[4] [Computational cost as a function of error tolerance] Recall from Lecture 19 that the error in the Composite Trapezoidal Rule (CTR) using n subintervals of width h is given by -12²2 (6- 12 -(b − a) ƒ" (µ) (1) for some μ € (a, b). (a) Determine the values of n and h that are sufficient to approximate 2 S² x ln(x) dx (2) to within an error tolerance of T = 10-5; that is, determine n and h so that the absolute error when applying the CTR to (2) is smaller that 7. (Hint: take the absolute value of (1) and then say that the result is less than or equal to (1) when f"(µ) is replaced with the maximum value of |ƒ"(x)| on [1,2]. Then, determine n and h that make the resulting quantity less than 7). (b) Repeat part (a) for the case of Composite Simpson's Rule. For this integral, which method is superior?
[4] [Computational cost as a function of error tolerance] Recall from Lecture 19 that the error in the Composite Trapezoidal Rule (CTR) using n subintervals of width h is given by -12²2 (6- 12 -(b − a) ƒ" (µ) (1) for some μ € (a, b). (a) Determine the values of n and h that are sufficient to approximate 2 S² x ln(x) dx (2) to within an error tolerance of T = 10-5; that is, determine n and h so that the absolute error when applying the CTR to (2) is smaller that 7. (Hint: take the absolute value of (1) and then say that the result is less than or equal to (1) when f"(µ) is replaced with the maximum value of |ƒ"(x)| on [1,2]. Then, determine n and h that make the resulting quantity less than 7). (b) Repeat part (a) for the case of Composite Simpson's Rule. For this integral, which method is superior?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
Composite trapezoidal rule
![[4] **Computational cost as a function of error tolerance**
Recall from Lecture 19 that the error in the Composite Trapezoidal Rule (CTR) using \( n \) subintervals of width \( h \) is given by
\[
-\frac{h^2}{12} (b-a) f''(\mu)
\]
for some \( \mu \in (a, b) \).
(a) Determine the values of \( n \) and \( h \) that are sufficient to approximate
\[
\int_{1}^{2} x \ln(x) \, dx
\]
to within an error tolerance of \( \tau = 10^{-5} \); that is, determine \( n \) and \( h \) so that the absolute error when applying the CTR to (2) is smaller than \( \tau \).
(Hint: take the absolute value of (1) and then say that the result is less than or equal to (1) when \( f''(\mu) \) is replaced with the maximum value of \( |f''(x)| \) on \([1, 2]\). Then, determine \( n \) and \( h \) that make the resulting quantity less than \( \tau \)).
(b) Repeat part (a) for the case of Composite Simpson’s Rule. For this integral, which method is superior?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7fd47556-f3ce-4f39-818d-be563d9523c8%2F2f268b73-e390-4850-aef1-9631e70a938b%2Fbvblqla_processed.png&w=3840&q=75)
Transcribed Image Text:[4] **Computational cost as a function of error tolerance**
Recall from Lecture 19 that the error in the Composite Trapezoidal Rule (CTR) using \( n \) subintervals of width \( h \) is given by
\[
-\frac{h^2}{12} (b-a) f''(\mu)
\]
for some \( \mu \in (a, b) \).
(a) Determine the values of \( n \) and \( h \) that are sufficient to approximate
\[
\int_{1}^{2} x \ln(x) \, dx
\]
to within an error tolerance of \( \tau = 10^{-5} \); that is, determine \( n \) and \( h \) so that the absolute error when applying the CTR to (2) is smaller than \( \tau \).
(Hint: take the absolute value of (1) and then say that the result is less than or equal to (1) when \( f''(\mu) \) is replaced with the maximum value of \( |f''(x)| \) on \([1, 2]\). Then, determine \( n \) and \( h \) that make the resulting quantity less than \( \tau \)).
(b) Repeat part (a) for the case of Composite Simpson’s Rule. For this integral, which method is superior?
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