Mathematical Approximation and Documentation

Harry A. Watson, Jr. Quality Assessment Directorate Naval Warfare Assessment Center Corona, California 91718-5000 (909) 273-4787 This book and the associated software fit the description in the U. S. Copyright Act of a “United States Government Work.” It was written as a part of the author’s official duties as a government employee. This means it cannot be copyrighted. This document and the software are freely available to the public for use without a copyright notice, and there are no restrictions on its use, now or subsequently.

Preface to Mathematical Approximations and Documentation

There are several elementary mathematical approximations that occur on every scientific and engineering project at irregular intervals. Frequently the same approximations appear again and again as ad hoc report requests (and often with short notice). While the programming and documentation is straightforward, the more elaborate and expensive commercial software is often unable to deliver suitable output. This text includes the basic theory and elementary computer programs in BASIC (Beginners All-purpose Symbolic Instruction Code^{1}) and “C.”^{2} Documentation, including graphs, is provided in TEX and PICTEX. A later chapter is devoted to more complex graphs done in the device-independent software METAFONT, which can be included in TEX files. No deep knowledge of programming is required, nor is the user expected to be a “TEX-nician;” indeed, the user is expected to lift the coding “as is” and make the obvious substitutions. This book and the associated software fit the description in the U. S. Copyright Act of a “United States Government Work.” It was written as a part of the author’s official duties as a government employee. This means it cannot be copyrighted. This document and the software are freely available to the public for use without a copyright notice, and there are no restrictions on its use, now or subsequently. Anyone using this document or these programs and files does so entirely at her/his own risk. Anyone who demands payment for the distribution of this material must make clear that the charge is for distribution only and is in no sense a license fee or purchase fee.

The three topics covered in this textbook are (1) Simpson’s rule for numeric integration,

(2) Newton’s method for approximation of roots, and (3) least-squares curve fit with a test to verify the choice of the underlying statistical distribution. Many examples are provided with hand-calculations and tables. It is expected that the user will find portions of the TEX file useful. With this in mind, the macros (\def) are kept to a minimum and the displayed portions are self-contained.

As often as possible, references are made to textbooks universally available, such as outline series books found in any college bookstore or encyclopedia articles. The author has attempted to locate examples of particular interest rather than pathological cases. This is to ensure a greater likelihood that an example may be copied directly into a scientific or engineering paper.

The graph of y = sin x_{_{−π}  −\frac{π}{2}  O  _{\frac{π}{2}}  _{π}  _{\frac{3π}{2}}  _{2π}}_{y=sin x}_{.}_{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}

FIGURE 1

A word about fonts is in order. The only fonts employed are from the standard distribution (16 fonts) for TEX. All computer program statements are typeset in the so-called typewriter font to provide contrast and to allow alignment of the characters for keyboard input.

The copyright for the program TEX belongs to D. E. Knuth; the trademark TEX belongs to the American Mathematical Society; and MS-DOS is a trademark of the Microsoft Corporation.

1 The coding is actually done in MS-DOS QBasic (MicroSoft-Disk Operating System). There should be little problem in conversion to GW-BASIC.

2 The “C” language is written for the Microsoft QuickC Compiler.

TABLE OF CONTENTS

Preface  i
Table of Contents  ii

Chapter 1—Simpson’s Rule

1.0  Introduction  1
1.1  Simpson’s Rule  2
1.2  Integration Computer Programs  4
1.3  Vivasection of the Programs  7
1.4  Numerical versus Exact Integration  9
1.5  Truncation Error  11
1.6  An Example  12
1.7  Verifying Derivatives  14
1.8  Generalizations  19

Chapter 2—Newton’s Method

2.0  Introduction  21
2.1  Theory  22
2.2  Applications  23
2.3  The Algorithm  26
2.4  A Second Opinion  27
2.5  The Quasi-Newton Method  28
2.6  Pathological Examples  30

Chapter 3—Linear Least-squares Line

3.0  Introduction  31
3.1  Dissection of a Graph  32
3.2  Formula Derivation  37
3.3  Goodness of Fit  39
3.4  More Theory  42
3.5  Problems  44

Chapter 4—An Adaptive Quadrature Routine

4.0  Introduction  47
4.1  A Simple Approach  48
4.2  An Adaptive Quadrature Routine  49

Chapter 5—METAFONT

5.0  Introduction  51
5.1  A Metafont Program  52

Chapter 6—An Important Improper Integral

6.0  Introduction  54
6.1  Theoretical Evaluation  55
6.2  Numerical Evaluation  57
Appendix A—Mathematical Preliminaries  58
Appendix B—References  61
Appendix C—A Geometric Persuasion  62
Appendix D—A Contour Integral  64
Appendix E—The Malgrange-Ehrenpreis Theorem  66

Simpson’s Rule

Chapter 1

1.0 Introduction. When one finally finishes algebra and geometry and embarks into the realm of calculus, all problems seem solvable. The universe appears to be reduced to mathematical constructs and it is only a matter of time until mathematics can define everything. There is a conceit among those who first master the art and discipline of The Calculus. The tyro^{3} feels a superiority to those who are ignornant of this powerful tool, replete with its repertoire of skills, special symbols, concise language, and geometric persuasions. The euphoria which is closely associated with the mastery of The Calculus leaves its pupil craving a repeat experience. The classical development of calculus has evolved over several centuries; the problems and examples are thus arranged so as to direct the student along a path devoid of many of the harsh pitfalls in applied science and engineering. Embedded in this development are the two classic encounters with the dirty reality of numerical approximation: Newton’s method for approximating roots and Simpson’s rule for approximating integrals. These two techniques are ubiquitous^{4} in numerical analysis, calculating devices, and engineering in general. Their study and development is the subject of the first two chapters. In this, the first chapter, the topic of Simpson’s rule is considered. The coding is done in several different computer languages and the text edited in different formats to ensure complete generality. Since Simpson’s rule depends on properties of the parabola, a general graph of a parabola is displayed below.

PARABOLA

O  x y  V^{•}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}

y = f(x) = ax^{2} + bx + c, a > 0,

Vertex = V = (−\frac{b}{2a}, \frac{b^{2}}{2a} (\frac{1}{2a} − 1) + c) .

FIGURE 2

3 A beginner in learning, a novice. Don’t rush to your dictionary—footnotes will be provided for many words.

4 Being found everywhere at the same time.

1.1 Simpson’s rule. Most scientists and engineers have completed elementary calculus and are aware of the technique for numerical approximation by fitting arcs of parabolas; however, even someone who has only completed high school mathematics (algebra and plane geometry) will be able to apply this technique with success (and produce a professional output that will amaze even the most erudite^{5} pedant on the staff). There are pitfalls; the expert as well as the tyro can be trapped by some speciously simple-looking functions.^{6} Having said all of this, let us begin the basic theoretical development of Simpson’s rule.^{7}

First of all, let [α, β] be a number interval, that is

[α, β] = { x | α ≤ x ≤ β }.

We’ll use the Greek letters α and β because later we’ll want to use a, b, and c as coefficients of the quadratic equation y = ax^{2} + bx + c. We will let f denote a function, f: [α, β] → R, where R = { x | x is a real number }. For convenience, we will use let y_{α} = y_{0} = f(α), y_{β} = y_{2} = f(β), and y_{1} = f((β + α)/2). This is all bookingkeeping. It is one of those mathematical concepts that causes “math anxiety;” however, it doesn’t need to. Just consider y to be the left-hand side of some expression, for instance

y = x^{2} − 4x + 2.

When x = 2 we might have y = 2 · 2 − 4 · 2 + 2 = −2. This could be written as y(2) or y(x = 2) or even y|_{x=2}, which is also written y|_{2}. We will just simplify things and write it as y_{2}. This is a fundamental idea, if the reader is confused or lost on this matter, it would be a good idea to review an algebra book (or consult appendix A).

Simpson’s rule follows from the so-called prismoidal formula (for areas)

A_{p} = \frac{h}{3}[ y_{0} + 4y_{1} + y_{2}] = \frac{β − α}{6} [f(α) + 4 · f (\frac{β + α}{2}) + f(β)] .

