The secant method is an iterative technique used to find the roots of a function. It starts with two initial approximations and iteratively improves the estimate by using a line through these points.

Given:

• $x_0 = 0$
• $x_1 = 0.045 + 0.345i$
• $f(x) = x^3 - 1$

Secant Method Formula

The formula for the secant method is:

$x_{n+1} = x_n - \frac{f(x_n) (x_n - x_{n-1})}{f(x_n) - f(x_{n-1})}$

Where:

• $x_0$ and $x_1$ are the initial guesses.
• $f(x)$ is the function whose root we want to find.

Steps to Apply the Secant Method

1. Initialize the first two guesses:

   • $x_0 = 0$
   • $x_1 = 0.045 + 0.345i$

2. Evaluate the function $f(x) = x^3 - 1$ at these points:

   • $f(x_0) = f(0) = 0^3 - 1 = -1$
   • $f(x_1) = f(0.045 + 0.345i)$

3. Use the secant formula to find $x_2$:

Repeat this process iteratively until the values converge to a root. I'll first compute the next few steps for you.

Let's start by calculating the first few iterations of the secant method. numpy as np

Define the function f(x) = x^3 - 1

def f(x): return x**3 - 1

Initialize values for x0 and x1

x0 = 0 x1 = 0.045 + 0.345j

Iterations for the secant method

iterations = 5 x_vals = [x0, x1] for i in range(2, iterations): # Secant method formula x_new = x_vals[-1] - f(x_vals[-1]) * (x_vals[-1] - x_vals[-2]) / (f(x_vals[-1]) - f(x_vals[-2])) x_vals.append(x_new)

x_valsThe first few iterations of the secant method are as follows:

1. $x_0 = 0$
2. $x_1 = 0.045 + 0.345i$
3. $x_2 = -7.985 - 2.119i$
4. $x_3 = 0.0587 + 0.3385i$
5. $x_4 = 0.0724 + 0.3319i$

It appears the values are oscillating, and it seems that the method might take more iterations to converge properly, or it might be struggling due to the choice of initial guesses.

Would you like to explore further iterations, adjust the initial guesses, or analyze the convergence behavior more deeply?