Problem 101 - Thomas Calculus

Question

Newton's method The following sequences come from the recursion formula for Newton's method, $$x_{n+1}=x_{n}-\frac{f\left(x_{n}\right)}{f^{\prime}\left(x_{n}\right)}$$ Do the sequences converge? If so, to what value? In each case, begin by identifying the function \(f\) that generates the sequence. $$ \begin{array}{l}{\text { a. } x_{0}=1, \quad x_{n+1}=x_{n}-\frac{x_{n}^{2}-2}{2 x_{n}}=\frac{x_{n}}{2}+\frac{1}{x_{n}}} \\\ {\text { b. } x_{0}=1, \quad x_{n+1}=x_{n}-\frac{\tan x_{n}-1}{\sec ^{2} x_{n}}} \\ {\text { c. } x_{0}=1, \quad x_{n+1}=x_{n}-1}\end{array} $$

Step-by-Step Solution

Answer

Sequence (a) converges to \(\sqrt{2}\). Sequence (b) converges to \(\frac{\pi}{4}\). Sequence (c) does not converge.

Step 1: Identify the Function for Sequence (a)

For sequence (a), the update formula is \( x_{n+1}=\frac{x_{n}}{2}+\frac{1}{x_{n}} \). This means \( f(x) = x^2 - 2 \), since its derivative is \( f'(x) = 2x \). Substituting in the Newton's method formula \( x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \) confirms it matches the given sequence.

Step 2: Analyze Convergence of Sequence (a)

We are seeking the square root of 2, where \( f(x) = x^2 - 2 \) becomes zero. Newton's method will converge to \( \sqrt{2} \) if it is initially 'close enough'. Starting with \( x_0 = 1 \), iteration will refine \( x_n \) towards \( \sqrt{2} \).

Step 3: Identify the Function for Sequence (b)

For sequence (b), given \( x_{n+1} = x_{n} - \frac{\tan(x_{n}) -1}{\sec ^{2}(x_{n})} \), the function is \( f(x) = \tan(x) - 1 \), as \( f'(x) = \sec^2(x) \).

Step 4: Analyze Convergence of Sequence (b)

We solve \( \tan(x) = 1 \), occurring at \( x = \frac{\pi}{4} \) with periodicity \( n\pi \), for any integer \( n \). With \( x_0 = 1 \), \( x_n \) converges to \( \frac{\pi}{4} \) because it is the closest solution.

Step 5: Identify and Analyze Sequence (c)

For sequence (c), \( x_{n+1} = x_n - 1 \) implies a linear decrease without stopping, since there is no \( f(x) \) to evaluate. The sequence does not converge to a finite value, as it continuously decreases by 1.

Key Concepts

Convergence Analysis

Convergence analysis is a crucial part of understanding how and if sequences generated by Newton's Method approach a specific value, known as a solution or root of an equation. In practice, convergence refers to whether iterative methods, like Newton's Method, actually zero in on a root over successive iterations or not.
Newton's Method is quite powerful because its convergence is typically quadratic, meaning the error between the estimated root and the actual root decreases exponentially as iterations progress. However, this rapid convergence is guaranteed only under specific conditions:

• The initial guess must be sufficiently close to the true root.
• The function must be differentiable near the root.
• The derivative at the initial guess must not be zero.

For example, in sequence (a), with an initial guess of 1 for finding \(\sqrt{2}\), the method effectively converges because these conditions are met. Convergence in such cases is not only fast but reliable, ensuring accuracy in calculations. Understanding these criteria helps decide how suitable Newton's Method is for solving a given problem.

Numerical Methods

Numerical methods are techniques for approximating solutions to mathematical problems that are otherwise difficult or impossible to solve analytically. Newton's Method is a prime example of a numerical method used for finding roots of real-valued functions.
These methods are particularly useful because they provide solutions even when an equation cannot be rearranged into a simple algebraic form. The core idea is to turn complex problems into simpler, iterative procedures. Newton's Method does this by using the function's derivative to successively get better approximations of the root:
1. Start with an initial guess close to the expected root.
2. Improve the guess using the formula \(x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}\).
3. Repeat until the change is smaller than a predetermined threshold.
This approach allows for solving problems like sequence (b), which involves trigonometric functions that have periodic solutions, demonstrating the flexibility of numerical methods in various mathematical contexts.

Root-Finding Algorithms

Root-finding algorithms, such as Newton's Method, are designed to locate zeros or "roots" of a function. A root is where the function evaluates to zero. Finding these values is fundamental to many areas of science and engineering because they often represent meaningful quantities, such as equilibrium points or solutions to physical systems.
Newton's Method stands out because of its efficiency and speed. It works by approximating the root through a tangent line approach. This mimics finding where the tangent line of the function crosses the x-axis, iteratively honing in on the root. However, unlike the method's example (c) with constant subtraction (which doesn't seek a root), Newton's Method requires a differential function to make adjustments.
With roots often not directly observable, especially in complex functions, root-finding offers a framework to estimate them effectively. Recognizing when and why an algorithm like Newton’s fails or succeeds is key to leveraging these powerful tools appropriately.