Torus (4th degree) - Mathematics of the donut

• Binary name: 105torus
• Repository name: 105torus_2017
• Repository rigths: ramassage-tek
• Language: C, C++, perl 5, python 3 (>= 3.5), ruby 2 (>= 2.2), php 5.6, bash 4
• Choosed language: PHP
• Compilation: via Makefile, including re, clean and fclean rules

Subject

Drawing circles, cylinders and cones is a good point for a software generating synthesis images, but one have to admit it is not fully satisfying... This project is the continuation of the previous project, and should allow you to draw more complex forms, such as tores, which do not emerge from 2nd degree equations, but from superior degree equations (4th degree in the tore case).

The objective of this very project is to solve a 4th degree equation : a4x4 + a3x3 + a2x2 + a1x + a0 = 0. A direct resolution method does exist (the Ferrari’s method), but does not generalize. Thus, we will rather compare here 3 iteratives algorithms :

1. bisection method,
2. Newton’s method,
3. secant method.

💡 Equations to be solved here will all have one and only one solution, in the [0;1] interval. This is the solution we are looking for. The initial value for Newton’s method will be 0.5, those for the 2 other methodes will be 0 and 1.

💡 Just in case you would need it, the derivative of the polynomial function x → a4x4 + a3x3 + a2x2 + a1x + a0 is the function x → 4a4x3 + 3a3x2 + 2a2x + a1

Usage

~/B-MAT-100> ./105torus opt a0 a1 a2 a3 a4 n

• opt: number of the option:
  • (1) - bisection method
  • (2) - Newton's method
  • (3) - secant method
• a0, a1, a2, a3, a4: coefficients of the equation
• n: precision (meaning the application of the polynomial to the solution should be smaller than 10^-n)

Bonus

• graphical interface to compare the speed of convergence.
• solving higher degree equations.

Examples

∼/B-MAT-100> ./105torus 1 -1 0 6 -5 1 6
x = 0.5
x = 0.75
x = 0.625
x = 0.5625
x = 0.53125
x = 0.515625
x = 0.523438
x = 0.519531
x = 0.521484
x = 0.522461
x = 0.522949
x = 0.522705
x = 0.522827
x = 0.522766
x = 0.522736
x = 0.522751
x = 0.522743
x = 0.522739
x = 0.522741
x = 0.522740

💡 The maximum number of displayed decimals is the same as the precision (n)

∼/B-MAT-100> ./105torus 2 -1 0 6 -5 1 12
x = 0.5
x = 0.522727272727
x = 0.522740003514
x = 0.522740003526

∼/B-MAT-100> ./105torus 3 -1 0 6 -5 1 8
x = 0.5
x = 0.52941176
x = 0.52274853
x = 0.52274000
x = 0.52274000