Bisection Method Project

A robust and educational implementation of the Bisection Method for finding roots of nonlinear equations, with strong emphasis on correctness, workability checks, and clear diagnostics.

This project goes beyond a basic numerical method implementation by carefully validating mathematical assumptions such as continuity, definedness, and denominator safety, using both symbolic and numeric techniques.

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✨ Features

• Intelligent expression preprocessing

  • Converts user-friendly math input into valid Python expressions
  • Examples:
    • xsinx → x*sin(x)
    • sinx → sin(x)
    • 3x → 3*x
    • 4(x+1) → 4*(x+1)

• Safe function evaluation

  • Restricted evaluation environment
  • Only math-safe functions are allowed
  • Prevents arbitrary code execution

• Advanced workability checks

  • Symbolic denominator zero detection (using SymPy)
  • Numeric fallback sampling for difficult cases
  • Continuity and definedness verification
  • Endpoint root detection
  • Sign-change test
  • Single-root heuristic (sampling-based)

• Guaranteed iteration count

  • Computes the required number of iterations using:

    N > (log10(b-a) - log10(0.5 * 10^-d)) / log10(2) - 1
    

• GUI Interface

  • Built using tkinter
  • Displays:
    • Root approximation
    • Iteration table
    • Convergence explanation
    • Diagnostic messages
  • Scrollable summary and results table

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📁 Repository Structure

├── bisection.py ├── bisection_testing.py ├── gui.py ├── main.py ├── utils.py └── README.md

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📄 File Descriptions

bisection.py

The core backend of the project.

Contains:

• Expression preprocessing
• Safe function construction
• Symbolic denominator analysis using SymPy
• Numeric sampling checks
• Root estimation heuristics
• Required iteration computation
• Main bisection() implementation

This file enforces all mathematical assumptions required by the bisection method.

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bisection_testing.py

Used for testing and validation.

Includes:

• Discontinuous function tests
• Continuous function tests
• Edge cases (endpoint roots, invalid intervals)
• Debug-oriented outputs

Helps verify correctness independently of the GUI.

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gui.py

Graphical User Interface for the project.

Features:

• Function input
• Interval input
• Precision (digits) input
• Sample function selector
• Iteration table (TreeView)
• Scrollable summary box
• CSV export of iterations

The GUI communicates directly with the backend and displays human-readable diagnostics.

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main.py

Entry point of the project.

Typically responsible for:

• Launching the GUI
• Or running backend logic directly (depending on configuration)

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utils.py

Utility functions shared across the project.

Includes:

• CSV export helpers
• Directory handling
• Number formatting utilities

Keeps the core logic clean and modular.

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🔍 Why Symbolic + Numeric Checks?

Why not sampling only?

• Sampling can miss exact roots if they lie between sample points.
• Sampling cannot prove absence of roots.
• Sampling is heuristic by nature.

Why use SymPy?

• Detects exact denominator zeros
• Handles algebraic expressions safely
• Provides mathematical guarantees when possible

Why keep numeric fallback?

• Some expressions (e.g., sin(x), tan(x)) produce infinite or implicit root sets
• Symbolic solvers may return:
  • ImageSet
  • ConditionSet
• Numeric scanning provides a safe fallback

➡️ Result: correctness first, performance second.

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🧠 Mathematical Guarantees

The implementation ensures:

• The function is defined on [a, b]
• The function is continuous on [a, b]
• f(a) * f(b) < 0
• Required assumptions of the Intermediate Value Theorem are satisfied

Only then does the algorithm proceed.

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🎓 Educational Focus

This project is designed for:

• Numerical Analysis courses
• Algorithm correctness discussions
• Demonstrating the gap between theory and implementation
• Understanding why checks matter before algorithms