We will denote the left-hand side of this equation as A_{p}, which stands for the area under the arc of the parabola. This simple formula gives the exact area if the function, f(x), is a polynomial of degree not higher than 3.^{8} In particular, A_{p}, gives the area under the arch of the parabola

y = f(x) = ax^{2} + bx + c

between x = −h and x = +h, as in Fig. 3.^{9}

−h  O  _{h}_{FIGURE 3}  _{x}^{y}^{y_{0}}  ^{y_{1}}  ^{y_{2}}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}

5 Possessing or displaying excessive knowledge acquired mainly from books.

6 One pathological example is the so-called Quadratrix of Hippias, y = x tan (πy/2).

7 George B. Thomas, Jr., Calculus and Analytic Geometry, (Reading, Massachusetts: Addison-Wesley Publishing Company, Inc., 1966), pages 385-388.

8 Glenn James and James, Robert C., Mathematics Dictionary, (Princeton, NJ: D. Van Nostrand Company, Inc., 1964), pages 358-359.

9 Without loss of generality, we choose the interval [a, b] to be [−h, h]. This simplifies the algebra. Everything is still completely general because we can always “shift” a graph back and forth along the x-axis without changing the area under the curve.

This result is readily established as follows: in the first place, we have

A_{p} = ∫_{−h}^{h} (ax^{2} + bx + c) dx = \frac{ax^{3}}{3} + \frac{bx^{2}}{2} + cx |_{x=−h}^{x=h} = \frac{2ah^{3}}{3} + 2ch.

Since the curve passes through the three points (−h, y_{0}), (0, y_{1}), and (h, y_{2}), we also have

y_{0} = ah^{2} − bh + c, y_{1} = c, y_{2} = ah^{2} + bh + c,

from which it follows that

\begin{aligned}c & = y_{1}, \\ ah^{2} − bh & = y_{0} − y_{1}, \\ ah^{2} + bh & = y_{2} − y_{1}, \\ 2ah^{2} & = y_{0} + y_{2} − 2y_{1}. \end{aligned}

Hence, expressing the area A_{p} in terms of the ordinates y_{0}, y_{1}, and y_{2}, we have

A_{p} = \frac{h}{3} [2ah^{2} + 6c] = \frac{h}{3} [(y_{0} + y_{2} − 2y_{1}) + 6y_{1}]

or

A_{p} = \frac{h}{3} [y_{0} + 4y_{1} + y_{2}] = \frac{h}{3} [f(−h) + 4 · f(0) + f(h)] .  (1)

Simpson’s rule is derived by applying the prismoidal formula to successive subintervals of [α, β]. Each separate subinterval has length 2h. So, we may write [α, α + h] as [x_{0}, x_{1}], [α + h, α + 2h] as [x_{1}, x_{2}], and so on. Now the curve y = f(x) between x = α and x = β is really n/2 curves. Each separate piece of the curve covers an x-subinterval of width 2h; it is approximated by an arch of a parabola through its ends and its mid-point. The area under each parabolic arc is computed as in Eq. (1) and the resulting areas are summed together to give the rule

A_{S} = \frac{h}{3}[ y_{0} + 4y_{1} + 2y_{2} + 4y_{3} + ⋯ + 2y_{n−2} + 4y_{n−1} + y_{n}].  (2)

This result, A_{S}, is an approximate value of ∫_{a}^{b} f(x) dx. Recall our bookkeeping. We let y_{0}, y_{1}, y_{2}, …, y_{n} stand for f(x_{0}), f(x_{1}), …, f(x_{n}). In other words, y_{0}, y_{1}, …, y_{n} are the ordinates of the curve y = f(x) corresponding to the abscissas x_{0} = a, x_{1} = a + h, x_{2} = a + 2h, …, x_{n} = a + nh = b. Each abscissa point then corresponds to an endpoint of a subdivision of the interval α ≤ x ≤ β into n equal subintervals each of width h = (β − α)/n. (See Fig. 4.) The number n of subdivisions must be an even integer in order for this method to work. Not all authors use this same scheme. Some rely on subintervals of length h instead of 2h. When consulting a table book, be careful to examine the setup for this rule.

O  α  ←h→  β_{FIGURE 4} y = f(x)  x y  ^{1}  ^{4}  ^{2}  ^{4}  ^{2}  ^{4}  ^{1}^{y_{0}}  ^{y_{1}}  ^{y_{2}}  ^{y_{n−1}}  ^{y_{n}}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}

It looks like the more subintervals we take, that is the smaller we let h become, the more accurate the approximation will be to the area under the curve (the value of the integral). This is part of error estimation and will be done in a later paragraph. In the next paragraph we will do numerical examples and see how the method behaves numerically.

1.2 Integration Computer Programs. We’ll list two computer programs to do the same job— one in BASIC and one in “C”. One integral that crops up again and again is the so-called elliptic integral. We will look at a complete elliptic integral of the second kind, usually denoted

E (k, \frac{π}{2} ) = ∫_{0}^{π/2} \sqrt{1 − k^{2} · sin^{2} θ} dθ.  (3)

Finally, we need to choose a value for k. If we let e denote the base of the natural logarithms, that is e ≈ 2.7182828459045 …, then assign k to be k = sin (e/2), we will have a good example to experiment with.^{10}

100 DEFDBL A-H, O-Z : REM DEF FNA defines the integrand (function).
105 DEF FNA (x AS DOUBLE) = SQR(1!-(SIN(EXP(1)/2)*SIN(x))^2)
110 A = 0!:  REM This is the lower endpoint of [a,b].
120 B = 3.141592653589793#/2!:  REM This is the upper endpoint of [a,b].
130 SUM = 0!:  REM This is an accumulator.
135 REM The user is asked for a number of intervals, n, even and > 0.
140 INPUT "Enter number of intervals (must be even) "; N
145 IF N <= 0 OR (2 * FIX(N / 2) - N < 0) THEN PRINT "Input error":  GOTO 140
150 H = (B - A) / N: REM The sub-interval size h = (b-a)/n.
160 FOR I = 1 TO N STEP 2:  REM The "FOR" loop is done n/2 times.
170 SUM = SUM + FNA(A + (I - 1) * H)
175 PRINT A + (I - 1) * H, FNA(A + (I - 1) * H)
180 SUM = SUM + 4!  * FNA(A + I * H)
185 PRINT A + I * H, FNA(A + I * H)
190 SUM = SUM + FNA(A + (I + 1) * H)
195 PRINT A + (I + 1) * H, FNA(A + (I + 1) * H)
200 NEXT I: REM After loop, the sum is y 0+4*y 1+2*y 2+...+4*y {n-1}+y n.
210 PRINT "Sum = "; SUM: PRINT " Value of integral = "; SUM * H / 3
220 REM This short program will integrate a function FNA(x) from x = a to b.

The “C” program is more complicated because it has the “#include” statements. However, the “C” program executes faster and is portable to other platforms. In addition, the programs below both use a math co-processor, which gives floating-point arithmetic. Minor differences in the last digit may be due to the print (format) algorithm. The BASIC program would not accept the longer value of π. If one examines the TEX file for these listings, it will be noted that the control characters are input as \char92 for the backspace (\), \char123 for the left brace ({), \char125 for the right brace (}), and \char94 for the hat or circumflex (^).

#include <stdio.h> /* Header for input/output subroutines.  */
#include <math.h> /* Header for math subroutines.  */
#include <float.h> /* Header for floating point subroutines.  */
#define pi 3.141592653589793238462643383279 /* Accurate value for pi.  */
/* Simpson's rule for approximating integrals.
    a:  left endpoint
    b:  right endpoint
    fc:  pointer to function to integrate
    n:  number of subintervals
*/
double fc (double x);

10 Harry A. Watson, Jr., “An approximation to Planck’s constant,” Abstracts of the American Mathematical Society (AMS) # 81T–181, June 1981.

main()
{
double a,b,h,sum,x,y; /* In 'C' all variables must be assigned */
double p1, p2, p3;
int i, n;
printf("\007"); /* Sound bell in computer.  */
a = (double) 0.0;
printf("\nLeft end point = %.16lf",a);
b = (double) pi/2.0;
printf("\nRight end point = %.16lf",b);
i = -1;
while (i < 0){
printf("\nEnter number of subintervals (must be even) ");
scanf("\%d",&n);
i = n/2; i = 2*i - n; /* Don't allow odd values of n.  */
if (n<=0) i = -1; /* Don't allow zero or negative.  */
}
printf("\nNumber of subintervals %d",n);
h = (double) (b-a)/n;
for (i=1, sum=0.0; i<=n; i = i+ 2){
  p1 = fc((double) a+(i-1)*h);
  p2 = fc((double) a+i*h);
  p3 = fc((double) a+(i+1)*h);
  sum += p1 + 4.0 * p2 + p3;
/* printf("\n x, f(x) %.16lf %.16lf",a+(i-1)*h,p1); */
/* printf("\n x, f(x) %.16lf %.16lf",a+i*h,p2); */
/* printf("\n x, f(x) %.16lf %.16lf",a+(i+1)*h,p3); */
}
printf("\nValue of sum = %.16lf", (double) sum);
y = (double) h*sum/3.0;
printf("\nValue of integral = %.16lf", (double) y);
printf("\n");
return(0);
}
double fc (double x)
{
    double y;
    y = sqrt(1.0-sqrt(exp(1.)/2.)*sqrt(exp(1.)/2.)*sqrt(x)*sqrt(x));
    return (y);
}
/* End of file */

