Question

For the following exercises, determine whether the statement is true or false. Justify your answer with a proof or a counterexampleIn numerical integration, increasing the number of points decreases the error.

Short answer

True, increasing points typically decreases error in numerical integration.

Step-by-step solution

Understand the Statement

The statement claims that, in numerical integration, using more points (increasing the number of points) reduces the error of the approximation.

Analyze Numerical Integration Methods

Numerical integration methods such as Trapezoidal Rule, Simpson's Rule, and numerical quadrature use a finite number of sample points to approximate an integral. The accuracy often depends on how well the function behavior is captured with these points.

Consider Common Integration Methods

For the Trapezoidal Rule and Simpson’s Rule, as the number of subdivisions (or points) increases, the approximation typically becomes more accurate. This is because a finer partition of the interval provides a better representation of the curve, reducing the approximation error.

Examine Convergence and Error Reduction

Most numerical integration methods have error terms inversely related to the number of points, meaning as you increase points, the error term is reduced. For example, the error in the Trapezoidal Rule is inversely proportional to the square of the number of points.

Evaluate through Counterexamples

However, if the function is highly oscillatory or ill-behaved, simply increasing the number of points might not necessarily reduce the error proportionately. Still, typically the error is expected to decrease with more points, except in those special cases.

Conclude Based on Typical Cases

In general, increasing the number of points in numerical integration decreases the error, confirming the statement as usually true for well-behaved functions.

Key concepts

Trapezoidal Rule

The Trapezoidal Rule is a straightforward and widely used method for approximating the definite integral of a function. It works by dividing the area under the curve into a series of trapezoids, hence the name. By summing the areas of these trapezoids, we get an estimate of the integral over a specified interval.
Calculating the integral with the Trapezoidal Rule involves taking the average of the function values at two consecutive points, multiplying by the distance between these points (known as the subinterval width), and adding these products up.

• The basic formula for the Trapezoidal Rule is given by: \[\int_a^b f(x) \, dx \approx \frac{h}{2} \left[ f(x_0) + 2f(x_1) + 2f(x_2) + \ldots + 2f(x_{n-1}) + f(x_n) \right]\]where:
  • \(h\) is the width of each subinterval: \( h = \frac{b-a}{n} \)
  • \(n\) is the number of subintervals.
  • \(f(x_i)\) are the function values at each point.

The accuracy, or error, of the Trapezoidal Rule is determined by how well the linear segments approximate the curve. Generally, as you increase the number of points \(n\), the approximation becomes more precise because each trapezoid better aligns with the curve, reducing the overall error.

Simpson's Rule

Simpson’s Rule provides another method of numerical integration that often yields superior accuracy compared to the Trapezoidal Rule. Rather than using straight lines to approximate the subareas, Simpson’s Rule uses parabolas. This method is particularly effective when the function behaves smoothly across the interval.
The formula for Simpson’s Rule is as follows:\[\int_a^b f(x) \, dx \approx \frac{h}{3} \left[ f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + \ldots + 4f(x_{n-1}) + f(x_n) \right]\]where:

• \(h\) is the width of each subinterval, and \(n\) must be even.
• Alternate coefficients 4 and 2 are multiplied by the function values, hinting at the weighting of the areas.

Simpson's Rule typically yields better results than the Trapezoidal Rule because parabolas can more accurately mimic the curvature of many functions. As with the Trapezoidal Rule, increasing the number of sampled points typically decreases the error in the approximation, offering a more exact integral estimation.

Error Analysis

In numerical integration, understanding how error behaves is crucial to improving approximation accuracy. Error Analysis examines these errors and how factors such as the number of points and the behavior of the function impact the result.
Both the Trapezoidal and Simpson’s Rule have associated error terms:

• For the Trapezoidal Rule, the error can be expressed as:\[E_T = - \frac{(b-a)^3}{12n^2} f''(\xi)\]This indicates that the error is inversely proportional to the square of the number of points \(n\), and heavily depends on the second derivative of the function \(f''(\xi)\).
• Simpson’s Rule has an error term given by:\[E_S = - \frac{(b-a)^5}{180n^4} f^{(4)}(\xi)\]which reveals that the error decreases much faster as \(n\) increases due to the power of four in the denominator.

However, if the function is very oscillatory, or if there are significant changes in curvature, these approximations can introduce substantial errors. In such cases, simply increasing \(n\) doesn't always guarantee a better result. But typically, for functions without erratic behavior, increasing the number of subdivisions indeed leads to more precise approximations by reducing the error associated with each method.