The Trapezoidal Rule Explained

The trapezoidal rule serves as the simplest method for calculating an integral. You begin by dividing the interval [a, b] into N equal segments, creating trapezoids, as illustrated below with N=5.

For those interested in the matplotlib script utilized for this figure, please refer to my supplementary materials.

The width of each trapezoid is defined as Δx, which is calculated by:

The integral, represented as the area beneath the blue curve, is approximately the sum of the trapezoidal areas. You may recall from high school that the area of a trapezoid is determined by multiplying the width by the average height, leading to the formula for the area of the n-th trapezoid:

The total area of all N trapezoids can be expressed as:

If you expand the sum term by term, you'll notice that neighboring terms can be combined:

Consequently, the sum simplifies to:

This method is straightforward: while using a limited number of trapezoids results in a rough approximation, increasing the number of trapezoids will enhance accuracy.

However, there's a notable drawback. The error associated with the trapezoidal rule diminishes with an increase in N, but this reduction is relatively slow. Specifically, when N is doubled, the error decreases only by a factor proportional to 1/N², as depicted in the figure below.

When function evaluations are computationally intensive, simply increasing the number of grid points is not feasible, which often necessitates the use of more complex methods. Nevertheless, you can achieve greater accuracy with the trapezoidal rule by incorporating a straightforward enhancement—this is the focus of the next section.

Romberg Integration: Enhancing Accuracy

The principle behind Romberg integration is quite simple. First, compute the integral using the trapezoidal rule with N slices, yielding T_N. Then, perform the same calculation with 2N slices, resulting in T_2N. The widths of the trapezoids in these two cases are Δx and Δx/2, respectively. From the previous section, we know that the error associated with the trapezoidal rule is of the order of 1/N².

Hence, we can express the exact value of the integral, I, as follows:

We can also represent it as:

By equating both expressions for I and rearranging the terms, we arrive at:

Therefore, our approximation improves to:

How effective is this approximation? By evaluating the integral for various N values of a known integral, we can visualize the results in the following plot:

As observed, the slope of the Romberg integration curve is twice as steep as that of the trapezoidal rule. Given that this is a double-logarithmic plot, where the error for the trapezoidal rule behaves as N², it follows that the error for the Romberg extension behaves as N⁴—comparable to the more complex Simpson's rule!

But the story doesn't end here. With our understanding of error behavior, we can derive additional extrapolations to further enhance accuracy, requiring merely another application of the trapezoidal rule with a different number of slices. Thus, as I mentioned earlier, Romberg integration enables the use of the trapezoidal rule with exceptional precision, sparing us from the intricacies of more advanced methods.

For a deeper dive into the Python code used to generate the above figures, please visit my supplementary page.

The first video titled "Romberg Integration" provides a comprehensive overview of this technique, enhancing your understanding of its application and benefits.

The second video, "Romberg Integration Process with Trapezoidal Rule," delves into how initial estimates from the trapezoidal rule enhance the Romberg integration process.