Newton's method, applied to a differentiable function f(x),...

Newton's method, applied to a differentiable function $f(x),$ begins with a starting value $x_{0}$ and constructs from it a sequence of numbers $\left\{x_{n}\right\}$ that under favorable circumstances converges to a zero of $f .$ The recursion formula for the sequence is $$x_{n+1}=x_{n}-\frac{f\left(x_{n}\right)}{f^{\prime}\left(x_{n}\right)}$$. a. Show that the recursion formula for $f(x)=x^{2}-a, a>0$ can be written as $x_{n+1}=\left(x_{n}+a / x_{n}\right) / 2$ b. Starting with $x_{0}=1$ and $a=3,$ calculate successive terms of the sequence until the display begins to repeat. What number is being approximated? Explain.

Newton's method, applied to a differentiable function $f(x),$ begins with a starting value $x_{0}$ and constructs from it a sequence of numbers $\left\{x_{n}\right\}$ that under favorable circumstances converges to a zero of $f .$ The recursion formula for the sequence is $$x_{n+1}=x_{n}-\frac{f\left(x_{n}\right)}{f^{\prime}\left(x_{n}\right)}$$.
a. Show that the recursion formula for $f(x)=x^{2}-a, a>0$ can be written as $x_{n+1}=\left(x_{n}+a / x_{n}\right) / 2$
b. Starting with $x_{0}=1$ and $a=3,$ calculate successive terms of the sequence until the display begins to repeat. What number is being approximated? Explain.

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Key Concepts

Convergence of Iterative Methods

Convergence refers to the property that the iteratively generated approximations get arbitrarily close to the true solution as the number of iterations increases. In Newton's method, under appropriate conditions, the convergence near a zero of a function is typically fast, often doubling the number of correct digits in each step.

Newton's Method

Newton's method is an iterative algorithm used to find successively better approximations to the roots of a real-valued function. By employing the function and its derivative, each iteration refines the approximation, and under favorable conditions, the sequence converges quickly (often quadratically) to an actual root.

Recursion Formula

A recursion formula defines each term of a sequence as a function of its previous term(s). In iterative algorithms like Newton's method, the recursion formula is central because it provides a systematic way to obtain a better approximation based on current information.

Application to Root Finding

The process of approximating a zero of a function, such as finding a square root by setting up a corresponding equation, illustrates how iterative methods can be specialized to solve particular types of problems. This application includes reformulating the problem so that the iterative method yields a well-known computational algorithm.

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