Linear Interpolation Formula

Linear Interpolation Formula

Interpolation Formula: Interpolation is the process of finding a value between two points on a line or curve. To help us remember what it means, we should think of the first part of the word, ‘inter,’ as meaning ‘enter,’ which reminds us to look ‘inside’ the data we originally had. This tool, interpolation, is not only useful in statistics, but is also useful in science, business or any time there is a need to predict values that fall within two existing data points.

Interpolation is a method for estimating the value of a function between two known values. Often some relationship is measured experimentally or traced with Dagra at a range of values. Interpolation can be used to estimate the function for untabulated points.

Linear Interpolation Formula

Linear Interpolation Formula

For example, suppose we have tabulated data for the thermal resistance of a transistor tabulated for air velocity from 0 to 1800 FPM in 200 FPM steps. Interpolation can be used to estimate the thermal resistance at non-tabulated values such as 485 FPM.

The method of finding new values for any function using the set of values is done by interpolation. The unknown value on a point is found out using this formula. If linear interpolation formula is concerned then it should be used to find the new value from the two given points. If compared to Lagrange’s interpolation formula, the “n” set of numbers should be available and Lagrange’s method is to be used to find the new value.

The following is  Linear Interpolation Formula

To interpolate the y2 value:
x1, x3, y1 and y3 need to be entered/copied from the table.
x2 defines the point to perform the interpolation.
y2 is the interpolated value and solution.

x1 y1
x2 y2
x3 y3

linear interpolation equation

 

Interpolation Formula Calculator

Interpolation Formula Calculator

If the two known points are given by the coordinates {\displaystyle (x_{0},y_{0})} and {\displaystyle (x_{1},y_{1})}, the linear interpolant is the straight line between these points. For a value x in the interval {\displaystyle (x_{0},x_{1})}, the value y along the straight line is given from the equation of slopes

which can be derived geometrically from the figure on the right. It is a special case of polynomial interpolation with n = 1.

Solving this equation for y, which is the unknown value at x, gives

which is the formula for linear interpolation in the interval {\displaystyle (x_{0},x_{1})}. Outside this interval, the formula is identical to linear extrapolation.

This formula can also be understood as a weighted average. The weights are inversely related to the distance from the end points to the unknown point; the closer point has more influence than the farther point. Thus, the weights are {\textstyle {\frac {x-x_{0}}{x_{1}-x_{0}}}} and {\textstyle {\frac {x_{1}-x}{x_{1}-x_{0}}}}, which are normalized distances between the unknown point and each of the end points.

Interpolation Formula Excel

Here’s an example that will illustrate the concept of interpolation. A gardener planted a tomato plant and she measured and kept track of its growth every other day. This gardener is a curious person, and she would like to estimate how tall her plant was on the fourth day.

Her table of observations looked like this:

interpolation data table

Based on the chart, it’s not too difficult to figure out that the plant was probably 6 mm tall on the fourth day. This is because this disciplined tomato plant grew in a linear pattern; there was a linear relationship between the number of days measured and the plant’s height growth. Linear pattern means the points created a straight line. We could even estimate by plotting the data on a graph.

interpolation graph 1a

But what if the plant was not growing with a convenient linear pattern? What if its growth looked more like this?

interpolation curve 1

What would the gardener do in order to make an estimation based on the above curve? Well, that is where the interpolation formula would come in handy.

Lagrange Interpolation Formula

The interpolation formula looks like this:

Interpolation Formula 1

Now, let’s explore how to use this formula. The two sets of points between which the estimate can be found are:

data points 1
interpolation data points 2

Going back to the tomato plant example, the first set of values for day three are (3,4), the second set of values for day five are (5,8), and the value for x is 4 since we want to find the height, y, on the fourth day. After substituting these values into the formula, calculate the estimated height of the plant on the fourth day.

interpolation problem

Based on the calculations, the estimated height of the plant on the fourth day is 6 mm.

What is linear interpolation method?

In mathematicslinear interpolation is a method of curve fitting using linear polynomials to construct new data points within the range of a discrete set of known data points.

How do you interpolate with two points?

Linear Interpolation Equations

Linear interpolation involves estimating a new value by connecting two adjacent known values with a straight line. If the two known values are (x1, y1) and (x2, y2), then the y value for some point x is: Linear interpolation is a straight line fit between two data points.

What does it mean to interpolate data?

Extrapolation is an estimation of a value based on extending a known sequence of values or facts beyond the area that is certainly known. … Interpolation is an estimation of a value within two known values in a sequence of values. Polynomial interpolation is a method of estimating values between known data points.