\[_{1}\] + \[\frac{(x-x_{1})(y_{2}-y{1})}{x_{2}-x_{1}}\]
It is using the coordinates of two given points to find the best fit curve as a straight line. Then it will give us any required value of y at a known value of x.
In this formula, we have the following terms:
x1 and y1 are the first coordinates
x2 and y2 are the second coordinates
x is the point where we perform the interpolation
y is the interpolated value.
Where is This Method Used?
Linear interpolation is helpful while searching for a value between a given set of points. Therefore mathematicians consider it as “filling in the gaps” for a given set of values in tabular format. Linear interpolation uses a strategy which implies the use of a straight line to connect the given set of points on positive as well as the negative side of the unknown point.
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Sometimes, it is not accurate for non-linear data. If the points in the data set change by a large value, then it may not give a good estimate. Also, it uses an estimation of a new value by connecting two adjacent known values with a straight line.
Problems with Solutions
Problem 1 : Newton- Gregory Forward interpolation formula can be used for _____________
a) equally spaced intervals
b) unequally spaced intervals
c) both equally and unequally spaced intervals
d) unequally intervals
View Answer
Answer: a
Reason:- Newton – Gregory Forward Interpolation formula is given below
f(x) = y0 + nΔy0 + n(n-1)Δ2y0/2! + n(n-1)(n-2) Δ3y0 /3! + …..
It is obtained by Newton’s Divided difference formula by putting the value of intervals as h. It is done because we assume the intervals to be constant, which is equally spaced.
Problem 2. Identify n for the following set of data when F(0.2) is asked.
a) 0.4
b) 0.2
c) 1
d) 0.1
Answer: b
Explanation: The formula is
x = x0 + nh.
Here x0 is 0 (first element) and h is 1 (length of the interval).
we have to find F(0.2), so x will be equal to 0.2.
Putting the values in the formula we get,
0.2 = 0 + n(1) .
therefore n= 0.2.
Problem 3. identify n for the following set of data when F(1.8) is asked.
a) 2.4
b) 3.4
c) 2.6
d) 3.6
View Answer
Answer: d
Here, x0 (first element) is 0, h(length of interval). is 0.5, x is 1.8.
puing the values in the formula
x = x0 + nh,
1.8 = 0 + n(0.5)
n = 3.6.
FAQs on Linear Interpolation Formula
1. What is Linear Interpolation?
Linear interpolation is the simplest method which is used for estimating a channel from the vector of the given channel’s estimates. It is very helpful in data prediction, data forecasting, market research, mathematical and scientific applications. It is a method of curve fitting by using linear polynomials to plot new data points which lie in the range of a discrete set of known data points. Linear interpolation uses a strategy which implies the use of a straight line to connect the given a set of points on positive as well as the negative side of the unknown point.
2. What is Newton – Gregory Forward Interpolation Formula?
Newton – Gregory Forward Interpolation formula is given below-
f(x) = y0 + nΔy0 + n(n-1)Δ2y0/2! + n(n-1)(n-2) Δ3y0 /3! + …..
It is obtained by Newton’s Divided difference formula by putting the value of intervals as h. It is done because we assume the intervals to be constant, which is equally spaced.
3. Write Down the Formula for Linear Interpolation?
y = y₁ + (x - x₁)(y₂ - y₁)/(x₂ - x₁)
4. What is the Use of Linear Interpolation?
Linear interpolation is helpful while searching for a value between a given set of points. Therefore mathematicians consider it as “filling in the gaps” for a given set of values in tabular format. Linear interpolation uses a strategy which implies the use of a straight line to connect the given set of points on positive as well as the negative side of the unknown point.