Root-Finding Methods: Bisection, Newton-Raphson, Secant, and False-Position

Overview

This program implements four numerical methods for root-finding:

• Bisection Method
• Newton-Raphson Method
• Secant Method
• False-Position Method

The program ensures solutions converge within a given tolerance and detects divergent or slow-converging solutions.

Implementation Details

This program is implemented in Java using a single main file:

1. Main.java - Handles user input, applies root-finding methods, and exports results.

Input Format

The user provides:

1. Starting values for each method.
2. Error tolerance (ea < 1%).
3. Maximum iterations (100 iterations max).

Functions Used

The program evaluates roots for two functions: (a) f(x) = 2x³ – 11.7x² + 17.7x – 5

• This function has three positive roots in the interval [0,4]. (b) f(x) = x + 10 – x cosh(50/x)
• The root lies in the interval [120,130].

Program Output

1. Root estimates for each method at each iteration.
2. Approximate relative percent error for each method.
3. Final root value once convergence is reached or after 100 iterations.
4. Exports error values for plotting.

Example Run

User Input

Please choose the first starting value for the methods to use on function 1: 0
Please choose the second starting value for the methods to use on function 1: 4

The program applies all four methods to find the roots.

Example Output

Relative error array exported to bisectFunction1_R34.txt
Relative error array exported to falsePosFunction1_R34.txt
Relative error array exported to newtonRaFunction1_R34.txt
Relative error array exported to secantFunction1_R34.txt

If the solution does not converge within 100 iterations, the program displays:

Sorry solution did not converge to desired relative percent error within 100 iterations.

User Input

Please choose the first starting value for the methods to use on function 2: 1
Please choose the second starting value for the methods to use on function 2: 3

The program applies all four methods to find the roots.

Example Output

Relative error array exported to bisectFunction1.txt
Relative error array exported to falsePosFunction1.txt
Relative error array exported to newtonRaFunction1.txt
Relative error array exported to secantFunction1.txt

If the solution does not converge within 100 iterations, the program displays:

Sorry solution did not converge to desired relative percent error within 100 iterations.

Plotting the Results

The program exports error values to text files, which can be plotted using Google Sheets, Excel, Python, or MATLAB.

Exported Files

• bisectFunction1_R34.txt
• falsePosFunction1_R34.txt
• newtonRaFunction1_R34.txt
• secantFunction1_R34.txt
• bisectFunction1.txt
• falsePosFunction1.txt
• newtonRaFunction1.txt
• secantFunction1.txt

Each file contains iteration vs. relative percent error, which can be plotted with:

• Logarithmic scale of error values.

Java Code Structure

Main.java

• Handles user input for starting values.
• Implements all four root-finding methods.
• Checks for convergence and stops if error criteria are met.
• Exports iteration data for visualization.

Usage Instructions

1. Compile the program:

   javac Main.java

2. Run the program:

   java Main

3. Follow the prompts to input starting values.
4. View the step-by-step solution and final computed results.
5. Plot the error data using external software.