Numerical Analysis

An Introduction

Timo Heister, Leo G. Rebholz, Fei Xue

• Publisher: De Gruyter
• Year: 2019
• ISBN:
  9783110573336
• Edition: 1
• Topic: Matematica
• Subtopic: Analisi matematica

Numerical analysis deals with the development and analysis of algorithms for scientific computing, and is in itself a very important part of mathematics, which has become more and more prevalent across the mathematical spectrum. This book is an introduction to numerical methods for solving linear and nonlinear systems of equations as well as ordinary and partial differential equations, and for approximating curves, functions, and integrals.

1Computer representation of numbers and roundoff error

In this chapter, we will introduce the notion and consequences of a finite number system. Each number in a computer must be physically stored. Therefore, a computer can only hold a finite number of digits for any number. Decades of research and experimentation has led us to a (usually) reasonable approximation of numbers by representing them with (about) sixteen digits in base 10. While this approximation may seem at first to be reasonable, we will explore its consequences in this chapter. In particular, we will discuss how to not make mistakes that can arise from using a finite number system. The representation of numbers is not specific to a program or programming language like MATLAB but they are part of the hardware of the processors of virtually every computer.

1.1Examples of the effects of roundoff error

To motivate the need to study computer representation of numbers, let us consider first some examples taken from MATLAB—but we note that the same thing happens in C, Java, etc.:

1.The order in which you add numbers on a computer makes a difference!

Note: AAAeBBB is a common notation for a floating-point number with the value AAA × 10BBB. So 1e-16 = 10−16.

As we will see later in this chapter, the computer stores about 16 base 10 digits for each number; this means we get 15 digits after the first nonzero digit of a number. Hence, if you try to add 1 e-16 to 1, there is nowhere for the computer to store the 1 e-16 since it is the 17th digit of a number starting with 1. It does not matter how many times you add 1e-16; it just gets lost in each intermediate step, since operations are always done from left to right. So even if we add 1e-16 to 1, 10 times in a row, we get back exactly 1. However, if we first add the 1e-16’s together, then add the 1, these small numbers get a chance to combine to become big enough not to be lost when added to 1.

2.Consider

$f\left(x\right)=\frac{e^x-e^{-x}}{x}.$

Suppose we wish to calculate

$\underset{x\to 0}{\lim }f\left(x\right).$

By L’Hopital’s theorem, we can easily determine the answer to be 2. However, how might one do this on a computer? A limit is an infinit...