Parabolic curve fits unequally spaced abscissas O  x f(x) ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}

FIGURE 5

We’ll make a table of values for various integer values of n, that is, various numbers of subdivisions. Much can be learned from comparing the outputs of the two programs and examining their behavior as n becomes large.

n =  BASIC Program  “C” Program  Difference
2  1.073441760234934  1.0734417602349340  0.00000000000000
4  1.057800052054470  1.0578000520544700  0.00000000000000
8  1.054898082319603  1.0548980823196030  0.00000000000000
10  1.054750510268630  1.0547505102686300  0.00000000000000
20  1.054686162116484  1.0546861621164840  0.00000000000000
40  1.054685849666779  1.0546858496667790  0.00000000000001
80  1.054685849645436  1.0546858496454370  0.00000000000001
100  1.054685849645437  1.0546858496454360  0.00000000000001
200  1.054685849645437  1.0546858496454370  0.00000000000000
500  1.054685849645437  1.0546858496454370  0.00000000000000
1,000  1.054685849645437  1.0546858496454370  0.00000000000000
2,000  1.054685849645436  1.0546858496454370  0.00000000000001
10,000  1.054685849645435  1.0546858496454330  0.00000000000002

Notice, if you will, two things are happening: (1) after n = 40 there is no change in the first five decimal places; and, (2) after n = 80 there is almost no change at all! So, after n = 80, successive terms become closer and closer together and the calculated value converges to a given number. This phenomenon is called, obviously enough, convergence. It would be nice if we could tell beforehand just what value of n to choose to ensure the accuracy we need. Clearly, if all we want is five decimal place accuracy, n = 40 will suffice. If we are using a double-precision calculation, then n = 80 will do. Of course, you might say why not just take n = 10000 to start with? Just run the program and see how long it takes to get an answer! It is possible that time is also a factor—even the fastest computers take time to do the involved mathematical operations required in numerical integration.

The next item to be introduced is error estimation. It will be shown that for n subintervals, there is a number ξ ∈ (α, β) such that the truncation error is

−\frac{(β − α)^{5}}{180n^{4}} f^{(4)}(ξ).

If the fourth derivative of f(x) is bounded on [α, β], that is, if there exists a number M such that

|f^{(4)}(x)| ≤ M for all x ∈ [α, β],

then the error is bounded by

\frac{(β − α)^{5} M}{180n^{4}} .

At this point in our development, the importance of the error bound is that we will be assured that the larger the value of n, the closer the answer will be to the exact solution. This says that for any given degree of accuracy, there is a positive integer N such that for all n ≥ N, each approximation computed by Simpson’s rule with n terms will lie within the given degree of accuracy to the exact solution.

This error estimation sounds like pure theory, and so it could be skipped without losing anything. That’s really not the case. What we have is exactly equivalent to the mechanical “tolerance.” If you want something within a particular tolerance, say ϵ, then you must calibrate your tools to within some accuracy, say δ. It is exactly this so-called ϵ-δ philosophy that is the basis for theoretical analysis. One of the great failings of pure analysis has been its inability to relate to the industrial quality efforts and statistical error estimations.

1.3 Vivasection^{11} of the Programs. The first item to note in the computer programs is the definition of the function f(x)^{12} In the BASIC program the function is defined by the statement

105 DEF FNA (x AS DOUBLE) = SQR(1!-(SIN(EXP(1)/2)*SIN(x))^2)

while in the “C” program the function is defined as a separate subroutine. “C” is fashioned after subroutines, so this should be no surprise. The function definition subroutine in “C” is thus

double fc (double x)
{
    double y;
    y = sqrt(1.0-sin(exp(1.)/2.)*sin(exp(1.)/2.)*sin(x)*sin(x));
    return (y);
}

Notice, further, that BASIC supports exponentiation, that is A^B means A^{B} whereas “C” does not (it requires a separate subroutine call to pow()). Now we will focus on the interval endpoints, α and β, of the interval [α, β]. Since the letters A, B and a, b are not used elsewhere inside the programs, they can be used for endpoint values. BASIC is not case sensitive; “C” is. We define the endpoints (or limits of integration) as follows:

110 A = 0!:  REM This is the lower endpoint of [a,b].
120 B = 3.141592653589793#/2!:  REM This is the upper endpoint of [a,b].
a = (double) 0.0;
printf("\nLeft end point = %.16lf",a);
b = (double) pi/2.0;
printf("\nRight end point = %.16lf",b);

Notice that in the “C” program the value of the endpoints is displayed via a printf() function. It is always a good practice to display such variables. The next item of interest is the input of the number of intervals, n. This also determines the width (length) of the subintervals, h = (β − α)/n. Each program does a check to ensure that the integer n is positive and not odd. There are many ways to make such a determination, so it is not necessary to dwell on the matter. The main portion of the Simpson’s rule is the iteration. Here a variable named sum or SUM is initialed to zero and functions as an accumulator. The values y_{i−1} +4·y_{i} +y_{i+1} are added to the accumulator for i between 1 and n − 1 in steps of 2. The final value of the accumulator is multiplied by h/3 to get the approximation to the integral ∫_{α}^{β} f(x) dx.

130 SUM = 0!:  REM This is an accumulator.
150 H = (B - A) / N: REM The sub-interval size h = (b-a)/n.
160 FOR I = 1 TO N STEP 2:  REM The "FOR" loop is done n/2 times.
170 SUM = SUM + FNA(A + (I - 1) * H)
180 SUM = SUM + 4!  * FNA(A + I * H)
190 SUM = SUM + FNA(A + (I + 1) * H)
200 NEXT I: REM After loop, the sum is y 0+4*y 1+2*y 2+...+4*y {n-1}+y n.

Notice in the “C” coding below the statement sum += . . .. This is identical to the coding sum

= sum + . . ..
h = (double) (b-a)/n;
for (i=1, sum=0.0; i<=n; i = i+ 2){
  p1 = fc((double) a+(i-1)*h);
  p2 = fc((double) a+i*h);

11 Disection of a living (as opposed to inoperable) computer program. 12 Mathematicians prefer to call a function by its single letter abbreviation, e.g. f, g, etc.; however, since the independent variable, say x, is required in the computer definitions, it seems better to use the convention of complex analysis and keep the notation f(x), g(x), etc., so that there is no ambiguity in the computer coding.

  p3 = fc((double) a+(i+1)*h);
  sum += p1 + 4.0 * p2 + p3;
}

The final calculation and the display is given below. First for the BASIC program and then for the “C” program. The BASIC program does its calculation in the PRINT statement.

210 PRINT "Sum = "; SUM: PRINT " Value of integral = "; SUM * H / 3

Notice that the “C” program uses a subroutine (printf()) to display its output on the video monitor. Notice further that in the “C” program the last statement is a print with the argument "\n". This is a carriage return/line feed that is needed after the last print statement. Otherwise, the program will type “Hit any key to continue” on the same line as the output.

printf("\nValue of sum = %.16lf", (double) sum);
y = (double) h*sum/3.0;
printf("\nValue of integral = %.16lf", (double) y);
printf("\n");

We will examine the error estimation for this integral. Let’s write down the function and its first four derivatives:

f(x) = \sqrt{1 − k^{2} sin^{2} x}; where k = sin (e/2);
f′(x) = −\frac{k^{2}}{2} \frac{sin(2x)}{\sqrt{1 − k^{2} sin^{2} x}};
f′′(x) = −k^{2} cos 2x/\sqrt{X} − k^{4} sin^{2}(2x)/(4 · X^{3/2}); where X = (1 − k^{2} sin^{2}(x))
f^{(3)}(x) = 2k^{2} sin(2x)/\sqrt{X} − 3k^{4} sin(4x)(4 · X^{3/2})
− 3k^{6} sin^{3}(2x)/(8 · X^{5/2});
f^{(4)}(x) = 4k^{2} cos(2x)/\sqrt{X} + k^{4} sin^{2}(2x)/X^{3/2} − 3k^{4} cos(4x)/X^{3/2}
− 9k^{6} sin(4x) sin(2x)/(8 · X^{5/2}) − 18k^{6} sin^{2}(2x) cos(2x)/(8 · X^{5/2})
− 15k^{8} sin^{4}(2x)/(16 · X^{7/2}).

