MCS 471 Project One: Summing Sequences of Numbers using Maple, Study Guides, Projects, Research of Mathematical Methods for Numerical Analysis and Optimization

Download MCS 471 Project One: Summing Sequences of Numbers using Maple and more Study Guides, Projects, Research Mathematical Methods for Numerical Analysis and Optimization in PDF only on Docsity! MCS 471 Project One due Friday 12 September, 1PM Fall 2003 MCS 471 Project One: Summing Sequences of Numbers The goal of this project is to investigate numerical methods to sum sequences of numbers using floating- point arithmetic. To experiment with various precisions, we will use Maple. This document is processed from a Maple worksheet, available from the web site for this class. It is highly recommended to start the project by downloading this worksheet. 0. Summing floating-point numbers In this project we consider numerical methods to sum sequences of floating-point numbers. We use two methods. The first method is the plain addition (prep school method). The second method first sorts the numbers before adding them. In Maple, these two methods are respectively provided in the procedures “plain add” and “sort add”. [> plain_add := n -> add(k,k=n): # plain addition [> sort_add:= n->add(k,k=sort(n)): # first sort then add For example, we use these procedures as follows: [> our_list := [1.0,9.0,0.7]: [> plain_add(our_list); 10.7 [> sort_add(our_list); 10.7 The input of these Maple procedures is a list of numbers. A list in Maple is a sequence of elements, separated by commas and enclosed by square brackets. For integer numbers, the two methods will always return the same result. Working with floating-point arithmetic, the summing of long lists will give different results. To compare our numerical results with the exact sum, we avoid roundoff errors by converting first to rational numbers: [> exact_add := n -> plain_add(map(x->convert(x,rational),n)): Then the exact sum is also a rational number: [> s := exact_add(our_list); 107 s := --- 10 which we convert to a floating-point number with evalf: [> evalf(s); 10.70000000 As the problem of summing numbers is so simple, lots of real-world applications can be found. For example, suppose we monitor the stock prize as it is going up and down almost continuously during the day. At the end of the day we want to sum up all the increments and decrements to know the final value of the sum. A similar calculation must be made at a cash register in a store, adding up the money going in and the money going out. University of Illinois at Chicago, Department of Mathematics, Statistics and Computer Science page 1 MCS 471 Project One due Friday 12 September, 1PM Fall 2003 1. Summing random sequences We first generate a random sequence of uniformly distributed numbers. [> randomize(): # initialize the seed [> random_data := [stats[random,uniform[-1,1]](5)]: We just generated 5 random numbers, uniformly distributed between −1 and 1. For longer lists, we may want to suppress the output, replacing the semicolon at the end of the command by a colon. [> exact_sum := exact_add(random_data): To get a floating-point approximation, we use the evalf command: [> approx_sum := evalf(exact_sum); approx_sum := -1.019136378 Maple converts to its default precision of 10 digits. To see the error made, we use evalf with as extra argument the number of digits we wish to calculate with: [> evalf(exact_sum-approx_sum,20); -9 -.4009374938 10 The magnitude of the error equals the default precision of Maple. We can restrict the precision to 3 digits [> Digits := 3: [> plain_sum := plain_add(random_data); plain_sum := -1.01 [> sort_sum := sort_add(random_data); sort_sum := -1.02 Already for such as small sequence we sometimes observe a significant difference in the errors: [> error_of_plain_add := abs(evalf(exact_sum-plain_sum,8)); error_of_plain_add := .0091364 [> error_of_sort_add:= abs(evalf(exact_sum-sort_sum,8)); error_of_sort_add := .0008636 Observe that we compute the error with higher precision than the precision used to obtain the sums. Assignment One Repeat the little experiment above with longer lists of numbers and for various levels of precision. Take lists of length 100, 200, 300,... and calculate the sums with values for Digits equal to 20, 30, 40,..., each time comparing the exact sum with the values returned by plain add and sort add. Do enough experiments until you see a pattern emerging. In particular, what is the influence of the length of the list and the value of Digits on the errors of plain add and sort add? Assignment Two 1. Explain the difference between the two summation algorithms. Why is it better to first sort the numbers before adding them? 2. Maple sorts in increasing order. Illustrate with an example why for the summing problem this is numerically better than sorting in decreasing order. University of Illinois at Chicago, Department of Mathematics, Statistics and Computer Science page 2