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# 数学代写|数值分析代写Numerical analysis代考|STAT434 The bisection method

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## 数学代写|数值分析代写Numerical analysis代考|The bisection method

The bisection method is a very robust method for solving nonlinear equations in one variable. It is based on a special case of the intermediate value theorem, which we recall now from first semester Calculus.

Theorem 32 (Intermediate Value Theorem). Suppose a function $f$ is continuous on $[a, b]$, and $f(a) \cdot f(b)<0$. Then there exists a number $c$ in the interval $(a, b)$ such that $f(c)=0$.

The theorem says that if $f(a)$ and $f(b)$ are of opposite signs, and $f$ is continuous, then the graph of $f$ must cross the $x$-axis $(y=0)$ between $a$ and $b$. Consider the following illustration of the theorem in Figure 5.1. In the top picture, two points are shown whose $y$-values have opposite signs. Do you think you can draw a continuous curve that connects these points without crossing the dashed line? No, you cannot! This is precisely what the intermediate value theorem says.

Now that we have established that any continuous curve that connects the points must cross the dashed line, let us name the $x$-point where it crosses to be $c$, as in Figure 5.1. Note that a curve may cross multiple times, but we are guaranteed that it crosses at least once.

## 数学代写|数值分析代写Numerical analysis代考|Fixed-point theory and algorithms

We consider now how to find solutions to $f(x)=0$ by finding solutions to $g(x)=x$. Solutions $x^$ of the latter equation are called “fixed points” because they are fixed with respect to the function $g$; when you plug $x^$ into $g$, you get back $x^*$.

At first, it may seem strange why we would consider the equation $g(x)=x$, but it turns out that the mathematical theory for finding solutions to fixed-point problems is easier to decipher than for rootfinding. Of course, it is equivalent mathematically, for example, finding the roots of $x^2-x-2=0$ is equivalent to finding the fixed points of $g(x)=x^2-2$, and also $g(x)=1+\frac{2}{x}$.

We will consider only the scalar case herein: $g: \mathbb{R} \rightarrow \mathbb{R}$. However, almost all of the theory generalizes to $\mathbb{R}^n$, and even general metric spaces, without significant difficulties.

We will denote by $I$ a generic closed interval, that is, $I=[a, b]$. We will assume the function $g: I \rightarrow I$ is continuous. In some of the theory that follows, $g$ will have additional smoothness properties.
Define the fixed-point iteration by
$$x_{k+1}=g\left(x_k\right) .$$
Much of this section will be concerned with when and how quickly this algorithm converges to a fixed point. Clearly, the fixed-point (FP) iteration is easy to code up and use. The difficulty with fixed-point methods usually comes from creating the function $g$ for which one would get fast and robust convergence of the FP iteration.

## 数学代写|数值分析代写NUMERICAL ANALYSIS代 考|FIXED-POINT THEORY AND ALGORITHMS

$$x_{k+1}=g\left(x_k\right) .$$

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