Observing that if 0.0 ≤ x ≤ 1.572, then −313.8 ≤ f^{(4)}(x) ≤ 85.0, we get a bound for the fourth derivative on [0, π/2]. If M = 320, then

|f^{(4)}(x)| ≤ M, ∀ x ∈ [0, \frac{π}{2} ] .

The error is thus bounded by

\frac{π^{5} · M}{180 · 32 · n^{4}} ≈ \frac{17}{n^{4}} ,

where n is the number of subintervals of [α, β] = [0, π/2]. Thus, to ensure that our answer is within say .5% of the true answer, we would have to pick n to be greater than 7 (and even!). But this is a conservative estimate. We have already seen that a much smaller value of n will suffice. In the examples chosen for textbooks, the fourth derivative is generally a constant (a constant-values function, that is, a function f^{(4)}(x) = constant for all x under consideration). This builds the student’s confidence, but does little to reflect the real-world situation. The integral we have chosen is an integral that crops up again and again in engineering problems. Is there a satisfactory resolution to this (common) problem? Indeed, there is. The idea is to examine the approximating values for increasing values of the integer n which determines the step size. This technique is based on the theoretical consideration that a Cauchy sequence converges. In computer programming, it is called an adaptive quadrature routine (AQR). We will cover the adaptive quadrature routine in a later paragraph.

1.4 Numerical versus Exact Integration. The sophomore who has just completed an introduction to calculus might be surprised to discover that an integral as simple as

∫_{0}^{π} \sqrt{1 + cos^{2} x} dx  (4)

cannot be solved by elementary techniques. (The value of this integral is the length of one “arch” of the sine curve, y = sin x. See Fig. 1 in the preface for a graph of the sine curve.) It can be expressed in terms of a complete elliptic integral of the second kind. If we write

E (k, \frac{π}{2} ) = ∫_{0}^{π/2} \sqrt{1 − k^{2} sin^{2} θ} dθ,  (5)

then we may make the simple trigonometric substitution

cos^{2} θ = 1 − sin^{2} θ

to obtain

∫_{0}^{π} \sqrt{1 − cos^{2} θ} dθ = 2 · \sqrt{2} · E(π/4, π/2)
= \sqrt{2} ∫_{0}^{π/2} \sqrt{1 − sin^{2}(π/4) · sin^{2}(θ)} dθ
+ \sqrt{2} ∫_{π/2}^{π} \sqrt{1 − sin^{2}(π/4) · sin^{2}(θ)} dθ,

where sin(π/4) = \sqrt{2}/2. (Recall that \frac{π}{4} radians^{13} is 45^{∘}. Lots of tables books give values in degrees and not in radians. You do remember radians, don’t you? If not, check out a trigonometry book or look in Appendix A for a review.) We look up the value of E(45^{∘}, π/2) in a tables book^{14} and find that

E (45^{∘}, \frac{π}{2} ) ≈ 1.35006.

This says that the value of the integral in equation (6) should be approximately

∫_{0}^{π} \sqrt{1 + cos^{2}(θ)} dθ ≈ 3.82007.

Now we apply our integration method (Simpson’s rule) and see what the results are:

n =  BASIC Program  “C” Program  Difference
2  3.575356081779317  3.5753560817793170  0.000000000000000
4  3.829178925615088  3.8291789256150880  0.000000000000000
8  3.820282406120432  3.8202824061204320  0.000000000000000
16  3.820197813575163  3.8201978135751630  0.000000000000000
32  3.820197789027719  3.8201977890277180  0.000000000000001
64  3.820197789027712  3.8201977890277120  0.000000000000000
128  3.820197789027713  3.8201977890277120  0.000000000000001
256  3.820197789027711  3.8201977890277110  0.000000000000000
512  3.820197789027712  3.8201977890277110  0.000000000000001

13 A radian is a unit a angular measure. Basically, 360^{∘} = 2π radians. This means that 1 radian is approximately 57.29578 degrees. 14 Dr. M. Fogiel, Handbook of Mathematical, Scientific, and Engineering Formulas, Tables, Functions, Graphs, Transforms, (Piscataway, NJ: Research and Education Association, 1984), page 623.

There is a lesson to be learned here. If we use a more accurate value for E(45^{∘}, π/2) than 1.3506, namely 1.3506438, then we will see that our computed answer agrees more closely with the theoretical answer. Now, lets look at the coding that was changed in the BASIC and in the “C” programs:

105 DEF FNA (x AS DOUBLE) = SQR(1!+COS(x)*COS(x))
double fc (double x)
{
    double y;
    y = sqrt(1.0+cos(x)*cos(x));
    return (y);
}

We added something else new. In the BASIC program and in the “C” program we added lines giving the (expected) theoretical value.

230 PRINT "Theoretical approx = "; 2*SQR(2)*1.35064388
printf("Theoretical approx = printf("\n");

We want to delve^{15} more deeply into the whole matter of numerical versus exact integration. To do this, we will evaluate an integral exactly and numerically and compare the results. Lets integrate one arch of the cosine curve:

∫_{0}^{π/2} cos(x) dx = sin(x)|_{0}^{π/2} = sin(π/2) − sin(0) = 1.  (6)

And, numerically, setting A_{p}(n) to be the value of Simpson’s rule approximation for n subintervals

n =  Simpson’s rule  |1 − A_{p}(n)|  Error bound
2  1.00228  0.00228  0.00332
4  1.00013  0.00013  0.00021
6  1.00003  0.00003  0.00004
8  1.00001  0.00001  0.00001
10  1.0000033922209010  0.0000033922209010  0.000005312
20  1.0000002115465910  0.0000002115465910  0.000000332
50  1.0000000054122520  0.0000000054122520  0.000000008
100  1.0000000003382360  0.0000000003382360  0.0000000005

Now we see how the error estimate really works. If f(x) = cos(x), then f^{(4)}(x) = cos(x), −1 ≤ cos(x) ≤ 1, for all real x.

f′(x) = − sin(x)
f′′(x) = − cos(x)
f^{(3)}(x) = sin(x)
f^{(4)}(x) = cos(x)

(7)

|f^{(4)}(x)| ≤ 1. We have an error bound

|\frac{(β − α)^{5}}{180 · n^{4}} f^{(4)}(ξ)| ≤ \frac{(π/2)^{5}}{180 · n^{4}} 1 = \frac{π^{5}}{32 · 180 · n^{4}} .

15 To delve—to search carefully and painstakingly for information.

1.5 Truncation Error. In this section we will derive the truncation error for Simpson’s rule. This section has some theory and may be omitted by someone only interested in applications. One might note that in the previous section that the actual error was much less than the error bound. The reason for the “looseness” in the error bound has to do with bounding the fourth derivative over the entire interval, [α, β].

We will assume that the function f(x) has a Taylor series expansion. This is a reasonable assumption for theoretical purposes; however, in the real world it may present a problem. Derivation with weaker conditions may be found in standard texts. Using equation (1) for A_{p}

\begin{aligned}A_{p} & = \frac{1}{3}h(y_{0} + 4y_{1} + y_{2}) \\ & = \frac{1}{3}h[y_{0} + 4 (y_{0} + hy′_{0} + \frac{1}{2}h^{2}y_{0}^{(2)} + \frac{1}{6}h^{3}y_{0}^{(3)} + \frac{1}{24}h^{2}y_{0}^{(4)} + …) \\ & + (y_{0} + 2hy′_{0} + 2h^{2}y_{o}^{(2)} + \frac{4}{3}h^{3}y_{0}^{(3)} + \frac{2}{3}h^{4}y_{0}^{(4)} + …) ] \\ & = \frac{1}{4}h (6y_{0} + 6hy′_{0} + 4h^{2}y_{0}^{(2)} + 2h^{3}y_{0}^{(3)} + \frac{5}{6}h^{4}y_{0}^{(4)} + …) \end{aligned}  (8)

Now do the same thing to the integral itself.

\begin{aligned}∫_{x_{0}}^{x_{2}} y(x) dx & = F (x_{2}) − F (x_{1}) \\ & = 2hF′(x_{0}) + \frac{1}{2}(2h)^{2}F^{(2)}(x_{0}) + \frac{1}{6}(2h)^{3}F^{(3)}(x_{0}) \\ & + \frac{1}{24}(2h)^{4}F^{(4)}(x_{0}) + \frac{1}{120}(2h)^{5}F^{(5)}(x_{0}) + … \\ & = 2hy_{0} + 2h^{2}y_{0} + \frac{4}{3}h^{3}y_{0}^{2} + \frac{2}{3}h^{4}y_{0}^{(3)} + \frac{4}{15}h^{5}y_{0}^{(4)} + …. \end{aligned}  (9)

Subtract equation (8) from equation (9) to get

∫_{x_{0}}^{x_{2}} y(x) dx − \frac{1}{3}h(y_{0} + 4y_{1} + y_{2}) = (\frac{4}{15} − \frac{5}{18}) h^{5}y − 0^{(4)} = −\frac{h^{5}y_{0}^{(4)}}{90}  (10)

Equation (10) follows from the arithmetic calculation

\frac{4}{15} − \frac{5}{18} = \frac{24 − 25}{90} = −\frac{1}{90}.

