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 […]

# Category Archives: Mathematics

## 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) […]

## 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 […]

## 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: […]

## 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 \ […]

## Measures on Algebras of Sets

FoldUnfold Table of Contents Measures on Algebras of Sets Measures on Algebras of Sets Definition: Let $X$ be a set and let $mathcal A$ be an algebra of sets (not necessarily a $sigma$-algebra) on $X$. A Measure on $mathcal A$ is a set function $mu : mathcal A to [0, infty]$ with the following properties: […]

## Outer Measurable Sets

FoldUnfold Table of Contents Outer Measurable Sets Outer Measurable Sets Recall from the Outer Measures on Measurable Spaces page that if we have the measurable space $(X, mathcal P(X))$ then an outer measure on this space is a set function $mu^* : mathcal P(X) to [0, infty]$ with the following properties: 1) $mu^*(emptyset) = 0$. […]

## Outer Measures on Measurable Spaces

FoldUnfold Table of Contents Outer Measures on Measurable Spaces Outer Measures on Measurable Spaces Definition: Let $(X, mathcal P(X))$ be a measurable space. An Outer Measure on this space is a set function $mu^* : mathcal P(X) to [0, infty]$ with the following properties: 1) $mu^*(emptyset) = 0$. 2) If $A$ and $B$ are subsets […]

## The Dominated Convergence Theorem for Measurable Functions

FoldUnfold Table of Contents The Dominated Convergence Theorem for Measurable Functions The Dominated Convergence Theorem for Measurable Functions Recall from The Lebesgue Dominated Convergence Theorem that if $(f_n(x))_{n=1}^{infty}$ is a sequence of Lebesgue measurable functions defined on a Lebesgue measurable set $E$ such that: 1) There exists a nonnegative Lebesgue integrable function $g$ such that […]

## The Comparison Test for Integrability

FoldUnfold Table of Contents The Comparison Test for Integrability The Comparison Test for Integrability Recall from The Comparison Test for Lebesgue Integrability that if $f$ is a Lebesgue measurable function defined on a Lebesgue measurable set $E$ and if there exists a nonnegative Lebesgue measurable function $g$ on $E$ such that: 1) $|f(x)| leq g(x)$ […]