First Semester in Numerical Analysis with Julia, 2020a

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CHAPTER 1. INTRODUCTION 49 Arya and the unexpected challenges of data analysis Meet Arya! Arya is a college student interested in math, biology, literature, and acting. Like a typical college student, she texts while walking on campus, complains about demanding professors, and argues in her blogs that homework should be outlawed. Arya is taking a chemistry class, and she performs some experiments in the lab to find the weight of two substances. Due to difficulty in making precise measurements, she can only assess the weights to four-significant digits of accuracy: 2.312 grams and 0.003982 grams. Arya’s professor wants to know the product of these weights, which will be used in a formula. Arya computes the product using her calculator: 2.312 × 0.003982 = 0.009206384, and stares at the result in bewilderment. The numbers she multiplied had four-significant digits, but the product has seven digits! Could this be the result of some magic, like a rabbit hopping out of a magician’s hat that was only a handkerchief a moment ago? After some internal deliberations, Arya decides to report the answer to her professor as 0.009206. Do you think Arya was correct in not reporting all of the digits of the product?
CHAPTER 1. INTRODUCTION 50 Sources of error in applied mathematics Here is a list of potential sources of error when we solve a problem. 1. Error due to the simplifying assumptions made in the development of a mathematical model for the physical problem. 2. Programming errors. 3. Uncertainty in physical data: error in collecting and measuring data. 4. Machine errors: rounding/chopping, underflow, overflow, etc. 5. Mathematical truncation error: error that results from the use of numerical methods in solving a problem, such as evaluating a series by a finite sum, a definite integral by a numerical integration method, solving a differential equation by a numerical method. Example 23. The volume of the Earth could be computed using the formula for the volume of a sphere, V =4/3πr 3 , where r is the radius. This computation involves the following approximations: 1. The Earth is modeled as a sphere (modeling error) 2. Radius r ≈ 6370 km is based on empirical measurements (uncertainty in physical data) 3. All the numerical computations are done in a computer (machine error) 4. The value of π has to be truncated (mathematical truncation error) Exercise 1.3-6: The following is from "Numerical mathematics and computing" by Cheney & Kincaid [7]: In 1996, the Ariane 5 rocket launched by the European Space Agency exploded 40 seconds after lift-off from Kourou, French Guiana. An investigation determined that the horizontal velocity required the conversion of a 64-bit floating-point number to a 16-bit signed integer. It failed because the number was larger than 32,767, which was the largest integer of this type that could be stored in memory. The rocket and its cargo were valued at $500 million. Search online, or in the library, to find another example of computer arithmetic gone very wrong! Write a short paragraph explaining the problem, and give a reference.
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CHAPTER 3. INTERPOLATION 104 Exerci
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CHAPTER 3. INTERPOLATION 120 Exerci
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CHAPTER 4. NUMERICAL QUADRATURE AND
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Chapter 5 Approximation Theory 5.1
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Index Absolute error, 38 Beasley-Sp
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INDEX 220 Chebyshev, 203 Julia code
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First Semester in Numerical Analysis with Julia, 2020a

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