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# 数学代写|数值分析代写Numerical analysis代考|MATH/CS514 Composite quadrature rules

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## 数学代写|数值分析代写Numerical analysis代考|Composite quadrature rules

In this section, we will derive the composite trapezoid and Simpson rules. The idea of composite quadrature rules is to increase the number of nodes in the interval of integration and thus decompose it into a union of disjoint subintervals. Then the related quadrature rule is applied on each of the subintervals. Naturally, this approach will reduce the error in the computation of the approximate value of the integral.

Suppose that $a, b \in \mathbb{T}, a<b$, and $f:[a, b] \rightarrow \mathbb{R}$ is a given continuous function.
Let
$$a=x_0<x_1<\cdots<x_n=b .$$
Then
$$\int_a^b f(x) \Delta x=\sum_{j=0}^{n-1} \int_{x_j}^{x_{j-1}} f(x) \Delta x$$

## 数学代写|数值分析代写Numerical analysis代考|σ-Composite quadrature rules

In a similar way, we can define the $\sigma$-composite trapezoid and Simpson rules. First, we derive the $\sigma$-composite trapezoid rule. Suppose that $a, b \in \mathbb{T}, a<b, f:[a, b] \rightarrow \mathbb{R}$ is rd-continuous, $n \in \mathbb{N}$ and that $x_j \in[a, b] \subset \mathbb{T}, j \in{0,1, \ldots, n}$ are $\sigma$-distinct, i. e.,
$$a=x_0 \leq \sigma\left(x_0\right)<x_1 \leq \sigma\left(x_1\right)<\cdots<x_n \leq \sigma\left(x_n\right)=b .$$

Then
\begin{aligned} \int_a^b f(x) \Delta x=& \int_{x_0}^{\sigma\left(x_1\right)} f(x) \Delta x+\int_{\sigma\left(x_1\right)}^{\sigma\left(x_2\right)} f(x) \Delta x+\cdots+\int_{\sigma\left(x_{n-1}\right)}^{\sigma\left(x_n\right)} f(x) \Delta x \ =& \int_{x_0}^{\sigma\left(x_1\right)} f(x) \Delta x+\left(\int_{x_1}^{\sigma\left(x_2\right)} f(x) \Delta x-\int_{x_1}^{\sigma\left(x_1\right)} f(x) \Delta x\right) \ &+\cdots+\left(\int_{x_{n-1}}^{\sigma\left(x_n\right)} f(x) \Delta x-\int_{x_{n-1}}^{\sigma\left(x_{n-1}\right)} f(x) \Delta x\right) \ =& \sum_{j=0}^{n-1} \int_{x_j}^{\sigma\left(x_{j+1}\right)} f(x) \Delta x-\sum_{j=0}^{n-1} \int_{x_j}^{\sigma\left(x_j\right)} f(x) \Delta x \ =& \sum_{j=0}^{n-1} \int_{x_j}^{\sigma\left(x_{j+1}\right)} f(x) \Delta x-\sum_{j=0}^{n-1} \mu\left(x_j\right) f\left(x_j\right) . \end{aligned}
We apply the $\sigma$-trapezoid rule and obtain
\begin{aligned} \int_a^b f(x) \Delta x \approx & \sum_{j=0}^{n-1} \frac{1}{\sigma\left(x_{j+1}\right)-\sigma\left(x_j\right)}\left(f\left(x_j\right) g_2\left(x_j, \sigma\left(x_{j+1}\right)\right)+f\left(x_{j+1}\right) g_2\left(\sigma\left(x_{j+1}\right), \sigma\left(x_j\right)\right)\right) \ &-\sum_{j=0}^{n-1} \mu\left(x_j\right) f\left(x_j\right) . \end{aligned}

## 数学代写|数值分析代写数值分析代考|复合求积规则

Let
$$a=x_0<x_1<\cdots<x_n=b .$$
Then
$$\int_a^b f(x) \Delta x=\sum_{j=0}^{n-1} \int_{x_j}^{x_{j-1}} f(x) \Delta x$$

## 数学代写|数值分析代写数值分析代考|σ-复合求积规则

$$a=x_0 \leq \sigma\left(x_0\right)<x_1 \leq \sigma\left(x_1\right)<\cdots<x_n \leq \sigma\left(x_n\right)=b .$$

\begin{aligned} \int_a^b f(x) \Delta x=& \int_{x_0}^{\sigma\left(x_1\right)} f(x) \Delta x+\int_{\sigma\left(x_1\right)}^{\sigma\left(x_2\right)} f(x) \Delta x+\cdots+\int_{\sigma\left(x_{n-1}\right)}^{\sigma\left(x_n\right)} f(x) \Delta x \ =& \int_{x_0}^{\sigma\left(x_1\right)} f(x) \Delta x+\left(\int_{x_1}^{\sigma\left(x_2\right)} f(x) \Delta x-\int_{x_1}^{\sigma\left(x_1\right)} f(x) \Delta x\right) \ &+\cdots+\left(\int_{x_{n-1}}^{\sigma\left(x_n\right)} f(x) \Delta x-\int_{x_{n-1}}^{\sigma\left(x_{n-1}\right)} f(x) \Delta x\right) \ =& \sum_{j=0}^{n-1} \int_{x_j}^{\sigma\left(x_{j+1}\right)} f(x) \Delta x-\sum_{j=0}^{n-1} \int_{x_j}^{\sigma\left(x_j\right)} f(x) \Delta x \ =& \sum_{j=0}^{n-1} \int_{x_j}^{\sigma\left(x_{j+1}\right)} f(x) \Delta x-\sum_{j=0}^{n-1} \mu\left(x_j\right) f\left(x_j\right) . \end{aligned}

\begin{aligned} \int_a^b f(x) \Delta x \approx & \sum_{j=0}^{n-1} \frac{1}{\sigma\left(x_{j+1}\right)-\sigma\left(x_j\right)}\left(f\left(x_j\right) g_2\left(x_j, \sigma\left(x_{j+1}\right)\right)+f\left(x_{j+1}\right) g_2\left(\sigma\left(x_{j+1}\right), \sigma\left(x_j\right)\right)\right) \ &-\sum_{j=0}^{n-1} \mu\left(x_j\right) f\left(x_j\right) . \end{aligned}

## MATLAB代写

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