From Newton to Newton Fractal part-1

Introduction

Posted on September , 2023 in JayanthBlog

When it comes to text rendering and sketching, achieving clear and smooth curves is paramount. To attain this level of precision, the use of polynomials is essential. Let's consider a scenario: imagine you open MS Paint and attempt to draw a curve. The initial result may not resemble a perfect curve, but beneath the surface, there's intricate mathematics at play. Polynomials, in particular, play a crucial role in this process. They provide a mathematical framework for defining and rendering curves with precision and clarity. These mathematical expressions allow us to generate curves that appear smooth to the human eye, creating text and drawings that are aesthetically pleasing and easy to read or interpret. In essence, polynomials serve as the mathematical backbone, helping us translate our creative intentions into visual representations that are far more precise and refined than what we can achieve with simple hand-drawn lines.

Other:-

POV :- Math and CS

Newton Method

The Newton Method is a powerful numerical technique primarily employed for approximating the roots of real-valued functions. I won't delve into the details of the Newton Method's process since you are already familiar with its main steps:

1. Start with an initial point.
2. Calculate the derivative at the initial point.
3. Perform a linear extrapolation.
4. Repeat steps 2 and 3 until the difference is less than 0.2.

These fundamental steps constitute the core of the Newton Method's procedure, allowing us to iteratively hone in on the roots of a function with increasing accuracy."

Sample code :)

1. from math import *
2. import time
3. def der(x): #derivative
4. c = 2*x #optional you can also sympy
5. return c
6. def func(x): #evalute
7. d = x**2 - 4
8. return d
9. x =0.5
10. c = time.time()
11. for i in range(20):
12. iter_start_time = time.time()
13. print("as",x)
14. x = x-(func(x)/der(x))
15. iter_end_time = time.time()
16. iter_time = iter_end_time - iter_start_time #updatefunction
17. print(f"Iteration {i+1} took {iter_time:.6f} seconds\n ")
18. d = time.time()