Truncation and Rounding Error Analysis

This project provides functions to calculate and analyze truncation error, rounding error, and total error for finite difference formulas. It includes a main function to plot the errors using a log-scale and print the optimal h values.

Overview

In scientific computations, it is crucial to understand the sources of errors and their impact on the results. This project focuses on two types of errors:

• Truncation Error: The error introduced by approximating a mathematical procedure.
• Rounding Error: The error introduced by the finite precision of floating-point arithmetic.

By analyzing these errors, we can gain insights into the accuracy and stability of numerical methods.

Features

• Finite Difference Formulas: Calculate the derivative of a function using finite difference and central difference formulas.
• Error Analysis: Compute truncation error, rounding error, and total error for different step sizes (h).
• Optimal Step Size: Determine the optimal step size that minimizes the total error.
• Visualization: Plot the errors using a log-scale to visualize their behavior as the step size changes.

Usage

To run the program, execute the Truncation and Rounding.py script. The main function will calculate and plot the errors for different step sizes and print the optimal h values.

python "Truncation and Rounding.py"

Conclusions

From the program, we can draw the following conclusions about errors in numerical methods:

1. Truncation Error: As the step size (h) decreases, the truncation error generally decreases. However, it may increase again if h becomes too small due to the limitations of floating-point precision.
2. Rounding Error: The rounding error increases as the step size (h) decreases because the finite precision of floating-point arithmetic becomes more significant.
3. Total Error: The total error is the sum of truncation and rounding errors. There is an optimal step size (h) that minimizes the total error, balancing the trade-off between truncation and rounding errors.

By understanding these errors, we can make informed decisions about the choice of step size and improve the accuracy of numerical computations.

Dependencies

• Python 3.x
• NumPy
• Matplotlib

Install the dependencies using pip:

pip install numpy matplotlib