Consider the problem of evaluating the integrals Yn = dx x+10 for n = 1,...
Fantastic news! We've Found the answer you've been seeking!
Question:
Transcribed Image Text:
Consider the problem of evaluating the integrals хп Yn = dx x+10 for n = 1, 2,... Analytically we notice that Also, 1 Yn + 10yn-1 = L x+10x-1 x+10 1 dx = xn-1 dx n Yo = S 1 1 dx = ln(11) - In(10). x+10 This gives us the following algorithm for computing Yo, Y1,...: 1. Compute yo ln(11) — In(10). = 2. For n = 1, 2, ..., evaluate yn = 1 - 10yn-1. (a) Show that yn < 1 for all n and yn decreases monotonically to 0 as n → ∞. (A picture would suffice for this!) (b) Program the algorithm in a computer and find yn for n = 1,2, Why does this happen? 30. What happens? (c) Derive an algorithm for computing the values of these integrals based on evaluating the value of Yn-1 from the value of Yn. (d) Suppose you want to calculate the values of yo, Y1, YN for some number N with an absolute error of less than ε for some ε > 0. Since the naive algorithm above produces large errors (even though it is exact in theory), you will use the algorithm in (c). But you need a starting value. Show that there exists M = N such that if you take YM O and use the algorithm in (c), then, if calculations are performed with infinite precision, the absolute errors in the calculations of Yo, Y1, ..., YN will be less than ɛ. = 3 (e) Explain why rounding errors in the computer do not produce excessive errors using this algorithm. (f) Use your algorithm (and the computer) to find the value of 30 to an accuracy of 105. Explain how you chose M in this case. Consider the problem of evaluating the integrals хп Yn = dx x+10 for n = 1, 2,... Analytically we notice that Also, 1 Yn + 10yn-1 = L x+10x-1 x+10 1 dx = xn-1 dx n Yo = S 1 1 dx = ln(11) - In(10). x+10 This gives us the following algorithm for computing Yo, Y1,...: 1. Compute yo ln(11) — In(10). = 2. For n = 1, 2, ..., evaluate yn = 1 - 10yn-1. (a) Show that yn < 1 for all n and yn decreases monotonically to 0 as n → ∞. (A picture would suffice for this!) (b) Program the algorithm in a computer and find yn for n = 1,2, Why does this happen? 30. What happens? (c) Derive an algorithm for computing the values of these integrals based on evaluating the value of Yn-1 from the value of Yn. (d) Suppose you want to calculate the values of yo, Y1, YN for some number N with an absolute error of less than ε for some ε > 0. Since the naive algorithm above produces large errors (even though it is exact in theory), you will use the algorithm in (c). But you need a starting value. Show that there exists M = N such that if you take YM O and use the algorithm in (c), then, if calculations are performed with infinite precision, the absolute errors in the calculations of Yo, Y1, ..., YN will be less than ɛ. = 3 (e) Explain why rounding errors in the computer do not produce excessive errors using this algorithm. (f) Use your algorithm (and the computer) to find the value of 30 to an accuracy of 105. Explain how you chose M in this case.
Expert Answer:
Answer rating: 100% (QA)
a To show that yn decreases monotonically to 0 as n increases we can observe the plot of yn for n 1 ... View the full answer
Related Book For
Statistics Principles And Methods
ISBN: 9780470904114
7th Edition
Authors: Richard A. Johnson, Gouri K. Bhattacharyya
Posted Date:
Students also viewed these programming questions
-
Consider the problem of testing H0: μ = 10 versus H1: μ > 10 with n = 64, Ï = 2 (known), and α = .025. The rejection region of this test is given by...
-
You have just been given a $454,000, which you decide to invest at an APR of 6.7 percent. If you were to withdraw $38,500 at the end of each year, starting at the end of this year, how many years...
-
You are looking to open up a cupcake shop in a high-traffic tourist area. In order to get your business open you will need investors to provide you with $250,000 dollars. You are going to be making a...
-
Alsup Consulting sometimes performs services for which it receives payment at the conclusion of the engagement, up to six months after services commence. Alsup recognizes service revenue for...
-
Consider the following cash flow profile and assume MARR is 10 percent/year. a. What does Descartes' rule of signs tell us about the IRR(s) of this project? b. What does Norstrom's criterion tell us...
-
Advanced Products Corporation has supplied the following data from its activity-based costing system: During the year, Advanced Products completed one order for a new customer, Shenzhen Enterprises....
-
A 47-kg pole vaulter running at 10 m/s vaults over the bar. Her speed when she is above the bar is 1.5 m/s. Neglect air resistance, as well as any energy absorbed by the pole, and determine her...
-
Bookkeeping is primarily A) preparing financial reports. B) analyzing the company's financial position. C) record-keeping. D) preparing tax reports.
-
All Seasons invested $150 million in acquiring the land and $500 million in building and furnishing the 10 hotels to a guest-ready stage. Each hotel has 150 rooms. Each room has a base rate of $200...
-
Let's turn to our financing needs. It looks like we need to raise $100 million dollars this quarter. Our bank will lend to us at 10% interest. This would be a loan that is secured by our inventory....
-
A girl is sledding down a slope that is inclined at 30.0 with respect to the horizontal. The wind is aiding the motion by providing a steady force of 113. N that is parallel to the motion of the...
-
Why should the contemporary manager be interested in the concept of collective action? Discuss a range of ways in which contemporary workplaces are impacted by collective action, using examples.
-
Ivanhoe Corp. purchased equipment for their diagnostic services division. The invoice price was $223000. This did not include the 7% sales tax. Transportation costs of $4000 were incurred to ship the...
-
2. Consider the chemical equation: 1 N, + 3 H, 2 NH, For the 3 scenarios below, predict which one will produce the most ammonia, and predict which ones will have leftovers. 10 A: resctarts B:...
-
Show that the block upper triangular matrix A in Example 5 is invertible if and only if both A 11 and A 22 are invertible. Data from in Example 5 EXAMPLE 5 A matrix of the form A = [ A11 A12 0 A22 is...
-
Table 13, on page 77, gives data collected from the students attending an elementary statistics course at the University of Wisconsin. These data include sex, height, number of years in college, and...
-
Referring to Exercise 15.30, calculate: (a) The sign test statistic. (b) The significance probability when the alternative is that Judge 2 gives higher scores than Judge 1.
-
Construct the 90% multiple-f confidence intervals using the sound distortion data in Example 1.
-
Define human communication.
-
Perceiving your communication as effective while those around you perceive it as ineffective is known as ______________.
-
Visit any of the websites for major social media companies (Facebook, Twitter, LinkedIn, etc.). Under the About Us section, read the company mission statement and goals for the organization. Do you...
Study smarter with the SolutionInn App