Question

Let us continue Exercise 4 . (a) Derive the composite corrected trapezoidal method. How does it relate to the composite trapezoidal method? Use both composite trapezoidal and corrected composite trapezoidal to evaluate approximations for \(I_{f}=\int_{0}^{1} e^{-x^{2}} d x\) with \(r=10\) subintervals. What are your observations? [The exact value is \(I_{f}=0.746824133 \ldots .\) ] (b) Show that the error in the uncorrected composite trapezoidal method can be written as $$ E(f)=I_{f}-I_{I r}=K_{1} h^{2}+\mathcal{O}\left(h^{4}\right) $$ where \(K_{1}\) is independent of \(h\).

Short answer

Question: Derive the composite corrected trapezoidal method, relate it to the composite trapezoidal method, and use both methods to approximate \(I_f\). Next, show the error in the composite trapezoidal method and express it in terms of \(h\). Answer: The composite corrected trapezoidal method is an improvement over the composite trapezoidal method by adding a curvature correction term. It is given by the formula: $$ I_{cc}=\frac{h}{2}\sum_{i=1}^{n} f(x_{i-1})+f(x_i) - \frac{h^2}{12}(f'(x_n)-f'(x_0)) $$ The composite trapezoidal method is given by the formula: $$ I_{c}=\frac{h}{2}\sum_{i=1}^{n} f(x_{i-1})+f(x_i) $$ Both methods can be used to approximate \(I_f\), with the composite corrected trapezoidal method providing more accurate results. The error in the composite trapezoidal method is expressed in terms of \(h\) as: $$ E(f)=-\frac{h^{3}}{12n}\sum_{i=1}^{n} f''(\xi_i) = K_1 h^2 + O(h^4) $$ where \(\xi_i\) is a point in the interval \([x_{i-1}, x_i]\), and \(K_1\) is a constant independent of \(h\).

Step-by-step solution

Deriving the Composite Corrected Trapezoidal Method

Let's first derive the composite corrected trapezoidal method, which is an improvement over the basic trapezoidal method by taking into account the curvature of the function. For the composite corrected trapezoidal method, the integral is approximated as follows: $$ I_{cc}=\frac{h}{2}\sum_{i=1}^{n} f(x_{i-1})+f(x_i) - \frac{h^2}{12}(f'(x_n)-f'(x_0)) $$ Here, we're adding a curvature correction term to the standard composite trapezoidal rule.

Relationship to Composite Trapezoidal Method

The composite corrected trapezoidal method includes a curvature correction term compared to the standard trapezoidal method. The composite trapezoidal method approximates the integral as follows: $$ I_{c}=\frac{h}{2}\sum_{i=1}^{n} f(x_{i-1})+f(x_i) $$ Comparing the two, we can establish that the composite corrected trapezoidal method is an improved version of the composite trapezoidal method.

Using Both Methods to Evaluate \(I_f\)

Let's evaluate the given integral using both the composite trapezoidal and composite corrected trapezoidal methods with 10 subintervals. 1. Calculate the width of the subintervals, \(h\): $$ h = \frac{1-0}{10} = 0.1 $$ 2. Define the function and its derivative: $$ f(x) = e^{-x^2} \quad \text{and} \quad f'(x) = -2xe^{-x^2} $$ 3. Calculate the sum of the function values using the composite trapezoidal rule: $$ I_{c} = 0.1 \times \frac{1}{2} \left[f(0)+f(1) + 2 \sum_{i=1}^{9} f(0.1 i) \right] $$ 4. Calculate the curvature correction term using the composite corrected trapezoidal rule: $$ -\frac{(0.1)^2}{12}(f'(1)-f'(0)) $$ 5. Obtain the composite corrected trapezoidal integral approximation: $$ I_{cc}= I_c + \text{(curvature correction term)} $$ Compute the values for \(I_c\) and \(I_{cc}\), and compare them with the exact value of \(I_f = 0.746824133\ldots\).

Deriving the Error in the Composite Trapezoidal Method

Now, we want to derive the error in the composite trapezoidal method and express it in terms of \(h\). The error in the composite trapezoidal method is defined as \(E(f)=I_{f}-I_{Ic}\). By using the error formula for the composite trapezoidal method, we have: $$ E(f)=-\frac{h^{3}}{12n}\sum_{i=1}^{n} f''(\xi_i) = K_1 h^2 + O(h^4) $$ where \(\xi_i\) is a point in the interval \([x_{i-1}, x_i]\), and \(K_1\) is a constant independent of \(h\). This establishes the relationship between the error and \(h\).

Key concepts

Numerical Integration

Numerical integration is a cornerstone of computational mathematics, providing a way to approximate the definite integral of a function, especially when an analytical solution is difficult or impossible to obtain. It involves calculating the area under a curve by summing up the areas of shapes that approximate the region beneath the curve.

One common scenario where numerical integration applies is in physics, where it may be necessary to determine the distance traveled over time given a complex velocity function, or in engineering when assessing the strength of materials under variable forces. Students often struggle with the theoretical aspects of numerical integration, but understanding its practical applications can make the concept more relatable and easier to grasp.

Understanding the Basics

Numerical integration methods, such as the composite corrected trapezoidal method discussed in our exercise, are iterative techniques. They divide the function's domain into smaller subintervals and compute the integral's approximate value over each subinterval, then sum these values to get a total approximation for the integral over the entire interval.

Students should note that the accuracy of these methods is closely tied to the number of subintervals used. More subintervals generally lead to a more accurate approximation, however, this increases computational effort. The choice of method and number of subintervals often depends on the trade-off between desired accuracy and computational resources available.

Trapezoidal Rule

The trapezoidal rule is a numerical technique to estimate the integral of a function. It is based on interpolating the integrand using straight-line segments, transforming the problem into one of adding up the areas of trapezoids. In its simplest form, this rule may sometimes yield a less precise approximation, hence the 'composite' version is introduced.

For many students, connecting the geometric visualization of trapezoids under the curve to the mathematical formula can be a major 'aha' moment improving their comprehension of the subject. By dissecting each trapezoid, they can see the direct relationship between the area they are calculating and the summation process in the composite rule.

From Theory to Practice

In practice, the composite trapezoidal rule involves calculating the sum of trapezoidal areas over each subinterval. This is done by multiplying the width of the subintervals (often referred to as 'h') with the average height of two consecutive points on the function. The more partitions or 'cuts' made (our exercise uses 10 subintervals), the closer the trapezoids follow the actual curve, and the better the approximation will be.

Students should be encouraged to use software tools to apply this rule for complex functions, as manual calculations can be labor-intensive and prone to errors.

Error Analysis

Error analysis in numerical integration is critical, as it helps assess the precision of the approximate value obtained. The error in methods such as the trapezoidal rule can be estimated using certain formulas, allowing students to understand the limitations of their calculations.

It is important for students to realize that numerical methods are not flawless and that the concept of error is inherently tied to them. Acknowledging and estimating the error lends credibility to the result and provides a measure for comparison when deciding which method to use for a given problem.

Reducing Computational Errors

In the context of our exercise, the error term is shown to be proportional to the square of the width of the subintervals, plus a term that becomes negligible as the subintervals get smaller. Essentially, the error decreases with the square of the partition size - emphasizing that making 'h' smaller (increasing the number of subintervals) reduces the error significantly.

The addition of the correction term in the composite corrected trapezoidal method further improves accuracy by accounting for the curvature of the function being integrated. This term is derived from the second derivative of the function and helps correct the estimate, particularly for functions with significant curvature. It's an excellent illustration for students of how understanding the nature of the function can inform the choice of numerical method and improve the accuracy of results.