Interpolation is a mathematical technique that estimates missing data points in a function or set of discrete data points. It involves finding a smooth curve or line that best fits the existing data points and then using that curve or line to estimate the missing points.

Types of Interpolation:

1. Linear Interpolation:– Estimates missing points based on the linear relationship between known points.- Simple and straightforward, but can be inaccurate for non-linear functions.

2. Quadratic Interpolation:– Estimates missing points based on a quadratic polynomial fitted to the known points.- More accurate than linear interpolation for non-linear functions.

3. Cubic Interpolation:– Estimates missing points based on a cubic polynomial fitted to the known points.- Even more accurate than quadratic interpolation for non-linear functions.

4. Trigonometric Interpolation:– Estimates missing points using trigonometric functions of the known points.- Useful for interpolating data with periodic patterns.

5. Spline Interpolation:– Uses spline functions to fit a smooth curve through the known points.- Can be more accurate than other methods, especially for complex functions.

6. Lagrange Interpolation:– Estimates missing points based on a set of Lagrange multipliers.- Can be accurate for complex functions, but can be computationally expensive.

7. Nearest Neighbor Interpolation:– Estimates missing points based on the nearest known points.- Simple and computationally efficient, but can be inaccurate for functions with a lot of noise.

Applications:

• Interpolating missing data points in datasets.
• Estimating missing values in functions.
• Creating smooth curves or lines through data points.
• Interpolating data for curve fitting and modeling.
• Filling in gaps in data sets.

Examples:

• Interpolating missing values in a table of population data.
• Interpolating missing points in a function that describes the relationship between temperature and pressure.
• Estimating the value of a stock at a future date based on historical data.