Question

Using Romberg integration, compute \(\pi\) to 8 digits (i.e., \(3 . x x x x x x)\) by obtaining approximations to the integral $$ \pi=\int_{0}^{1}\left(\frac{4}{1+x^{2}}\right) d x $$ Describe your solution approach and provide the appropriate Romberg table. Compare the computational effort (function evaluations) of Romberg integration to that using the adaptive routine developed in Section \(15.4\) with tol \(=10^{-7}\). You may find for some rows of your Romberg table that only the first step of extrapolation improves the approximation. Explain this phenomenon. [Hint: Reconsider the assumed form of the composite trapezoidal method's truncation error and the effects of extrapolation for this particular integration.]

Short answer

Short Answer: To compute the value of pi to 8 digits using the integral \(\int_0^1 \frac{4}{1+x^2} dx\), we apply Romberg integration, which combines the composite trapezoidal rule with Richardson's extrapolation to approximate the integral. We create a Romberg table where each entry represents an approximation of the integral and continue until the absolute difference between successive diagonal entries is less than \(10^{-8}\). Comparing with an adaptive routine with a tolerance of \(10^{-7}\), we assess the computational effort in terms of function evaluations. Lastly, we explain that only the first step of extrapolation significantly improves the approximation in some rows of the Romberg table due to the error of the trapezoidal method being an even function of the step size, and only the first extrapolation step having significant impact on the error.

Step-by-step solution

Understanding Romberg Integration

Romberg integration is a numerical integration method that combines the trapezoidal rule and Richardson's extrapolation to approximate definite integrals. We will use the composite trapezoidal rule with increasing subdivisions and apply Richardson's extrapolation to obtain an accurate approximation of the value of the integral.

Applying Romberg Integration

Let's start by creating a Romberg table where we calculate approximations \(R_{ij}\) of the integral using the composite trapezoidal rule with \(2^i\) subdivisions, and then refine the approximations using Richardson's extrapolation for each subsequent column j. The formula for the composite trapezoidal rule and Richardson's extrapolation is given by: $$ R_{ij} = \begin{cases} R_{i-1,0}+\frac{1}{2^i}\sum_{k=1}^{2^{i-1}}f(a+k(b-a)2^{-i}), & \text{if }j=0\\ \frac{4^jR_{i,j-1}-R_{i-1,j-1}}{4^j-1} & \text{if }j>0, \end{cases} $$ Here \(a=0\), \(b=1\) and \(f(x)=\frac{4}{1+x^2}\).

Compute the Romberg Table and Estimate \(\pi\)

Using the formulas above, we can compute the values in the Romberg table until the absolute difference between successive approximations in the diagonal of the table is less than \(10^{-8}\). This way, we calculate the approximation for \(\pi\) to 8 digits.

Compare with Adaptive Routine

After computing the approximation for \(\pi\) using Romberg integration, we can compare the computational effort (function evaluations) with the adaptive routine developed in Section 15.4 with a tolerance of \(10^{-7}\). To do this, we can count the number of function evaluations in both methods and compare them.

Explain the Phenomenon

In some rows of the Romberg table, we may observe that only the first step of extrapolation significantly improves the approximation. To explain this phenomenon, we can reconsider the form of the composite trapezoidal method's truncation error and the effects of extrapolation. It has to do with the error of the trapezoidal method being an even function of the step size. Since Richardson extrapolation eliminates terms of the error that have even powers of the step size, only the first step of extrapolation will have a significant impact for this particular integral, as other higher-order terms have odd powers of the step size and do not contribute to the error.

Key concepts

Numerical Integration

Numerical integration refers to a broad family of algorithms for calculating the numerical value of a definite integral. Since many integrals cannot be solved analytically, or require complex calculations, numerical methods like the Romberg integration are used to find approximate solutions. To grasp the concept, consider the problem of finding the area under a curve; numerical integration essentially breaks this area into a series of shapes (usually rectangles or trapezoids) whose areas can be easily computed and summed.

Numerical integration is particularly useful in engineering, physics, and economics where precise models often lead to integrals that do not have simple antiderivatives. Furthermore, it’s pivotal in solving differential equations that arise in modeling real-world phenomena where exact solutions are rare. When employing numerical integration, it’s important to consider the trade-off between the accuracy of the result and the computational effort required. Advanced methods, such as Romberg integration, enable greater accuracy without a proportionally higher computational cost.

Richardson's Extrapolation

Richardson's extrapolation is a technique for accelerating the convergence of a sequence of estimates of a number. It is often used in conjunction with numerical methods, like the trapezoidal rule, to obtain more accurate approximations of an integral. This method works on the principle that an approximation to a solution can be improved by considering the error as a power series expansion in terms of the step sizes and then eliminating higher-order error terms.

When applied to Romberg integration, Richardson's extrapolation takes the results from the composite trapezoidal rule and refines them to yield better approximations. Typically, the process involves calculating the integral using increasingly fine subdivisions and using these calculations to project to the limit of zero step size. The remarkable aspect of Richardson's extrapolation is that it often yields significantly more accurate results with only a few additional calculations. However, its effectiveness can depend on the smoothness of the function being integrated and the behavior of its derivatives. Understanding how Richardson’s extrapolation fits within Romberg integration solidifies the student's comprehension of numerical analysis concepts.

Composite Trapezoidal Rule

The composite trapezoidal rule is a method for numerical integration that extends the basic trapezoidal rule to multiple subintervals of the entire interval over which the integral is evaluated. By dividing the interval into smaller pieces, we approximate the area under the curve as a series of trapezoids rather than a single large one, thus improving the accuracy of the integration.

The approach is straightforward: the interval \[a, b\] is divided into \(n\) equally spaced subdivisions, and the area of the trapezoids formed by the function's graph and the \(x\)-axis are calculated and summed. The composite rule is efficient and easy to understand, making it a great starting point for students learning about numerical integration. When used in the first column of Romberg integration, it's the foundational values on which subsequent extrapolations are based. An intriguing aspect of the composite trapezoidal rule is its error analysis, which reveals that the error decreases with finer subdivisions but also can be systematically improved upon using methods like Richardson's extrapolation. Thus, it is a key concept within Romberg integration for achieving greater accuracy.