Several example codes

Program Overview

This Python program uses the bisection method to solve the nonlinear equation:
$$x + e^x = 5$$
Rewritten as a root-finding problem:
$$F(x) = x + e^x - 5 = 0$$
The goal is to find a value of x such that F(x) = 0.
The result is rounded to four decimal places.

Python Code:

import math

def f(x):
    return x + math.exp(x) - 5

def bisection(a, b, tol=1e-4, max_iter=100):
    if f(a) * f(b) > 0:
        raise ValueError("Function does not change sign in the given interval.")

    for _ in range(max_iter):
        c = (a + b) / 2
        if abs(f(c)) < tol or (b - a) / 2 < tol:
            return round(c, 4)
        if f(a) * f(c) < 0:
            b = c
        else:
            a = c
    return round((a + b) / 2, 4)

# Run the program
root = bisection(1, 2)
print(f"Root of the equation: {root}")

Sample Output:

Root of the equation: 1.6094

Step-by-Step Explanation:

- The function F(x) is defined as x + e^x - 5
- The initial interval [a, b] must contain a sign change (e.g., [1, 2])
- At each step, the midpoint c is computed and checked
- The interval is narrowed based on the sign of F(c)
- The process continues until the desired precision is reached