Interpolation

Definition

Interpolation is the process of estimating unknown values that fall between known data points. It is often used to construct new data points within the range of a discrete set of known data points.

5 Must Know Facts For Your Next Test

1. Linear interpolation assumes that changes between two data points are linear and can be represented by a straight line.
2. The formula for linear interpolation between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by $y = y_1 + \frac{(y_2 - y_1)}{(x_2 - x_1)} (x - x_1)$.
3. Interpolation can only be used within the range of known data points; extrapolation is required for estimating values outside this range.
4. Polynomial interpolation involves fitting a polynomial to a set of data points, but this method can suffer from Runge's phenomenon, where oscillations occur at the edges of the interval.
5. Spline interpolation uses piecewise polynomials called splines to estimate values and provides smoother results compared to polynomial interpolation.

Review Questions

• What is the main assumption behind linear interpolation?
• Write down the formula for linear interpolation between two points.
• Why might spline interpolation be preferred over polynomial interpolation?

Related terms

Extrapolation: The process of estimating values outside the range of known data points using trends or patterns in the data.

Polynomial Interpolation: A method of fitting a polynomial function through multiple known data points to estimate unknown values.

Spline Interpolation: A form of interpolation where piecewise polynomials called splines are used to estimate values, providing smooth transitions between intervals.