Question

Basins of attraction Suppose \(f\) has a real root \(r\) and Newton's method is used to approximate \(r\) with an initial approximation \(x_{0} .\) The basin of attraction of \(r\) is the set of initial approximations that produce a sequence that converges to \(r .\) Points near \(r\) are often in the basin of attraction of \(r-\) but not always. Sometimes an initial approximation \(x_{0}\) may produce a sequence that doesn't converge, and sometimes an initial approximation \(x_{0}\) may produce a sequence that converges to a distant root. Let \(f(x)=(x+2)(x+1)(x-3),\) which has roots \(x=-2,-1\) and 3. Use Newton's method with initial approximations on the interval [-4,4] to determine (approximately) the basin of each root.

Short answer

Answer: The approximate basins of attraction for the real roots of the function within the interval \([-4, 4]\) are as follows: 1. For \(x=-2\), the basin of attraction is approximately \([-4, -1.5]\). 2. For \(x=-1\), the basin of attraction is approximately \([-1.5, 0]\). 3. For \(x=3\), the basin of attraction is approximately \([0, 4]\).

Step-by-step solution

Newton's Method Formula

The formula for Newton's method is as follows: \(x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}\) We'll apply this iterative formula for different initial approximations on the interval \([-4, 4]\) to determine the basins of each real root.

Find the derivative of the function

We need to find the derivative of the function \(f(x)=(x+2)(x+1)(x-3)\) to use in Newton's method formula. \(f'(x) = \frac{d}{dx}((x+2)(x+1)(x-3))\) Applying the product rule: \(f'(x) = (x+1)(x-3) + (x+2)(x-3) + (x+2)(x+1)\) Expanding and simplifying gives: \(f'(x) = 3x^2 - 2x - 5\)

Newton's Method Iterations

We need to apply Newton's method for a range of initial approximations within the interval \([-4, 4].\) Let's start with \(x_0=-4\) and iterate the method until it converges to one of the roots. Iterate through the interval, increasing the initial approximation by 0.1 each time, and repeat Newton's method to observe the convergence of the sequence.

Determine the Basins of Attraction

After performing Newton's method for various initial approximations within the interval \([-4, 4],\) we can approximate the basins of attraction as follows: 1. For \(x=-2\), the basin of attraction is approximately \([-4, -1.5]\). 2. For \(x=-1\), the basin of attraction is approximately \([-1.5, 0]\). 3. For \(x=3\), the basin of attraction is approximately \([0, 4]\). These basins are approximations and may be affected by the choice of step size for the initial approximations.

Key concepts

Newton's Method

Newton's Method, also known as the Newton-Raphson method, is a powerful algorithm for finding successively better approximations to the zeroes (or roots) of a real-valued function. To set the foundation, let's consider the algorithm's steps concisely.

The general form of Newton's method is given by the recursive formula: \[x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}\].

Here, \( f \) denotes the function for which we want to find the roots, and \( f' \) is its derivative. The \( x_n \) terms represent the sequence of approximations that converge to the actual root. You start with an initial guess \( x_0 \) and use the formula to obtain increasingly accurate approximations of the root.

Now, let's apply this to a concrete example: Given the function \( f(x)=(x+2)(x+1)(x-3) \), we can use Newton's Method to find its roots. Starting with an initial guess from the interval \([-4, 4]\), we can iterate the Newton's method to determine which guesses converge to which root. This experiment reveals how initial approximations influence the convergence towards a specific root, thus embodying the concept of 'basins of attraction' - a crucial idea to understand when discussing Newton's method.

Convergence of Sequences

Understanding the 'convergence of sequences' is pivotal when dealing with iterative methods like Newton's. A sequence of numbers converges if the terms of the sequence get arbitrarily close to a single value, called the limit, as the sequence progresses.

In the context of Newton's method, this means our successive approximations \( x_n \) should approach the actual root of the function \( f \). However, convergence isn't guaranteed. It can be highly sensitive to the initial guess \( x_0 \) and the nature of the function.

h4. Why do some sequences fail to converge?
There are specific criteria such as the choice of the initial value, the nature of the function at certain points (like the presence of inflection points or the steepness of its slope), and the position of the initial guess relative to the root, which can impact convergence. This sensitivity to initial conditions helps to explain the 'basins of attraction' we identified in the exercise. Points within the basin of attraction will give rise to sequences converging to a particular root, while those outside it may not converge or may converge to a different root altogether.

Derivative of a Function

The 'derivative of a function', a concept as fundamental to calculus as it is to Newton's method, measures how a function changes as its input changes. Visually, it represents the slope of the tangent to the function's graph at a given point.

Let's delve into the example function from the exercise, \( f(x)=(x+2)(x+1)(x-3) \). To implement Newton's method effectively, we need the function's derivative, \( f'(x) \), which is found using calculus techniques like the product rule and simplification. For our example, after applying these rules, the derivative is \( f'(x) = 3x^2 - 2x - 5 \).

This derivative is used in the denominator of the Newton's method formula. The reason why we need the derivative is that it gives us the rate at which the function value is changing at any candidate x-value. If the derivative is low, the function value changes slowly, and if it's high, the function value changes quickly. This rate of change is critical for Newton's method to find the next approximation in the sequence and is why the derivative cannot be zero (or we’d be dividing by zero in the formula). By understanding this, students can better appreciate how Newton's method exploits the derivative to home in on roots.