Numerical Analysis Algorithms in Python

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Instead, Let’s get cracking with some practical examples and have a little fun along the way!

First things first: what is numerical analysis? Well, its basically the study of how computers can be used to solve mathematical problems that involve numbers (duh). And in Python, we have access to all sorts of awesome libraries like NumPy, SciPy, and SymPy that make this process a whole lot easier.

So let’s get started with one of my favorite numerical analysis algorithms: the bisection method! This is a simple yet powerful technique for finding roots (or solutions) to equations by repeatedly dividing an interval in half until we find what were looking for. And heres how it works in Python code:

# Import necessary libraries
import math # Importing the math library for mathematical operations
from sympy import Symbol, solve # Importing the sympy library for symbolic mathematics

# Define the function we want to solve
def f(x):
    return x**3 + 5*x + 1 # Corrected the syntax error, added a plus sign between x**3 and 5*x

# Set up our initial bounds (the interval where we think the root might be)
a = 0 # Lower bound of the interval
b = 2 # Upper bound of the interval

# Keep iterating until we find a solution within our tolerance level (in this case, 0.0001)
while abs(f((a+b)/2)) > 0.0001:
    # Calculate the midpoint of our current interval
    c = (a + b) / 2
    
    # Check if we've found a solution within our tolerance level
    if abs(f(c)) < 0.0001:
        print("Root found at:", c) # Print the root if it is within the tolerance level
        
    # Update our bounds based on whether the function is positive or negative in the middle of our interval
    else:
        if f(a)*f(c) > 0: # If the function has the same sign at both ends of the interval, update the lower bound
            a = c
        else: # If the function has different signs at both ends of the interval, update the upper bound
            b = c

With just a few lines of code, we’ve implemented one of the most basic and important numerical analysis algorithms in Python. And who knows maybe this will inspire you to explore other techniques like Newton’s method or Simpson’s rule!

But for now, let’s just sit back, relax, and enjoy the beauty of math (or at least try our best to understand it)!

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