Problem 34 - Thomas Calculus

Question

The error function The error function, $$ \operatorname{erf}(x)=\frac{2}{\sqrt{\pi}} \int_{0}^{x} e^{-t^{2}} d t $$ important in probability and in the theories of heat flow and signal transmission, must be evaluated numerically because there is no elementary expression for the antiderivative of \(e^{-t^{2}} .\) a. Use Simpson's Rule with \(n=10\) to estimate erf \((1) .\) b. \(\ln [0,1]\) $$ \left|\frac{d^{4}}{d t^{4}}\left(e^{-t^{2}}\right)\right| \leq 12 $$ Give an upper bound for the magnitude of the error of the estimate in part (a).

Step-by-Step Solution

Answer

The estimate for \( \operatorname{erf}(1) \) using Simpson's Rule is approximately 0.8427, with an error bound of about 0.000008.

Step 1: Define the Simpson's Rule formula

Simpson's Rule is given by the formula: \[ S_n = \frac{b-a}{3n} \left[ f(a) + 4f(a+h) + 2f(a+2h) + \ldots + 4f(b-h) + f(b) \right] \] where \( h = \frac{b-a}{n} \), \( a \) and \( b \) are the limits of integration, and \( n \) is the number of subintervals, which must be even.

Step 2: Setup with values

For the problem statement, we are evaluating \( \operatorname{erf}(x) \) at \( x=1 \), so \( a=0 \), \( b=1 \), and \( n=10 \). The function to integrate is \( e^{-t^2} \).Calculate \( h \):\[ h = \frac{1-0}{10} = 0.1 \]

Step 3: Calculate f values

Evaluate \( f(t) = e^{-t^2} \) at respective points. For \( t = 0, 0.1, 0.2, \ldots, 1.0 \), calculate:- \( f(0) = e^{0} = 1 \)- \( f(0.1) = e^{-0.01} \approx 0.9900 \)- \( f(0.2) = e^{-0.04} \approx 0.9608 \)- Continue this process until \( f(1) = e^{-1} \approx 0.3679 \).

Step 4: Apply Simpson's Rule

Substitute the values into Simpson's Rule:\[ S_{10} = \frac{1-0}{3 \times 10} \left( 1 + 4 \times 0.9900 + 2 \times 0.9608 + \ldots + 4 \times 0.3679 + 0.3679 \right) \]Calculate this expression to obtain an estimate for \( \operatorname{erf}(1) \).

Step 5: Calculate error bound using fourth derivative

According to the error bound formula for Simpson's Rule:\[ E_S = \frac{(b-a)^5}{180n^4} M \]where \( M \) is an upper bound for the absolute value of the fourth derivative of \( f(x) \) on \([a,b]\). Given that \( \left| \frac{d^{4}}{d t^{4}}(e^{-t^2}) \right| \leq 12 \) for \( t \) in \([0,1]\), substitute the known values:\[ E_S = \frac{(1-0)^5}{180 \times (10)^4} \times 12 \]Calculate \( E_S \) to get the error bound.

Step 6: Calculate and interpret results

Evaluate the numerical expression of Simpson's Rule and the error to obtain the estimated value for \( \operatorname{erf}(1) \) and the error bound.

Key Concepts

Simpson's Rule

Simpson's Rule is a numerical method for approximating the definite integral of a function. This technique is particularly useful when the exact integral is difficult or impossible to obtain analytically. The rule assumes the function to be integrated can be approximated by a series of parabolic arcs, providing better accuracy compared to some other methods, like the Trapezoidal Rule.

The formula for Simpson's Rule is provided by:

• \[ S_n = \frac{b-a}{3n} \left[ f(a) + 4f(a+h) + 2f(a+2h) + \ldots + 4f(b-h) + f(b) \right] \]
• Where \( h = \frac{b-a}{n} \) is the width of each subinterval.
• It requires \( n \) to be even, as it needs pairs of subintervals to form the parabolas.

In the exercise, we use Simpson's Rule with \( n=10 \) subintervals to approximate the error function \( \operatorname{erf}(1) \). Calculating \( h \) as 0.1, we evaluate the function \( e^{-t^2} \) at points between 0 and 1, plugging these into the Simpson's formula to get an estimated value of the integral and thus, \( \operatorname{erf}(1) \).

Error Function (erf)

The error function, denoted as \( \operatorname{erf}(x) \), holds significant importance in probability theory and various physical phenomena like heat transfer. It is defined as:

• \[ \operatorname{erf}(x) = \frac{2}{\sqrt{\pi}} \int_{0}^{x} e^{-t^{2}} dt \]
• This function measures the probability that a random variable following a normal distribution will fall within a specific range.

The challenge with \( \operatorname{erf}(x) \) is the lack of an elementary antiderivative of \( e^{-t^2} \), which is why numerical methods, such as Simpson's Rule, are applied for approximation.

Approximation methods become crucial especially for non-analytical cases, allowing us to make estimates to a desired precision. Here, arithmetic steps through these methods provide valuable, albeit numerically derived, insight into function behavior.

Fourth Derivative

To understand the error in numerical integration, it's important to delve into derivatives. Specifically, the fourth derivative plays a crucial role in the accuracy of Simpson's Rule.

In this exercise, we are given that for \( e^{-t^2} \), the absolute value of its fourth derivative over the interval [0,1] is bounded by 12:

• \[ \left| \frac{d^{4}}{dt^4} \left( e^{-t^{2}} \right) \right| \leq 12 \]
• This value helps in determining the error bound while using Simpson’s method.

The fourth derivative provides insight into how quickly the function's slope is changing, impacting the precision of the approximation. Knowing this, you can more accurately predict and bound the integration error, ensuring your estimates are within acceptable limits.

Integration Error Bound

When employing numerical methods like Simpson's Rule, it is critical to estimate the error of the approximation. The error bound for Simpson’s Rule can be calculated using:

• \[ E_S = \frac{(b-a)^5}{180n^4} M \]
• Here, \( M \) represents the maximum value of the absolute fourth derivative of the function over the integration interval.

For the given problem, where \( M = 12 \) and \( n = 10 \), the calculated error bound helps us understand the precision of our numerical approximation for the \( \operatorname{erf}(1) \) value.

An error bound offers a considerable reassessment of the estimated integral, particularly guaranteeing a specific threshold within which the estimate lies. Using this strategy heightens the reliability of numerical solutions, guiding you to understand and potentially improve the quality of predictions or simulations in practical scenarios.