Polynomial Interpolation

This Python notebook generate the polynomial function that passes through a given set of points, then explain the process and plot the resulting polynomial function.

Features

• A personal method for computing polynomial coefficients.
• Gaussian Elimination to solve systems of linear equations.
• Calculation of factorials for polynomial terms.
• Comprehensive tabulated outputs for:
  • Input values
  • Differences (Divided Differences Table)
  • Augmented matrix and its solution
  • Polynomial coefficients
  • Final polynomial function
  • Evaluated polynomial values
• Visualization of the polynomial function and input points using Matplotlib.

Getting Started

Installing Dependencies

Run the following command to install the required dependencies:

pip install -r requirements.txt

Using the Program

1. Define your input x and y values.
2. Set the range of values for the polynomial plot.
3. Run the interpolate() function with your inputs.

For example:

Y_VALUES = [1, 9.5, 43, 122.5, 269]
X_VALUES = [0, 1, 2, 3, 4]
PLOT_RANGE = range(-5, 6)

interpolate(X_VALUES, Y_VALUES, PLOT_RANGE)

Outputs

Tabulated Outputs

The program provides detailed tabular outputs:

• Input values:

  ┌─────┬───────┐
  │   x │     y │
  ├─────┼───────┤
  │   0 │   1   │
  │   1 │   9.5 │
  │   2 │  43   │
  │   3 │ 122.5 │
  │   4 │ 269   │
  └─────┴───────┘
  

• Differences Table:

  ┌───────┬───────┬──────┬──────┐
  │     y │    d1 │   d2 │   d3 │
  ├───────┼───────┼──────┼──────┤
  │   1   │   8.5 │   25 │   21 │
  │   9.5 │  33.5 │   46 │   21 │
  │  43   │  79.5 │   67 │      │
  │ 122.5 │ 146.5 │      │      │
  │ 269   │       │      │      │
  └───────┴───────┴──────┴──────┘
  

• Polynomial Function:

  f(x) = 3.5x^3 + 2x^2 + 3x + 1
  

• Evaluated Points:

  ┌─────┬────────┐
  │   x │   f(x) │
  ├─────┼────────┤
  │  -5 │ -401.5 │
  │  -4 │ -203   │
  │  -3 │  -84.5 │
  │  -2 │  -25   │
  │  -1 │   -3.5 │
  │   0 │    1   │
  │   1 │    9.5 │
  │   2 │   43   │
  │   3 │  122.5 │
  │   4 │  269   │
  │   5 │  503.5 │
  └─────┴────────┘
  

Graphical Outputs

The program plots the polynomial function along with the input points:

License

This project is licensed under the MIT License.