If we have an interval [α, β], which can be written as the union of intervals [x_{0}, x_{2}], [x_{2}, x_{4}], …, [x_{n−2}, x_{n}], then we may write the total truncation error as

−\frac{h^{5}}{90} (y_{0}^{(4)} + y_{2}^{(4)} + … + y_{n−2}^{(4)}) .

Now we make the assumption that the fourth derivative is continuous on the interval [α, β]. β − α = nh, so we have

−\frac{h^{5}}{90} (y_{0}^{(4)} + y_{2}^{(4)} + … + y_{n−2}^{(4)}) = −\frac{(β − α)^{5}}{180 · n} y^{(4)}(ξ),

where ξ ∈ (α, β) and

y_{0}^{(4)} + y_{2}^{(4)} + … + y_{n−2}^{(4)} = \frac{n}{2} · y^{(4)}(ξ).

1.6 An Example. In this section we will consider an application to the particular function

f(x) = (x + 4π) · (x + 4π − 1/π) · (x + 4π − 2/π)

This remarkable function has properties as follows^{16}

Ionized  \underline{mass}  relative
Isotope  electron mass  x  f(x)  deviation

  _{1}^{1}H^{+}  1836.152701  0  1836.15174  0.00003%
  _{1}^{2}H^{+}  3670.4830  π  3643.34824  0.37%
  _{2}^{4}He^{++}  7294.2995  π + 4  7287.74349  0.01%
  _{4}^{9}Be^{+4}  16424.2099  π + 10  16364.47397  0.18%
  _{6}^{12}C^{+6}  21868.6918  5π  21845.9892  0.05%
  _{29}^{63}Cu^{+29}  114684.6335  9π + 8  114237.3148  0.19%
  _{50}^{120}Sn^{+6}  218518.1598  14π + 4  218491.3263  0.05%

However, for our example, we will only be interested in computing the integral of the area between the curve and the x-axis between the roots α_{1} = −4π and α_{2} = −4π + 1/π. This will involve an application of Simpson’s rule. This example was chosen because it involves a polynomial (of degree three) which does not have integer coefficients. Indeed, the coefficients are not even algebraic numbers—they are transcendental numbers. We will observe how nicely our integration formula functions in this case.

Graph of f(x) = (x + 4π) · (x + 4π − 1/π) · (x + 4π − 2/π)_{−13  −12} −0.015 −0.010 −0.005 0.000 0.005 0.010 0.015  _{FIGURE 6}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{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16 The superscript before the chemical symbol gives the value of Z, the atomic weight; the subscript before the symbol gives the value of A, the atomic number; and, the superscript after the symbol gives the ionization. Values taken from Quantum Physics, 2nd Ed. by Eisberg and Resnick, (NY: John Wiley & Sons, 1985), page 520.

The value of the integral is calculated to be approximately 0.0025664955636711. This example points out one of the strong points of using Simpson’s rule. When the answer is computed, it is in numeric form. If one obtained a so-called “closed-form” solution, then it would be necessary to plug in the values for π to get a numeric result.

Following this same argument, let’s consider integrating the same function, f(x) = (x + 4π) · (x + 4π − 1/π) · (x + 4π − 2/π), over the (closed) interval [ − 4π + 2/π, 0]. Because of the variation of the range of f, we will use a logarithmic scale on the y-axis. The value “under the curve” is approximately 5618.52714450635. We do actually have this much accuracy because of the very precise value in our “C” program for the constant π. Recall the expression

#define pi 3.141592653589793238462643383279 /* Accurate value for pi.  */

The only changes necessary to the “C” program were

a = (double) -4.0*pi+2.0/pi;
printf("\nLeft end point = %.16lf",a);
b = (double) 0.0;
printf("\nRight end point = %.16lf",b);

And, now the graph.

The integral of f(x) = (x + 4π) · (x + 4π − 1/π) · (x + 4π − 2/π) over the interval [ − 4π + 2/π, 0]

−4π+8/π −8 −6 −4 −2 0 2 1 2 3 4

P o w e r s of 10

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • ⟵ (0, 1836.15) |

{↑ \atop −4π+2/π}

FIGURE 7

We might also consider some other interesting values for the function f(x), as defined above, for ratios of subatomic particles’ masses to the mass of the electron.

Subatomic  \underline{mass}  relative
Particle  electron mass  x  f(x)  deviation

  n  1838.683662  1/64π  1838.39048447 0.008%

τ^{−}  3491.4  3 − 1/4π  3488.5  0.039%
η^{0}  1073.97  −2 − 1/64π 1073.7  0.014%

  η′  1873.78  1/4π  1872.2  0.069%

Σ^{+}  2327.5  1 + 1/64π  2326.8  0.0064%
Σ^{0}  2333.76  1 + 1/16π  2334.3  0.034%
Σ^{−}  2343.31  1 + 1/8π  2344.8  0.049%

  Λ  2183.23  1 − 1/π  2160.26  0.53%

1.7 Verifying Derivatives. Calculus is traditionally partitioned into differential and integral calculus. This partition supposes that the two concepts are independent and mutually exclusive. Then, almost as a gift from the gods, we encounter the so-called Fundamental Theorem of Calculus,^{17} which unifies the two concepts (more or less—mostly less). In this era of digital computers and the future directed towards artificial intelligence, it might be a good idea to discard the artificial divisions of calculus in favor of a more pragmatic^{18} development. In this section we will examine a good technique for determining if a function, which is supposed to be the derivative of a given function, really is. The idea is simple. We have a given function and a candidate for its derivative. Looking at some interval, we choose four points and use Simpson’s rule to evaluate the integral of the derivative on subintervals, ending at the four points, and compare the results with the original function. This is awkward to verbalize but easy to write “in mathematics.” Let f(x) be a given function and let g(x) be a candidate for the derivative f′(x) of f(x). In the interval [α, β], choose four points, x_{1}, x_{2}, x_{3}, and x_{4}. (Let α < x_{1} < x_{2} < x_{3} < x_{4} < β.) We can eliminate g(x) if any of the four approximations are incorrect:

f(x_{1}) − f(α) ≈ ∫_{α}^{x_{1}} g(t) dt

f(x_{2}) − f(α) ≈ ∫_{α}^{x_{2}} g(t) dt

f(x_{3}) − f(α) ≈ ∫_{α}^{x_{3}} g(t) dt

f(x_{4}) − f(α) ≈ ∫_{α}^{x_{4}} g(t) dt

We want to examine this idea, using a non-trivial example. One such example comes from the

four derivatives of the function f(x) = \sqrt{1 − k^{2} sin^{2}(x)}. Recall it was the fourth derivative that was used in the error estimate. This function is easy to write; however, its derivatives become more and more difficult to expand algebraically. How can we be sure that we did not make some arithmetic error? It has been shown that, on the average, 1 out of every 300 human calculations is in error. This will be done via a computer program. The program evaluates all the integrals simultaneously and outputs the results in a rectangular array of real numbers—a matrix. By visually comparing the rows of the matrix, we can tell at once if our derivatives are correct. Without any more delay, let’s write down the program and its results and then do an analysis of the coding.

