FoldUnfold Table of Contents Applying The Fixed Point Method for Solving Systems of Two Nonlinear Equations Example 1 Applying The Fixed Point Method for Solving Systems of Two Nonlinear Equations Be sure to review the following pages regarding The Fixed Point method for solving systems of two nonlinear equations: The Fixed Point Method for Solving Systems of Two … [Read more...]

## The Algorithm for The Fixed Point Method for Solving Systems of Two Nonlinear Equations

FoldUnfold Table of Contents The Algorithm for The Fixed Point Method for Solving Systems of Two Nonlinear Equations The Algorithm for The Fixed Point Method for Solving Systems of Two Nonlinear Equations We will now summarize The Fixed Point Method for Solving Systems of Two Nonlinear Equations in the following algorithm. Let $\left\{\begin{matrix} f(x, y) = 0\\ g(x, y) … [Read more...]

## Convergence of The Fixed Point Method for Solving Systems of Two Nonlinear Equations

FoldUnfold Table of Contents Convergence of The Fixed Point Method for Solving Systems of Two Nonlinear Equations Convergence of The Fixed Point Method for Solving Systems of Two Nonlinear Equations We will now develop criterion to ensure that the successive iterations converge to $(\alpha, \beta)$. If $(\alpha, \beta)$ is a solution to the system prescribed above, then … [Read more...]

## The Fixed Point Method for Solving Systems of Two Nonlinear Equations

FoldUnfold Table of Contents The Fixed Point Method for Solving Systems of Two Nonlinear Equations The Fixed Point Method for Solving Systems of Two Nonlinear Equations We will now look at an extension to The Fixed Point Method for Approximating Roots. Suppose that a solution $(\alpha, \beta)$ exists to the system of two nonlinear equations: (1) \begin{align} \quad … [Read more...]

## Newton’s Method for Solving Systems of Many Nonlinear Equations

FoldUnfold Table of Contents Newton's Method for Solving Systems of Many Nonlinear Equations Newton's Method for Solving Systems of Many Nonlinear Equations We will now extend Newton's Method further to systems of many nonlinear equations. Consider the general system of $n$ linear equations in $n$ unknowns: (1) \begin{align} f_1(x_1, x_2, …, x_n) = 0 \\ f_2(x_1, x_2, …, … [Read more...]

## Applying Newton’s Method for Solving Systems of Two Nonlinear Equations

FoldUnfold Table of Contents Applying Newton's Method for Solving Systems of Two Nonlinear Equations Example 1 Applying Newton's Method for Solving Systems of Two Nonlinear Equations Recall from the Newton's Method for Solving Systems of Two Nonlinear Equations page that if we have a system of two nonlinear equations $\left\{\begin{matrix} f(x, y) = 0 \\ g(x, y) = 0 … [Read more...]

## The Algorithm for Newton’s Method for Solving Systems of Two Nonlinear Equations

FoldUnfold Table of Contents The Algorithm for Newton's Method for Solving Systems of Two Nonlinear Equations The Algorithm for Newton's Method for Solving Systems of Two Nonlinear Equations We will now look at the algorithm for Newton's method for solving systems of two nonlinear equations. Let $\left\{\begin{matrix} f(x, y) = 0\\ g(x, y) = 0 \end{matrix}\right.$ be a … [Read more...]

## Newton’s Method for Solving Systems of Two Nonlinear Equations

FoldUnfold Table of Contents Newton's Method for Solving Systems of Two Nonlinear Equations Newton's Method for Solving Systems of Two Nonlinear Equations Recall from the Newton's Method for Approximating Roots page that if we have a function $y = f(x)$ and $\alpha$ is a root of this function, then if we have an initial approximation $x_0$ of this root, then we can define … [Read more...]

## Applying The Power Method to Find a Dominating Eigenvalue

FoldUnfold Table of Contents Applying The Power Method to Find a Dominating Eigenvalue Example 1 Applying The Power Method to Find a Dominating Eigenvalue Recall from The Power Method that if $A$ is an $n \times n$ matrix with $n$ distinct eigenvalues, $\lambda_1, \lambda_2, ..., \lambda_n$ for which one is dominating, say $\lambda_1$ such that $\mid \lambda _1 \mid \: > … [Read more...]

## The Power Method

FoldUnfold Table of Contents The Power Method The Power Method We will now look at a numerical method for finding one eigenvalue of an $n \times n$ matrix - in particular, the largest eigenvalue. Assume that $A$ is an $n \times n$ matrix that has $n$ distinct eigenvalues $\lambda_1$, $\lambda_2$, …, $\lambda_n$. Further assume that amongst these eigenvalues there exists … [Read more...]