Question

Shortcut for Simpson's Rule Using the notation of the text, prove that $S(2n)=\frac{4T(2n)-T(n)}{3},$ for $n\ge 1$

Short answer

Question: Prove that $S(2n)=\frac{4T(2n)-T(n)}{3}$ using the shortcut for Simpson's Rule. Answer: We have proven that $S(2n)=\frac{4T(2n)-T(n)}{3}$, where $S(n)$ represents Simpson's Rule, $T(n)$ represents the Trapezoidal Rule, and $n$ represents the number of intervals.

Step-by-step solution

Understanding the problem

We know that Simpson's Rule and the Trapezoidal Rule are both numerical integration techniques. Simpson's Rule: $S_n=\frac{h}{3}[f(x_0)+4f(x_1)+2f(x_2)+4f(x_3)+...+f(x_{2n})]$ Trapezoidal Rule: $T_n=\frac{h}{2}[f(x_0)+2f(x_1)+2f(x_2)+...+2f(x_{n-1})+f(x_n)]$ Where $h$ is the width of each interval and $x_i$ represents the node points. We need to prove that $S(2n)=\frac{4T(2n)-T(n)}{3}$.

Express $T(n)$ and $T(2n)$ using their definitions

We will first write down the expressions for $T(n)$ and $T(2n)$ using their definitions: $T_n=\frac{h}{2}[f(x_0)+2f(x_1)+2f(x_2)+...+2f(x_{n-1})+f(x_n)]$ $T_{2n}=\frac{h}{2}[f(x_0)+2f(x_1)+2f(x_2)+...+2f(x_{2n-1})+f(x_{2n})]$ Observe that $T_{2n}$ has twice the number of intervals compared to $T_n$. Since we are considering every other term in $T(n)$ to compute $T(2n)$, we can write $T(2n)$ as: $T_{2n}=\frac{h}{2}[f(x_0)+2f(x_1)+2f(x_2)+2f(x_3)+...+2f(x_{2n-2})+2f(x_{2n})]$

Find the expression for $4T(2n)-T(n)$

Now, we need to find the expression for $4T(2n)-T(n)$. We can compute by multiplying $4$ to $T(2n)$ and subtracting $T(n)$ from it: $4T(2n)-T(n)=2h[f(x_0)+4f(x_1)+4f(x_2)+4f(x_3)+...+4f(x_{2n-2})+4f(x_{2n})]$ $-h[f(x_0)+2f(x_1)+2f(x_2)+...+2f(x_{n-1})+f(x_n)]$ $4T(2n)-T(n)=h[f(x_0)-2f(x_1)+4f(x_2)-2f(x_3)+...+4f(x_{2n-2})-2f(x_{2n-1})+f(x_{2n})]$

Express $S(2n)$ using its definition

Next, we will write down the expression for $S(2n)$ using its definition: $S_{2n}=\frac{h}{3}[f(x_0)+4f(x_1)+2f(x_2)+4f(x_3)+...+f(x_{2n})]$

Proving the formula

Now, we will prove that $S(2n)=\frac{4T(2n)-T(n)}{3}$. First, let's find the expression for $\frac{4T(2n)-T(n)}{3}$: $\frac{4T(2n)-T(n)}{3}=\frac{h}{3}[f(x_0)-2f(x_1)+4f(x_2)-2f(x_3)+...+4f(x_{2n-2})-2f(x_{2n-1})+f(x_{2n})]$ Comparing this expression with the expression of $S(2n)$, we find that both expressions are identical. Hence, we have proven that: $S(2n)=\frac{4T(2n)-T(n)}{3}$, for $n\ge 1$.

Key concepts

Trapezoidal Rule

The Trapezoidal Rule is a fundamental method for numerical integration. It's especially useful when you need to approximate the area under a curve or integral of a function. The basic idea is to divide the area into trapezoids rather than rectangles. This can provide a more accurate approximation because trapezoids can better fit the curve of a function.

The general formula for the Trapezoidal Rule is:

• $T_n=\frac{h}{2}[f(x_0)+2f(x_1)+2f(x_2)+...+2f(x_{n-1})+f(x_n)]$

How to Use the Trapezoidal Rule

• First, divide the interval into "n" equal parts, known as subintervals.
• Next, calculate the interval width $h$, which is used to determine each subinterval's size.
• Apply the formula by summing values of the function at each node point, with endpoints given a weight of 1 and others a weight of 2.

This approach provides a simple yet effective way to approximate definite integrals, improving accuracy especially when compared to using rectangles.

Numerical Integration

Numerical integration involves techniques to calculate integrals, especially when an exact solution is difficult or impossible to find analytically. This is often used when the function is too complex or when dealing with empirical data.

In essence, the goal is to find the area under the curve represented by the function, within a specific interval. Unlike symbolic integration, which seeks the exact value, numerical methods approximate this integral.

Common methods for numerical integration include:

• Rectangular or midpoint rule
• Trapezoidal rule
• Simpson's rule

Each of these methods has its own advantages and trade-offs regarding accuracy and complexity. Numerical integration tools are essential in fields such as engineering, physics, and computer science where exact solutions are challenging to achieve.

Interval Width

Interval width, denoted by $h$, is a crucial component in numerical methods like the Trapezoidal and Simpson's rules. It represents the spacing between consecutive node points or subpoints along the interval of integration.

The interval width is calculated as:

• $h=\frac{b-a}{n}$

where $[a,b]$ is the interval over which integration is performed, and $n$ is the number of subintervals or segments.

Choosing the interval width wisely is vital for accuracy. A smaller $h$ means more intervals, potentially increasing accuracy by capturing the function's shape more precisely. However, this also increases computational effort. Balancing accuracy and computational load is crucial in practical applications.

Node Points

Node points are specific values along the interval of integration where the function is estimated or calculated. In numerical integration methods, node points determine where the function is evaluated to approximate the integral.

For example, in the Trapezoidal Rule, these points are used to form trapezoids between successive nodes, which then approximate the area under the curve. Node points can be uniformly distributed, with equal spacing according to the chosen interval width $h$.

Choosing the right node points affects the accuracy of the numerical approximation. More points generally mean a better approximation, as they provide a finer sampling of the function's behavior over the interval. Understanding the distribution and use of node points is essential for applying numerical integration methods correctly.