First Semester in Numerical Analysis with Julia, 2020a

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.