If you begin the secant’s method near x = 2, which root is reached? How many iterations to achieve an error less than 10-5? solve

1. Break Down the Problem

To determine which root is reached by the secant method starting near $x = 2$, we'll follow these steps:

1. Identify the function for which we are finding the root.
2. Set the stopping criteria for the error.
3. Implement the secant method using an initial guess of $x = 2$.

2. Relevant Concepts

The secant method is an iterative numerical method for finding roots of a function. The formula for updating the approximations is given by:

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

Where:

• $f(x)$ is the function whose root we are looking for.
• $x_n$ is the current approximation.
• $x_{n-1}$ is the previous approximation.

We will apply the method until the absolute error:

$|x_{n+1} - x_n| < 10^{-5}$

3. Analysis and Detail

Let's assume we are trying to find the root of the function:

$f(x) = x^2 - 4$

This function has roots at $x = 2$ and $x = -2$. We will start near $x = 2$.

Iteration Steps

1. Choose initial points:

   • Let $x_0 = 2$ (initial guess)
   • Let $x_1 = 1.9$ (a nearby point)

2. Apply the secant method:

Iteration 1:

• Compute $f(x_0) = f(2) = 0$ and $f(x_1) = f(1.9) = 1.9^2 - 4 = -0.61$.
• Update $x_{2}$:

$x_2 = 1.9 - (-0.61) \frac{1.9 - 2}{-0.61 - 0} = 1.9 - (-0.61) \cdot \frac{-0.1}{-0.61} = 1.9 + 0.1 = 2$

Iteration 2:

• Since $x_2 = 2$ is reached, we can see that the method has converged immediately since it achieved $f(x_2) = 0$ in one iteration.

4. Verify and Summarize

• After starting at $x = 2$, it appears that the secant method converged to the root at $x = 2$ in just 1 iteration, with an error less than $10^{-5}$ achieved.

Final Answer

The root reached is $x = 2$ with 1 iteration required to achieve an error less than $10^{-5}$.