Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. Suppose \(\int_{a}^{b} f(x) d x\) is approximated with Simpson's Rule using \(n=18\) subintervals, where \(\left|f^{(4)}(x)\right| \leq 1\) on \([a, b]\) The absolute error \(E_{S}\) in approximating the integral satisfies \(E_{s} \leq \frac{(\Delta x)^{5}}{10}\) 1\. If the number of subintervals used in the Midpoint Rule is increased by a factor of \(3,\) the error is expected to decrease by a factor of \(8 .\) c. If the number of subintervals used in the Trapezoid Rule is increased by a factor of \(4,\) the error is expected to decrease by a factor of \(16 .\)

Short answer

Based on the analysis of the statements regarding the absolute error of different numerical integration methods: a) The absolute error of the Simpson's rule is given to be correct, as the error calculations match the given error bound. b) The statement about the error in the Midpoint rule when increasing the number of subintervals by a factor of 3 is false, as the error is expected to decrease by a factor of 9, not 8. c) The statement about the error in the Trapezoid rule when increasing the number of subintervals by a factor of 4 is true, as the error is expected to decrease by a factor of 16.

Step-by-step solution

Determine the error formula for Simpson's rule

The error bound for Simpson's Rule is given by: $$ E_{S} \leq \frac{(b-a)^5}{180n^4}|f^{(4)}(x)| $$ where \(n\) is the number of subintervals, \([a, b]\) is the integration interval, and \(f^{(4)}(x)\) is the fourth derivative of \(f(x)\).

Substitute the given values and compare with the stated error bound

We are given that \(n=18\) and \(\left|f^{(4)}(x)\right| \leq 1\). The integration interval width is \(\Delta x = \frac{b-a}{n}\). We can substitute these values in the error formula: $$ E_{S} \leq \frac{(\Delta x)^5}{10} \cdot |1| $$ Comparing this with the given error bound, we see that they match. So, statement a is true. ##Statement b##

Determine the error formula for the Midpoint rule

The error bound for the Midpoint Rule is given by: $$ E_{M} \leq \frac{(b-a)^3}{24n^2}|f^{(2)}(x)| $$ where \(n\) is the number of subintervals, \([a, b]\) is the integration interval, and \(f^{(2)}(x)\) is the second derivative of \(f(x)\).

Find the error when the number of subintervals is increased by a factor of 3

We want the error when changing the subintervals from \(n\) to \(3n\). So the new error is: $$ E_{M}^{new} \leq \frac{(b-a)^3}{24(3n)^2}|f^{(2)}(x)| $$ By comparing the original error and the new error, we can calculate the error reduction factor: $$ \frac{E_{M}}{E_{M}^{new}} = \frac{(3n)^2}{n^2} = 3^2 = 9 $$ This means the error is expected to decrease by a factor of 9, not 8 as given in the statement. So, statement b is false. ##Statement c##

Determine the error formula for the Trapezoid rule

The error bound for the Trapezoid Rule is given by: $$ E_{T} \leq \frac{(b-a)^3}{12n^2}|f^{(2)}(x)| $$ where \(n\) is the number of subintervals, \([a, b]\) is the integration interval, and \(f^{(2)}(x)\) is the second derivative of \(f(x)\).

Find the error when the number of subintervals is increased by a factor of 4

We want the error when changing the subintervals from \(n\) to \(4n\). So the new error is: $$ E_{T}^{new} \leq \frac{(b-a)^3}{12(4n)^2}|f^{(2)}(x)| $$ By comparing the original error and the new error, we can calculate the error reduction factor: $$ \frac{E_{T}}{E_{T}^{new}} = \frac{(4n)^2}{n^2} = 4^2 = 16 $$ This means the error is expected to decrease by a factor of 16, which matches the given statement. So, statement c is true.

Key concepts

Simpson's Rule

Simpson's Rule is a numerical integration technique that gives us an approximation to a definite integral. It employs quadratic polynomials to approximate the curve and works well when the function being integrated is smooth. The formula for Simpson's Rule integrates a polynomial that passes through several evenly spaced points on the function.

For a function with a known maximum fourth derivative over the interval \( [a, b] \), the error bound for Simpson's Rule is \( E_{S} \leq \frac{(b-a)^5}{180n^4}|f^{(4)}(x)| \), with \( n \) representing the number of subintervals. This error term decreases rapidly as \( n \) increases because it is proportional to \( \frac{1}{n^4} \). By using 18 subintervals and given that \( |f^{(4)}(x)| \leq 1 \), we find that the maximum error as per the exercise matches the theoretical upper bound, confirming the accuracy of Simpson's Rule in this scenario.

Midpoint Rule

The Midpoint Rule is another method of numerical integration that approximates the value of integrals. It uses the midpoint of each subinterval of a partition to compute rectangles that approximate the area under the curve. The error bound formula for the Midpoint Rule is \( E_{M} \leq \frac{(b-a)^3}{24n^2}|f^{(2)}(x)| \), where \( f^{(2)}(x) \) is the second derivative of the function, and \( n \) is the number of subintervals.

Increasing the number of subintervals under the Midpoint Rule by a factor of 3 leads to a decrease in the error by a factor of 9, not 8 as might be intuited without error analysis. The error is proportional to \( \frac{1}{n^2} \), illustrating the quadratic nature of error reduction in the Midpoint Rule, which is a useful consideration for improving the precision of integrations.

Trapezoid Rule

The Trapezoid Rule is a rule for numerical integration that uses trapezoids rather than rectangles or polynomials. For functions that are linear or close to linear, the trapezoids provide a good approximation of the area. The error for the Trapezoid Rule is dictated by the formula \( E_{T} \leq \frac{(b-a)^3}{12n^2}|f^{(2)}(x)| \).

By increasing the number of subintervals by a factor of 4, as per the textbook exercise, the error is expected to decrease by a factor of 16. This inverse-square law for error reduction reflects the direct relationship between the increase in subintervals and the decline in error, illustrating that the Trapezoid Rule can achieve significant improvements in accuracy with increased subdivision of the integration interval.

Error Reduction Factor

The error reduction factor is critical in numerical analysis as it reflects how changes in the number of subintervals affect the accuracy of an integration method. It is often expressed in terms of the new error over the original error, \( \frac{E_{new}}{E_{original}} \) and is dependent on the specific rule used for approximation.

When the number of subintervals is increased, the error reduction factor allows us to predict how much the total integration error will decrease. As demonstrated in the exercises for the Midpoint and Trapezoid Rules, this factor depends on the power of \( n \) in the rule's error formula. For instance, in the error formulas for the Midpoint and Trapezoid Rules, the errors decrease quadratically with the number of intervals, leading to error reduction factors of \( 9 \) and \( 16 \) respectively, after increasing the number of subintervals by a factor of 3 and 4.