100 DEFDBL A-H, O-Z
102 C2=SIN(EXP(1.)/2.)*SIN(EXP(1.)/2.)
105 DEF FNA(x AS DOUBLE)= SQR(1.-C2*SIN(x)^2)
106 DEF FNB(x AS DOUBLE)= -.5*C2*SIN(2.*x)/FNA(x)
107 DEF FNC(x AS DOUBLE)= -C2*COS(2.*x)/FNA(x)
        -.25* C2*C2*SIN(2.*x)^2/FNA(x)^3
108 DEF FND(x AS DOUBLE)= 2.*C2*SIN(2.*x)/FNA(x)
        -.75*C2*C2*SIN(4.*x)/FNA(x)^3
        -.375*C2^3*SIN(2.*x)^3/FNA(x)^5
109 DEF FNE(x AS DOUBLE)= 4.*C2*COS(2.*x)/FNA(x)
        + C2^2*SIN(2.*x)^2/FNA(x)^3
        -3.*C2^2*COS(4.*x)/FNA(x)^3
        -1.125*C2^3*SIN(4.*x)*SIN(2.*x)/FNA(x)^5
        -(18./8.)*C2^3*SIN(2.*x)^2*COS(2.*x)/FNA(x)^5

17 Theorem. Let f be continuous on [α, β] and x ∈ (α, β). If F (x) = ∫_{a}^{x} f(t) dt, then F′(x) = f(x). 18 Pragmatic—practical, especially in contrast to idealistic.

        -(15./16.)*C2^4*SIN(2.*x)^4/FNA(x)^7
110 A = 0.
112 INPUT "Left endpoint = "; B
114 IF (A>B) OR (B>3.141592653589793/2.)
        THEN PRINT "Input out of range":  GOTO 112
116 OPEN "92 12 14.txt" FOR APPEND AS #1
120 REM Left endpoint <= 3.141592653589793# / 2!
130 SUM1 = 0.:  SUM2 = 0.:  SUM3 = 0.:  SUM4 = 0.
150 N = 512:  H = (B - A) / N
160 FOR I = 1 TO N STEP 2:  REM The "FOR" loop is done n/2 times.
170 SUM1 = SUM1 + FNB(A + (I - 1) * H)
172 SUM2 = SUM2 + FNC(A + (I - 1) * H)
174 SUM3 = SUM3 + FND(A + (I - 1) * H)
176 SUM4 = SUM4 + FNE(A + (I - 1) * H)
180 SUM1 = SUM1 + 4.*FNB(A + I * H)
182 SUM2 = SUM2 + 4.*FNC(A + I * H)
184 SUM3 = SUM3 + 4.*FND(A + I * H)
186 SUM4 = SUM4 + 4.*FNE(A + I * H)
190 SUM1 = SUM1 + FNB(A + (I + 1) * H)
192 SUM2 = SUM2 + FNC(A + (I + 1) * H)
194 SUM3 = SUM3 + FND(A + (I + 1) * H)
196 SUM4 = SUM4 + FNE(A + (I + 1) * H)
200 NEXT I
210 L$ = "##.##### ##.##### ##.##### ##.##### ##.#####"
300 PRINT USING L$; FNA(B)-FNA(A); FNB(B)-FNB(A);
        FNC(B)-FNC(A); FND(B)-FND(A)
310 PRINT USING L$; SUM1*H/3.; SUM2*H/3.;
        SUM3*H/3.; SUM4*H/3.
320 PRINT #1, USING L$; FNA(B)-FNA(A); FNB(B)-FNB(A);
        FNC(B)-FNC(A); FND(B)-FND(A)
330 PRINT #1, USING L$; SUM1*H/3.; SUM2*H/3.;
        SUM3*H/3.; SUM4*H/3.
340 CLOSE #1
350 STOP
360 END

Of course, we want to look at the output. For input values, we used .25, .50, 1.00, and 1.50. Rows 1 and 2 of this 4 by 8 matrix correspond to x = .25, rows 3 and 4 correspond to x = .5, and so on.

  -0.02969            -0.23615            0.03387              0.27142
  -0.02969            -0.23615            0.03387              0.27142
  -0.11666            -0.45528            0.13655              0.55428
  -0.11666            -0.45528            0.13655              0.55428
  -0.43151            -0.76446            0.62760              1.73354
  -0.43151            -0.76446            0.62760              1.73354
  -0.77883            -0.30495            4.81402             17.17882
  -0.77883            -0.30495            4.81402             17.17882

All this goes to show that the equations we coded into the computer program are probably correct. This is not rigorous; however, in our days and times anything that fits this well would be a good approximation—that is, it would serve as the derivative for all practical purposes. A word of caution: there may be more than one way to express a given equation. For instance, y = x^{2} + 2x + 1 and y = (x + 1)^{2} are algebraically the same. Even though two expressions are algebraically they same, they may not yield the same value from a computer. This is due to several factors: (1) computers may evaluate the equations differently; (2) there may be more truncation (round-off) error with one equation than with the other (after all, computers don’t have infinite accuracy); (3) a coding error may affect one equation (especially doing an integer division where the fractional part is lost); and, (4) the grouping of successive multiplications and divisions may affect the answer, especially when two nearly equal numbers are being subtracted.

There are a few items worth mentioning about the computer program itself. We “captured” or “saved” the output by appending to a file (in this case named 92 12 14.txt. Later, we edited the file and merged it into this document. This gave two distinct advantages: (1) it was quicker and (2) there was no possibility of a human blunder in re-typing the numbers. This is always a good technique. Capturing output data and inputing the file directly into text ensures a correct copy. It may not always possible to do this. A second alternative is to compare the final text against the input via some compare routine. In Appendix B, a method of comparing text with output will be devised, developed, and discussed. In BASIC^{19} the coding line is 256 characters long, more or less (usually less, maybe 254 characters). In a text document, it is a good idea to have lines of 80 characters or less. To accommodate the document, it is necessary to “break” long lines into shorter segments. Whenever this is done, the line is indented. The sequence numbers are retained to aid in determining when a line has been broken. Some versions of BASIC write floating point integers using an exclamation point, e.g., 3.00 becomes 3!. All floating point integers have been written in the standard mathematical format using a decimal point. QBasic^{20} puts in spaces between the mathematical operators. While the spaces add to the readability of the code and aid the programmer, they do little for the documentation, which has to fit on an 8\frac{1}{2} × 11 inch page. The extra spaces were removed. For the “TEX-nicians,” they may note that some symbols cause the typesetting program to suffer. Before merging into a TEX file, the special symbols backslash (\) and hat (or circumflex) (^) were replaced by their character equivalences. The pound sign, dollar sign, and underscore (#, $, and ) were dealt with in the usual manner, by prefixing them with backslashes (\).

As usual, we will complement the BASIC program with a “C” program. The “C” program also saves its output to a file.

#include <stdio.h> /* Header for input/output subroutines.  */
#include <math.h> /* Header for math subroutines.  */
#include <float.h> /* Header for floating point subroutines.  */
#define pi 3.141592653589793238462643383279 /* Accurate value for pi.  */
#define k2 0.9558669573934826
/* Simpson's rule for approximating integrals.
    a:  left endpoint
    b:  right endpoint
    fa:  original function
    fb,fc,fd,fe:  pointers to functions to integrate
    n:  number of subintervals
*/
double fa (double x);
double fb (double x);
double fc (double x);
double fd (double x);
double fe (double x);
main()
{
double a,b,h,sum,x,y; /* In 'C' all variables must be assigned */

19 BASIC—Beginners’ All-purpose Symbolic Instruction Code. This elementary computer code is held in disdain by many programmers who prefer “C.” 20 QBasic—the version of BASIC copyrighted by the Microsoft Corporation.

