An Introduction to Numerical Computation

Wen Shen

• Publisher: WSPC
• Year: 2015
• ISBN:
  9789814730099
• Topic: Mathematik
• Subtopic: Mathematik Allgemein

Developed during ten years of teaching experience, this book serves as a set of lecture notes for an introductory course on numerical computation, at the senior undergraduate level. These notes contain the material that can be covered in a semester, together with a few optional sections for additional reading. Rather than surveying a large number of algorithms, the book presents the most important computational methods and emphasizes the underlying mathematical ideas. In most chapters, graphs and drawings are relied on, to build up intuition.

The notes are written in a rather colloquial style, presenting the subject matter in the same form as it can be explained in a classroom. For instructors, this will minimize the amount of effort required to prepare their blackboard presentations.

As prerequisites, the book only relies on standard calculus, an introductory course on matrices, and some basic computer programming skills. As a new feature, these notes are supplemented by two sets of videos from the author's Youtube channel. These videos contain a complete set of live lectures given in Spring 2015, together with a complete set of short tutorials, from 5 to 15 minutes each.

A set of homework problems is included at the end of each chapter. Homework projects cover a variety of applications, in connection with population dynamics, engineering, mechanics, image reconstruction, etc. A complete set of solutions is available for instructors, upon request.

Request Inspection Copy

Developed during ten years of teaching experience, this book serves as a set of lecture notes for an introductory course on numerical computation, at the senior undergraduate level. These notes contain the material that can be covered in a semester, together with a few optional sections for additional reading. Rather than surveying a large number of algorithms, the book presents the most important computational methods and emphasizes the underlying mathematical ideas. In most chapters, graphs and drawings are relied on, to build up intuition.

The notes are written in a rather colloquial style, presenting the subject matter in the same form as it can be explained in a classroom. For instructors, this will minimize the amount of effort required to prepare their blackboard presentations.

As prerequisites, the book only relies on standard calculus, an introductory course on matrices, and some basic computer programming skills. As a new feature, these notes are supplemented by two sets of videos from the author's Youtube channel. These videos contain a complete set of live lectures given in Spring 2015, together with a complete set of short tutorials, from 5 to 15 minutes each.

A set of homework problems is included at the end of each chapter. Homework projects cover a variety of applications, in connection with population dynamics, engineering, mechanics, image reconstruction, etc. A complete set of solutions is available for instructors, upon request.

Request Inspection Copy

Readership: Junior or senior undergraduate students interested in numerical computation and analysis, majoring in mathematics, computer science, physics, engineering, etc.
Key Features:

• Matlab is integrated throughout the book
• The book assumes minimum prior mathematical knowledge
• The book was class-tested through 5 semesters at Penn State University, with top reviews from students
• Supplementary videos offer an enrichment to these lecture notes

Chapter 1

Computer Arithmetic

1.1Introduction

What are numerical methods? They are algorithms that compute approximate solutions to a number of problems for which exact solutions are not available. These include, but are not limited to:

•evaluating functions,

•computing the derivative of a function,

•computing the integral of a function,

•solving a nonlinear equation,

•solving a system of linear equations,

•finding solutions to an ODE (ordinary differential equation),

•finding solutions to a PDE (partial differential equation).

Such algorithms can be implemented (programmed) on a computer.

The following diagram shows how various aspects are related.

Keep in mind that a course on numerical methods is NOT about numbers. It is about mathematical ideas and insights.

We shall study some basic issues:

•development of algorithms;

•implementation;

•a little bit of analysis, including error-estimates, convergence, stability etc.

Throughout the course we shall use Matlab for programming purposes.

1.2Representation of Numbers in Different Bases

Historically, there have been several bases for representing numbers, used by different civilizations. These include:

10:	decimal, which has become standard nowadays;
2:	binary, computer use;
8:	octal;
16:	hexadecimal, used in ancient China;
20:	vigesimal, used in ancient France (numbers 70 to 79 are counted as 60+10 to 60+19 in French, and 80 is 4 × 20);
60:	sexagesimal, used by the Babylonians.

Fig. 1.1The big picture

In principle, one can use any integer β as the base. The value of a number in base β is then written as

Taking β = 10, we obtain the standard decimal representation.

The above formula allows us to convert a number in any base β into decimal base.

In principle, one can convert the numbers between different bases. Here are some examples.

Example 1.1. Conversion of octal → decimal:

(45.12)₈ = 4 × 8¹ + 5 × 8⁰ + 1 × 8⁻¹ + 2 × 8⁻² = (37.15625)₁₀

Example 1.2. Conversion of octal ↔ binary:

Observe that, since 8 = 2³, we have

Then, the conversion between these two bases become much simpler. To convert from octal to binary, we can convert each digit in the octal base, and write it into binary base, using 3 digits in binary for each digit in octal. For example:

To convert a binary number to an octal one, we can group the binary digit in groups of 3, and write out the octal base value for each group. For example:

Example 1.3. Conversion of decimal → binary: write (12.45)₁₀ in binary base.

This example is of particular interest. Since the computer uses binary base, how would the number (12.45)₁₀ look like in binary base? How does a computer store this number in its memory? The conversion takes two steps.

First, we treat the integer part. The procedure is to keep dividing by 2, and store the remainders of each step, until one cannot divide anymore. Then, collect the remainder terms in the rev...