Numerical Solution of Nonlinear Systems

This program solves a system of two nonlinear algebraic equations:

x - y³ = 0
ln(x+2) - y = 0

using four iterative methods:

• Simple iteration (fixed-point iteration)
• Newton's method (with analytical Jacobian)
• Modified Newton's method (constant Jacobian matrix)
• Discrete Newton's method (finite-difference approximation of derivatives)

The program finds the two roots:

• Root 1: (-1.696595, -1.192686)
• Root 2: (97.07366, 4.59586)

Convergence is compared in terms of number of iterations. A Python script visualises the functions and the level lines of the residual sum of squares.

Requirements

• C++ compiler with C++17 support (e.g., g++, clang++)
• GNU Make
• Python 3 with matplotlib and numpy (for visualisation)

Quick Start

git clone <your-repo-url>
cd <project-folder>
make          # compile the program
make run      # compile and run

Alternatively:

make
./nonlinear_system

To clean up:

make clean

Example Output

After running ./nonlinear_system, the console shows a table comparing the four methods:

––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––
|                      |     x      |     y      | iter_num |
––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––
|   simple iteration   | 9.707e+01  | 4.596e+00  |    50    |
––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––
|        newton        | 9.707e+01  | 4.596e+00  |     6    |
––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––
|      mod newton      | 9.707e+01  | 4.596e+00  |     8    |
––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––
|     discr newton     | 9.707e+01  | 4.596e+00  |     6    |
––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––

(The exact values depend on the initial approximation and tolerance.)

Additionally, a Python script create_plot.py will be launched to display the graphs of the functions and the level lines of the residual sum of squares, with the roots marked.

Project Structure

.
├── Makefile
├── README.md
├── .gitignore
├── LICENSE
├── create_plot.py          # Python script for visualisation
├── src/
│   ├── main.cpp
│   ├── functions.cpp
│   └── functions.h
├── docs/
│   └── task.pdf
└── build/                  # object files (ignored)

Author

Name: @g30613740 Philip K.

Year: 2025

License

This project is distributed under the MIT License.