double p1, p2, p3;
int i, n;
FILE *fp;
fp = fopen("92 12 13.txt", "a");
printf("\007"); /* Sound bell in computer.  */
a = (double) 0.0;
printf("\nLeft end point a = %.16lf",a);
printf("\nEnter right end point ");
scanf("%lf",&b);
printf("\nRight end point b = %.16lf",b);
n = 512;
printf("\nNumber of subintervals %d",n);
h = (double) (b-a)/n;
for (i=1, sum=0.0; i<=n; i = i+ 2){
  p1 = fb((double) a+(i-1)*h);
  p2 = fb((double) a+i*h);
  p3 = fb((double) a+(i+1)*h);
  sum += p1 + 4.0 * p2 + p3;
}
y = (double) h*sum/3.0;
printf("\nValue of integral = %.16lf", (double) y);
printf("\nValue of f(b)-f(a) = %.16lf", (double) fa(b)-fa(a) );
fprintf( fp, " %.5lf %.5lf", y, fa(b)-fa(a) );
for (i=1, sum=0.0; i<=n; i = i+ 2){
  p1 = fc((double) a+(i-1)*h);
  p2 = fc((double) a+i*h);
  p3 = fc((double) a+(i+1)*h);
  sum += p1 + 4.0 * p2 + p3;
}
y = (double) h*sum/3.0;
printf("\nValue of integral = %.16lf", (double) y);
printf("\n f'(b) - f'(a) = %.16lf", (double) fb(b)-fb(a) );
fprintf( fp, " %.5lf %.5lf", y, fb(b)-fb(a) );
for (i=1, sum=0.0; i<=n; i = i+ 2){
  p1 = fd((double) a+(i-1)*h);
  p2 = fd((double) a+i*h);
  p3 = fd((double) a+(i+1)*h);
  sum += p1 + 4.0 * p2 + p3;
}
y = (double) h*sum/3.0;
printf("\nValue of integral = %.16lf", (double) y);
printf("\n f''(b) - f''(a) = %.16lf", (double) fc(b)-fc(a) );
fprintf( fp, " %.5lf %.5lf", y, fc(b)-fc(a) );
for (i=1, sum=0.0; i<=n; i = i+ 2){
  p1 = fe((double) a+(i-1)*h);
  p2 = fe((double) a+i*h);
  p3 = fe((double) a+(i+1)*h);
  sum += p1 + 4.0 * p2 + p3;
}
y = (double) h*sum/3.0;
printf("\nValue of integral = %.16lf", (double) y);
printf("\n f'''(b)-f'''(a) = %.16lf", (double) fd(b)-fd(a) );
fprintf( fp, " %.5lf %.5lf \n", y, fd(b)-fd(a) );

fclose(fp);
return(0);
}
double fa (double x)
{
    double y;
    y = (double) sqrt(1.0 -k2*sin(x)*sin(x));
    return (y);
}
double fb (double x)
{
    double y;
    y = (double) -0.5*k2*sin(2.0*x)/fa(x);
    return (y);
}
double fc (double x)
{
    double y;
    y = (double) -k2*cos(2.0*x)/fa(x)
      -.25*k2*k2*sin(2.0*x)*sin(2.0*x)/pow(fa(x),3.0);
    return (y);
}
double fd (double x)
{
    double y;
    y = (double) 2.0*k2*sin(2.0*x)/fa(x)
      -0.75*k2*k2*sin(4.0*x)/pow(fa(x),3.0)
      -.375*k2*k2*k2*pow(sin(2.0*x),3.0)/pow(fa(x),5.0);
    return (y);
}
double fe (double x)
{
    double y;
    y = (double) 4.0*k2*cos(2.0*x)/fa(x)
      +k2*k2*sin(2.0*x)*sin(2.0*x)/pow(fa(x),3.0)
      -3.0*k2*k2*cos(4.0*x)/pow(fa(x),3.0)
      -1.125*k2*k2*k2*sin(4.0*x)*sin(2.0*x)/pow(fa(x),5.0)
      -2.25*k2*k2*k2*sin(2.0*x)*sin(2.0*x)*cos(2.0*x)/pow(fa(x),5.0)
      -0.9375*k2*k2*k2*k2*pow(sin(2.0*x),4.0)/pow(fa(x),7.0);
    return (y);
}
/* End of file */

Of course, there is an output file to be examined also. Here we included values of 1.51, 1.52, and 1.55.

-0.02969  -0.02969  -0.23615  -0.23615  0.03387   0.03387    0.27142   0.27142
-0.11666  -0.11666  -0.45528  -0.45528  0.13655   0.13655    0.55428   0.55428
-0.43151  -0.43151  -0.76446  -0.76446  0.62760   0.62760    1.73354   1.73354
-0.77883  -0.77883  -0.30495  -0.30495  4.81402   4.81402   17.17882  17.17882
-0.78168  -0.78168  -0.26553  -0.26553  4.97895   4.97895   15.74172  15.74172
-0.78414  -0.78414  -0.22454  -0.22454  5.12756   5.12756   13.91650  13.91650
-0.78894  -0.78894  -0.09416  -0.09416  5.43883   5.43883    6.37637   6.37637

1.8 Generalizations. Mathematicians love to take a popular, useful concept and generalize it. Sometimes much can be learned by generalizing and by abstracting; more often than not, a generalization results in a more complicated, theoretical, and generally useless body of knowledge that exists solely as a requirement for a degree. This is especially true with Simpson’s rule. Simpson’s rule is popular and efficient. Its error calculation is straightforward and its convergence is assured. There are some examples where Simpson’s rule does poorly—but these are mostly “pathological” in nature. (They are artificially constructed simply to demonstrate the fallability of the rule.) There are also so-called improper integrals. The improper integrals are either defined over an interval such as [α, +∞], [−∞, β], or [−∞, +∞], or the function is unbounded in the interval of integration (or both!) The improper integrals have to be approached with common sense. Take for example the integral

∫_{0}^{1} \frac{dx}{\sqrt{x}} = 2\sqrt{x}|_{x=1} − lim_{x→0} 2\sqrt{x} = 2.

How could Simpson’s rule be applied to such an example? The answer is simple—make the computer do the work! Choose a small number ϵ, 0 < ϵ ≪ 1. By a proper determination of the number of subintervals and a sufficiently small value of ϵ, the desired answer can be obtained with the required accuracy.

FIGURE 8^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}

In the case of integrals of the form

\frac{1}{\sqrt{2π}} ∫_{0}^{∞} e^{−x^{2}/2} dx

(the classic normal curve), one needs only observe that the integral may be replaced by an integral

\frac{1}{\sqrt{2π}} ∫_{0}^{N} e^{−x^{2}/2} dx

for a suitable chosen integer N. We note here that N = 6 generally suffices for most engineering work. This is called the six-sigma (6-σ). We can compute this estimate by considering

∫_{1}^{N} e^{−x^{2}/2} dx ≤ ∫_{1}^{N} e^{−x/2} dx.

If x ≥ 1, then x^{2} ≥ x and e^{−x^{2}/2} ≤ e^{−x/2}.

∫_{N}^{∞} e^{−x/2} dx = 2e^{−N/2}.

The density φ(ζ) = e^{−ζ^{2}/2}/\sqrt{2π} of the

standard normal distribution.

−3  −2  −1  0  1 ζ 2  3

↙ φ(ζ)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

FIGURE 9

Having mentioned how to generalize this rule to improper integrals, it is time to examine the position of this integration scheme in mathematicians’ grand scheme of things. Simpson’s rule is the most popular of the so-called Newton-Cotes integration formulas. The first three of which are given below

\begin{aligned}∫_{x_{0}}^{x_{0}+h} f(x) dx & = \frac{h}{2}(y_{0} + y_{1}) − \frac{h^{3}}{12} f′′(ξ) \\ ∫_{x_{0}}^{x_{0}+2h} f(x) dx & = \frac{h}{3}(y_{0} + 4y_{1} + y_{2}) − \frac{h^{5}}{90} f^{(4)}(ξ) \\ ∫_{x_{0}}^{x_{0}+3h} f(x) dx & = \frac{3h}{8}(y_{0} + 3y_{1} + 3y_{2} + y_{3}) − \frac{3h^{5}}{90} f^{(4)}(ξ), \end{aligned}

where ξ is an intermediate value of x. The Newton-Cotes formulas, such as those above, are called closed because the interval end-points are used. If the end-points are not used, but only the interior points in the interval, the formula is called open. The Newton-Cotes formulas belong to a class of formulas known as polynomial approximations; the class of polynomial approximations is contained in a class of formulas of approximation of integrals by families of analytic functions, and so on.

0  x  1  2  3^{•} e^{−x} 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^{.} ^{.} 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FIGURE 10

Newton’s Method

Chapter 2

2.0 Introduction. One of the most exciting applications of differential calculus is the use of Newton’s method for the determination of roots of functions. Most students of algebra think that the only roots of interest are those of polynomials. An algebra student who has done all the assigned homework might think that it’s just a matter of factoring; after all, it was so clever just completing the square in the quadratic equation. (Wasn’t it fun being so smart and showing that off?) That’s simply not the case. In fact, there is no purely algebraic method for finding roots of polynomials of degree five or higher. The situation with the trigonometric functions is even more involved. What technique would one use to find the root or roots of x − cos x = 0?

Tangent line at (x_{0}, y_{0}) slope = f′(x_{0})

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•

•

(x_{0}, y_{0})

↖x-intercept f(x)↘

↖tangent line

FIGURE 11

All of the important details of Newton’s method are found in the above figure. The equation of the tangent line is determined by the so-called point-slope method from elementary algebra:

\frac{y − y_{0}}{x − x_{0}} = slope = \frac{Δy}{Δx} = \frac{rise}{run} = f′(x_{0}).

