91Ó°ÊÓ

The Trapezium Rule is a numerical integration method used to approximate the area under a curve. This technique involves dividing the area into small trapezoids, calculating the area of each trapezoid, and summing them up. Suppose you want to estimate an integral like \( \int_{0}^{1} e^{x^2} \mathrm{d} x \).

You would start by choosing a step size \( h \), and over the interval \([0, 1]\), calculate the function values at each point \( x_i \), with \( x_i = a + i \, h \) for \( i = 0, 1, \, ..., \, n \) where \( n \) is the number of segments. The trapezium rule formula for this integral would be:

\[ \int_a^b f(x) \, dx \approx \frac{h}{2} \left( f(a) + 2 \sum_{i=1}^{n-1} f(x_i) + f(b) \right) \]

Using this approach, we create an estimate for \( \int_{0}^{1} e^{x^2} \) which yields the value of \( 1.49068 \) with \( h = 0.25 \). The smaller \( h \) is, the more precise the estimate tends to be.

The global truncation error in the trapezium rule provides insight into the accuracy of our numerical approximation. It represents the total error accumulated when using a finite number of trapezoids to estimate the integral. This error stems from two factors: the method itself and the finite number of intervals.

To compute the global truncation error for the trapezium rule, we use:

\[ -\frac{(b-a)^3}{12n^2}f''(\xi) \]

where \( f''(\xi) \) is an evaluation of the second derivative at some point within the interval \([a, b]\). This value is crucial as it determines the size of the error.

After evaluating, we found that the second derivative's maximum on \([0, 1]\) occurred at \( x = 1 \), leading to \( 6e \) as the maximum value. This resulted in an estimated global truncation error of approximately \(-0.08541\), which helped verify the range of the integral from \(1.40527\) to \(1.57609\).

Determining derivatives is a core part of understanding and estimating the error in numerical methods like the trapezium rule. In our exercise, the function we're interested in is \( f(x) = e^{x^2} \). To calculate the global truncation error, we needed its second derivative.

First, we found the first derivative:

\[ f'(x) = 2x \cdot e^{x^2} \]

This was followed by the second derivative:

\[ f''(x) = 2e^{x^2} + 4x^2 \cdot e^{x^2} = 2e^{x^2} (1+2x^2) \]

With these derivatives, we can evaluate their values across the interval to identify maximum values, using them for calculating the global truncation error. Evaluating at \( x = 1 \), the second derivative peaked at \( 6e \), providing a key component in error analysis.

Precision in numerical calculus dictates how close our approximation is to the actual value. For the trapezium rule, step size \( h \) plays a major role in determining this precision. Smaller \( h \) reduces error, providing a more accurate approximation. However, a smaller step size also means more calculations, requiring a balance.

In our example, we wanted to find \( h \) that results in an error less than \( 0.00005 \) to ensure precision to four decimal places. After rearranging and solving the error formula:

\[ \frac{1}{12n^2} \cdot 6e = 0.00005 \]

We calculated \( n \approx 45.71 \), needing some approximation to the next whole number. Hence, \( n = 46 \). Consequently, the new step size \( h \approx 0.02174 \) satisfies the required precision, assuring our results are accurate within four decimal places.