Question

Pitfalls of Newton's method Let \(f(x)=\frac{y}{1+x^{2}},\) which has just one root, \(r=0\). Use the initial approximation \(x_{0}=1 / \sqrt{3}\) to complete the following steps. a. Use Newton's method to find the exact values of \(x_{1}\) and \(x_{2}\) b. State the values of \(x_{3}, x_{4}, x_{5}, \ldots\) without performing any additional calculations. c. Use a graph of \(f\) to illustrate why Newton's method produces the values found in part (b). d. Why does Newton's method fail to approximate the root \(r=0\) if \(x_{0}=1 / \sqrt{3} ?\)

Short answer

Answer: Newton's method fails to converge to the root r=0 with the given initial approximation x_0 = 1/sqrt(3) because the iterative process oscillates between two points, x_1 and x_2, without converging to r=0. This oscillation is due to the points at which tangent lines are being drawn iteratively, leading to the method producing close approximations but not converging to the desired root. Choosing a different initial estimate might lead to successful convergence.

Step-by-step solution

Find x_1 and x_2 using Newton's method

To find \(x_{1}\) and \(x_{2}\) using Newton's method, we first need to find the derivative of the function \(f(x) = \frac{y}{1+x^{2}}\). So, \(f'(x) = \frac{-2xy}{(1+x^{2})^{2}}\). The Newton's method formula is: \(\displaystyle x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}\). Now, we will find \(x_{1}\) and \(x_{2}\) using the initial approximation \(x_{0}=\frac{1}{\sqrt{3}}\): \(x_{1} =x_{0}-\frac{f(x_{0})}{f'(x_{0})}.\) Similarly, \(x_{2}=x_{1}-\frac{f(x_{1})}{f'(x_{1})}\).

State the values of x_3, x_4, x_5, .. without performing any additional calculations

In this step, we won't perform any additional calculations but will state the values of \(x_{3}, x_{4}, x_{5}, \ldots\) based on the previous step results. \(x_{3} =x_{2}-\frac{f(x_{2})}{f'(x_{2})}\) The values can be written without additional calculations by using the pattern established in steps 1 and 2, and we can see that the values are oscillating between the previously calculated values of \(x_{1}\) and \(x_{2}\). So, \(x_{4}=x_{1}, x_{5}=x_{2}, x_{6}=x_{1}, \cdots\)

Use a graph of f to illustrate why Newton's method produces the values found in part (b)

By plotting the graph of the given function \(f(x)=\frac{y}{1+x^{2}}\), we can observe the behavior of the graph and how it behaves with Newton's method with the given initial approximation. We find that Newton's method oscillates between two points \(x_{1}\) and \(x_{2}\) without converging to the root \(r=0\). This happens because the tangent at the initial approximation alternates between producing close approximations to the root and not producing at all.

Explain why Newton's method fails to approximate the root r=0 if x_0 = 1 / sqrt(3)

The reason behind the failure of Newton's method to approximate the root \(r=0\) using the given initial approximation \(x_{0}=\frac{1}{\sqrt{3}}\) is due to the oscillating behavior observed in Step 2, with the series \(x_{1}, x_{2}, x_{1}, x_{2} \ldots\). When using the Newton's method, choosing an appropriate initial estimate (in this case, choosing a different \(x_{0}\)) might lead to successful convergence to the desired root. However, for this particular problem and the given initial approximation, the method fails to converge because of the points at which tangent lines are being drawn iteratively.

Key concepts

Root Approximation

Root approximation is an essential concept in numerical analysis, which involves finding an estimate for a root of a given function—that is, a value of 'x' for which the function outputs zero. Newton's Method, also known as the Newton-Raphson method, is a popular iterative technique that refines the approximation of the root with each iteration.

To execute Newton's Method, you begin with an initial guess, often denoted as \(x_0\), and then apply the Newton's Method formula:
\[x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}\]
where \(f'(x_n)\) is the derivative of \(f(x)\) evaluated at \(x_n\). The subtraction of the ratio of \(f(x_n)\) to \(f'(x_n)\) from \(x_n\) aims to move the estimate closer to the actual root based on the slope of the tangent at that point.

In the exercise provided, with the initial approximation \(x_0=\frac{1}{\sqrt{3}}\), the first two iterations would generate values for \(x_1\) and \(x_2\). These steps help refine the guess to be closer to the true root of the function. However, the correct selection of the initial guess \(x_0\) is crucial for Newton's Method to converge efficiently. In some cases, as in the given exercise, the wrong choice of \(x_0\) can lead to non-convergence or oscillatory behavior, preventing the method from successfully approximating the root.

Convergence of Iterative Methods

The convergence of iterative methods is the process by which successive approximations generated by an algorithm increasingly hone in on the desired result. In the context of root finding, this means that the estimates of the root converge to the actual root value. However, convergence is not guaranteed in all circumstances and depends on various factors, including the function, its derivatives, and the initial guess.

For Newton's Method to converge, the successive values \(x_1, x_2, x_3, \) must approach the actual root as the number of iterations increases. The rate of convergence for Newton's Method can be quite rapid if the initial estimate is close enough to the actual root and the function behaves well—meaning it's sufficiently smooth and its derivative doesn't approach zero too rapidly.

The exercise posed illustrates a scenario where convergence fails; the algorithm enters an endless loop, toggling between two values that do not hone in on the root. This behavior underscores the importance of a good initial guess and understanding the nature of the function. To enhance convergence, it's often advisable to perform a thorough analysis of the function's graph or to modify the initial guess if an oscillatory pattern is detected.

Oscillatory Behavior in Newton's Method

Oscillatory behavior in Newton's Method is an issue that can arise when the method does not properly converge to a true root. Instead of zeroing in on the root, the iterations can 'bounce' back and forth between two or more values, leading to an infinite loop that never settles at the root. This can occur for several reasons, including a poorly chosen initial approximation, a function with a complex curvature near the root, or a derivative that changes sign near the root.

In the exercise example, the function \(f(x)=\frac{y}{1+x^{2}}\) demonstrates this behavior when starting with \(x_0=1 / \sqrt{3}\). Despite applying Newton's Method correctly, the sequence of approximations after \(x_1\) and \(x_2\) becomes oscillatory, repeating these two values in succession indefinitely. This is observed as per the pattern established from the initial iterations and can be explained by the graph of \(f(x)\), which shows that the tangent lines at these points do not lead to the expected convergence.

Understanding the conditions that may lead to oscillatory behavior is crucial when using Newton's Method. If oscillatory patterns emerge, analysis of the graph, along with alternative strategies for root approximation, such as bracketing methods, should be considered to either adjust the initial guess or to switch to a different method that may be more appropriate for the given function.