Last modified: August 18, 2023

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Linear interpolation

Linear interpolation is one of the most basic and commonly used interpolation methods. The idea is to approximate the value of a function between two known data points by assuming that the function behaves linearly (like a straight line) between these points. Although this assumption may be simplistic, it often provides a reasonable approximation, especially when the data points are close together or the underlying function is relatively smooth.

Conceptual Illustration:

Imagine you have two points on a graph:

Linear interpolation draws a straight line between the two known data points (xi,yi) and (xi+1,yi+1), and then estimates the value at x by following this line.

Mathematical Formulation

Given two known data points (xi,yi) and (xi+1,yi+1), and a target x-value with xi≤x≤xi+1, the line connecting these points has a slope α given by:

α=yi+1−yixi+1−xi.

To find the interpolated value y at x, start from yi and move along the line for the interval (x−xi):

y=yi+α(x−xi).

Substituting α:

y=yi+(x−xi)yi+1−yixi+1−xi.

This formula provides the interpolated y-value directly.

Derivation

I. Slope Calculation:

The slope α of the line passing through (xi,yi) and (xi+1,yi+1) is:

α=yi+1−yixi+1−xi.

II. Linear Equation:

A line passing through (xi,yi) with slope α is:

y−yi=α(x−xi).

III. Substitution:

Replace α with its expression:

y−yi=yi+1−yixi+1−xi(x−xi).

IV. Final Formula:

Simplifying:

y=yi+(yi+1−yi)xi+1−xi(x−xi).

Algorithm Steps

I. Identify the interval [xi,xi+1] that contains the target x.

II. Compute the slope:

yi+1−yixi+1−xi.

III. Substitute into the linear interpolation formula:

y=yi+(yi+1−yi)xi+1−xi(x−xi).

The result is the interpolated value y at the desired x.

Example

Given Points: A(−2,0) and B(2,2). Suppose we want to find y at x=1.

I. Compute the slope:

α=2−02−(−2)=24=0.5.

II. Substitute x=1:

y=0+0.5(1−(−2))=0.5×3=1.5.

So, the line passing through (−2,0) and (2,2) gives y=1.5 when x=1.

Advantages

• The method offers simplicity, as the calculation involves straightforward arithmetic, making it easy and quick to apply.
• Minimal data requirements make it practical, needing only two data points to estimate intermediate values.
• It provides a local approximation, working well when the function is nearly linear within the specified interval.

Limitations

• The linear assumption can lead to poor results if the actual relationship between points is not close to linear.
• Linear interpolation uses no derivative information, ignoring the slope or curvature of the function, which could enhance accuracy.
• Accuracy diminishes as the interval between points increases or as the function becomes more non-linear, leading to potential errors in approximation.