Newton’s Divided Difference Calculator
Newton's Divided Difference method is a powerful technique for polynomial interpolation. Here's a comprehensive guide on creating and using a divided difference table:
What is Newton's Divided Difference?
Newton's Divided Difference is an interpolation method that creates a polynomial to fit a set of data points. It's particularly useful when the data points are not evenly spaced.
Creating the Divided Difference Table
The table is constructed as follows:
- Start with two columns: x values and f(x) values.
- Create additional columns for each order of divided differences.
- Calculate the divided differences using the recursive formula.
Steps to Create the Table
- First column: List the x values in ascending order.
- Second column: List the corresponding f(x) values.
- Third column (1st order differences):
- Calculate [f(x₁) - f(x₀)] / (x₁ - x₀)
- Continue for each pair of adjacent points
- Fourth column (2nd order differences):
- Calculate [1st order difference₂ - 1st order difference₁] / (x₂ - x₀)
- Continue this process
- Continue creating columns until you reach a single value or desired precision.
Example Table
| x | f(x) | 1st Order | 2nd Order | 3rd Order |
|---|---|---|---|---|
| x₀ | f(x₀) | |||
| [x₀,x₁] | ||||
| x₁ | f(x₁) | [x₀,x₁,x₂] | ||
| [x₁,x₂] | [x₀,x₁,x₂,x₃] | |||
| x₂ | f(x₂) | [x₁,x₂,x₃] | ||
| [x₂,x₃] | ||||
| x₃ | f(x₃) |
Key Points to Remember
- Notation: [xᵢ,xⱼ] represents the divided difference between f(xᵢ) and f(xⱼ).
- Formula: The general formula for divided differences is:
[xᵢ,xⱼ] = (f(xⱼ) - f(xᵢ)) / (xⱼ - xᵢ) - Higher Order Differences: For higher orders, use:
[x₀,x₁,...,xₙ] = ([x₁,...,xₙ] - [x₀,...,xₙ₋₁]) / (xₙ - x₀) - Interpolation Polynomial: The coefficients in the table are used to construct the interpolation polynomial:
P(x) = f(x₀) + x₀,x₁ + x₀,x₁,x₂(x-x₁) + ... - Accuracy: The method becomes more accurate as you include more terms.
- Efficiency: It's computationally efficient, especially for adding new data points.
- Error Estimation: The error can be estimated using the next higher-order term.
- Application: Useful in numerical analysis, curve fitting, and approximating functions.
By following these steps and understanding the key points, you can effectively create and use Newton's Divided Difference table for polynomial interpolation. This method provides a flexible and powerful tool for approximating functions and analyzing data sets