FoldUnfold Table of Contents The Algorithm for The Fixed Point Method The Algorithm for The Fixed Point Method We will now look at the algorithm for the fixed point method in approximating a root of a function. Obtain a function $f$ in the appropriate form ($f(x) = 0 \Leftrightarrow x = g(x)$) and assume that […]

## The Convergence of The Fixed Point Method

FoldUnfold Table of Contents The Convergence of The Fixed Point Method The Convergence of The Fixed Point Method Recall from The Fixed Point Method for Approximating Roots page that if we want to solve the equation $f(x) = 0$, then if we can rewrite this equation as $x = g(x)$ then the fixed points of […]

## The Fixed Point Method for Approximating Roots

FoldUnfold Table of Contents The Fixed Point Method for Approximating Roots The Fixed Point Method for Approximating Roots We saw on the Fixed Points page that $\alpha$ is a fixed point of $g$ if $\alpha = g(\alpha)$. We will now discuss why fixed points are important in finding roots. Suppose that we want to solve […]

## Fixed Points Examples 1

FoldUnfold Table of Contents Fixed Points Examples 1 Example 1 Example 2 Example 3 Fixed Points Examples 1 Recall from the Fixed Points page that if we can rewritten $f(x) = 0$ as $x = g(x)$, then if $\alpha$ is such that $\alpha = g(\alpha)$, then $\alpha$ is called a fixed point of $g$ and […]

## Fixed Points

FoldUnfold Table of Contents Fixed Points Example 1 Fixed Points So far we have looked at the Bisection Method and Newton’s Method for approximating roots of functions. We are about to introduce another root finding method know as the Fixed Point Method, but before we do so, we will need to learn about special types […]

## Error Analysis of Newton’s Method for Approximating Roots

FoldUnfold Table of Contents Error Analysis of Newton’s Method for Approximating Roots Error Analysis of Newton’s Method for Approximating Roots Recall from the Newton’s Method for Approximating Roots page that if $f$ is a differentiable function that contains the root $\alpha$, and $x_0$ is an approximation of $\alpha$, then we can obtain a sequence of […]

## Applying Newton’s Method

FoldUnfold Table of Contents Applying Newton’s Method Example 1 Applying Newton’s Method Be sure to review the following pages regarding Newton’s Method: Newton’s Method for Approximating Roots Error Estimation and Error Verification of Newton’s Method The Convergence of Newton’s Method We will now look at an examples of approximating a root with Newton’s Method, approximating […]

## The Algorithm for Newton’s Method for Approximating Roots

FoldUnfold Table of Contents The Algorithm for Newton’s Method for Approximating Roots The Algorithm for Newton’s Method for Approximating Roots We will now look at the algorithm for Newton’s Method for approximating roots to functions. Obtain a function $f$ and assume that a root $\alpha$ exists. Obtain an initial approximation $x_0$ to this root. Also […]

## The Convergence of Newton’s Method

FoldUnfold Table of Contents The Convergence of Newton’s Method The Convergence of Newton’s Method Suppose that $f$ is a twice differentiable function on an interval containing the root of interest, $\alpha$ and suppose that $f'(\alpha) \neq 0$. Now consider the first order Taylor polynomial of $f$ about $x_n$ denoted $P_1(x) = f(x_n) + (x – […]

## Error Estimation and Error Verification of Newton’s Method

FoldUnfold Table of Contents Error Estimation and Error Verification of Newton’s Method Error Verification Error Estimation and Error Verification of Newton’s Method Let $f$ be a function that satisfies the conditions to apply Newton’s Method and converges to the root $\alpha$. We will now develop a way to estimate the error of the approximation iterates […]

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