Description

A simple C++ program which utilizes an OpenGL shader to visualize the convergence of the fixed-point iteration numerical method.

Fixed-point iteration root-finding method

Given any function $f(x)$, one can prove that the recursive formula: $$x_{n+1}=x_n + \alpha f(x_n)$$ will converge for some values of the parameter $\alpha$ and initial values of $x_0$ towards the roots of function $f(x)$.

This program visualizes which pairs of the above parameters: $\alpha$ and $x_0$ give rise to the convergence to the solution.

Main window

Example screens

$$f(x) = e^x - \frac{1}{x}$$

$$f(x) = \sin(e^x)$$

$$f(x) = x^2 \sin(x)$$

$$f(x) = x\cos(x) - x\sin(x)$$

Explanation

In the main window of the application a graph of $\alpha$ versus $x_0$ is shown. Each pixel is a color-coded representation of the logarithmic relative error between the computed result and the actual solution. Color values create a heat map with black or blue pixels being furthest from the solution and white or purple being closest.

Controls

• zoom in/out by scrolling the mouse or using a touchpad
• click and drag to move the view

Function input window

The secondary window is meant for entering the function $f(x)$. Any function that the GLSL compiler can compile can be used; input from this window is inserted directly into the shader code.

How to build

Prerequisites

• C++17 compliant compiler
• CMake (>= 3.22)
• SFML library, along with System, Window and Graphics components (>= 2.5)
• Boost library; system, filesystem, math and dll components needed (>= 1.74)
• pkg-config (>= 0.29)
• gtkmm (API version 3.0)

Make sure that all libraries are located in a directory, where CMake can find them.

Build instructions

git clone https://github.com/roghed/convergence
cd convergence
cmake -DCMAKE_BUILD_TYPE=Release -S . -B build
cmake --build build

Third party licenses

ExprTk

This program uses the C++ Mathematical Expression Toolkit Library:

File: exprtk.hpp

Author: Arash Partow (1999-2023) URL: https://www.partow.net/programming/exprtk/index.html

Copyright notice: Free use of the C++ Mathematical Expression Toolkit Library is permitted under the guidelines and in accordance with the most current version of the MIT License.

https://www.opensource.org/licenses/MIT

Ubuntu font

This program uses the UbuntuMono-Regular font which is distributed under the UBUNTU FONT LICENSE Version 1.0.

For more information see the Ubuntu Font License.txt file provided with the project.

FreeType

Portions of this software are copyright � 2006 The FreeType Project (www.freetype.org). All rights reserved.