Set y = 0 and determine the x-intercept of the tangent line, (x_{0} − y_{0}/f′(x_{0}), 0). From the above figure, we see that the point at which the tangent line crosses the x-axis is moving quickly towards the root—the point at which the curve f(x) crosses the x-axis. If we only had a better first approximation x_{0}, then the x_{0} − y_{0}/f′(x_{0}) would be even closer to the root. Why not just try again and define a new approximation?

x_{1} = x_{0} − \frac{f(x_{0})}{f′(x_{0})}  (11)

2.1 Theory. This paragraph is concerned with the theory behind Newton’s method. The user who is interested only in results may skip this entire section. This paragraph employs the so-called Taylor series (also known as the Taylor’s series)

f(x_{0}) + f′(x_{0})(x − x_{0}) + \frac{f′′(x_{0})}{2}(x − x_{0})^{2} + … + \frac{f^{(n)}}{n!} (x − x_{0})^{n} + ….  (12)

An alternate derivation using the Mean-Value Theorem will be presented in paragraph 2.2. If one looks in a dictionary,^{21} equation (12) might be written in the form

f(x) = f(a) + \frac{f^{[1]}}{1!} (x − a) + \frac{f^{[2]}}{2!} (x − a)^{2} + … + \frac{f^{n]}}{n!} (x − a)^{n} + …,  (12′)

where f^{[n]}(a) is the derivative of the n^{th} order of f(x) evaluated at a. Before delving into the mathematics, a digression^{22} might be in order. Whenever a textbook says “degree,” it means the highest exponent or power. For example: x^{2} is of second degree; the polynomial x^{3}−x^{2}+1 is of third degree; and, x^{2} + y^{2} = 1 is a second degree relation. Whenever a textbook says “order,” it generally refers to a more abstract concept. In general, x^{n} means the n^{th} power (degree n) of x, that is

x^{n} = \underbrace{x · x ⋯ x}_{n times} .

(Like every rule, this one has an exception. In tensor analysis, the i, j, and k in expressions like x^{i}, y^{jk}, etc., refer to contravariant indices, not powers.) On the other hand, whenever one sees such notations as x^{(n)}, y^{[n]}, and so on, a red flag ought to go up. Indeed, if a function does have a derivative, it is usually denoted by f′(x), the second derivative (the derivative of the derivative) is denoted by f′′(x). But, the third derivative is not generally denoted by f′′′(x). The third derivative of f at x is usually written f^{(3)}(x) and sometimes written as f^{[3]}(x). What does f^{(0)}(x) mean? Recall that (for all x ≠ 0) x^{0} ≡ 1. f^{(0)}(x) ≡ f(x). The symbol made up of three horizontal lines (≡) means more that just equals (=); this symbol, (≡), means equivalence and that means equal for all the values under consideration.

One of mathematics’ favorite devices, or tricks, is to truncate^{23} a Taylor series. In the Newton method, the series is truncated at the quadratic term, that is, at \frac{1}{2}f′′(x_{0}) · (x − x_{0})^{2}. So, the truncated series becomes

\begin{aligned}f(x) & = f(x_{0}) + \frac{f^{[1]}}{1!} (x − x_{0})^{1} \\ & = f(x_{0}) + f′(x_{0}) · (x − x_{0}). \end{aligned}  (13)

This is the derivation. So, after all that was said and done, there’s not much to do to get the result. Just pretend that f(x_{1}) = 0 and get

f(x_{1}) = 0 = f(x_{0}) + f′(x_{0})(x_{1} − x_{0}),

or

x_{1} = x_{0} − \frac{f(x_{0})}{f′(x_{0})},  (14)

provided, of course, that f′(x_{0}) ≠ 0. If f′(x_{0}) = 0, then we just choose another x_{0}, say x̃_{0} where f′(x̃_{0}) ≠ 0 and proceed.

21 Webster’s Ninth New Collegiate Dictionary, (Springfield, Massachusetts: Merriam-Webster, Inc., 1984), page 1209. 22 Digression—A swerving away from the main subject or the Leitmotif ; a turning aside from the main argument. 23 Truncate—To cut off; to make short by trimming or cutting off.

2.2 Applications. One of the most important applications of Newton’s method is in the calculation of square roots. Many small hand calculators have a “hard-wired” square root generator. The circuitry follows a simple algorithm. Since long division of decimal fractions may not be supported, we will derive an algorithm which does not involve division and finds the square root for any number a > 0.

It is clear that if we let h(x) = x^{2} − a and solve this equation for the root h(x_{0}) = 0, this will be the same as finding the square root of a. Newton’s formula becomes x_{n+1} = \frac{1}{2}(x_{n} − a/x_{n}).^{24}

The next thing that we will do is to plug the function F (x) = a − 1/x into Newton’s method. Then we will look at the fixed point formula. To determine the starting value, we’ll require ξ to satisfy the expression |f′(ξ)| < 1. After we’ve done those things, we will generate a short little BASIC program to check out some sample values. Then, to satisfy the academics, we will quote the necessary theorems and satisfy the hypotheses.

F (x) = a − \frac{1}{x}  (15)
F′(x) = \frac{1}{x^{2}}  (16)

Applying f(x) = x + g(F (x)), where g(y) = −y/y′,

\begin{aligned}f(x) & = x − F (x)/F′(x) \\ & = x − \frac{a − 1/x}{1/x^{2}} \\ & = x − ax^{2} + x \\ & = 2x − ax^{2} \end{aligned}  (17)

And we observe, substituting s = 1/a into f′(s) that

f′(x) = 2 − 2ax,  (18)
f′(1/a) = 2 − 2a(1/a) = 2 − 2 = 0.  (19)

Recall that a ≠ 0.

f′′(x) = −2a  (20)

tell us that f′′ does not vanish at 1/a so we have quadratic convergence in some (yet to be determined) neighborhood of 1/a. So, the required function is equation (17) and the iteration procedure is

x_{n+1} = 2x_{n} − ax_{n}^{2}.  (21)

All that remains is to find the necessary x_{0} = x_{0}(a) to get things started.

We would like to determine an interval [α, β] on which f is contractive. It isn’t enough just to know that one exists, we really need to compute it. But we have an explicit function, f, and f ∈ C^{1}. By the mean value theorem (MVT for derivatives), we seek values for ξ such that |f′(ξ)| < 1. This will ensure

|f(x_{1}) − f(x_{2})| = |f′(ξ)| |x_{1} − x_{2}|  (22)
|f′(x)| < 1
|2 − 2ax| < 1

24 If the following mathematics looks too deep, try Schaum’s Outline Numerical Analysis, 2nd Edition, page 334. In any case, the two BASIC computer programs ought to be useful.

−1 < 2 − 2ax < 1
−3 < −2ax < −1

If we multiply an inequality by −\frac{1}{2}, it changes the ’sense’ of the inequality:

\frac{3}{2} > ax > \frac{1}{2}  (23)

Either a > 0 or a < 0, so:

interval = \{ \begin{matrix}\frac{3}{2a} > x > \frac{1}{2a}, & if a > 0 \\ \frac{1}{2a} > x > \frac{3}{2a}, & if a < 0 \end{matrix}  (24)

Basic program:

10 REM ************************************************
20 REM * Harry A. Watson, Jr.
30 REM * Math 219b
40 REM * 27 April 1991
50 REM ************************************************
60 INPUT "a, x 0 = "; A, X0
70 FOR I = 1 TO 100
80 X1 = 2*X0 - A*X0*X0
90 IF ABS(X1-X0) < .0001 THEN PRINT X1,1/A,I:STOP
100 X0 = X1
110 NEXT I

Now we would like to have an iterative method for computing ^{n} \sqrt{a}, a > 0, which converges locally in second order. We only want to employ addition, subtraction, multiplication and division (the four arithmetic operations of the real number field.

Again, we will be successful in using Newton’s method. The fixed point theorem will guarantee that the iterative method converges locally in second order, that is, quadratically.

F (x) = a − x^{n}  (25)
F′(x) = −nx^{n−1}  (26)
\frac{F (x)}{F′(x)} = \frac{a − x^{n}}{−nx^{n−1}} = −\frac{a}{nx^{n−1}} + \frac{x}{n}  (27)

\begin{aligned}f(x) & = x − F (x)/F′(x) \\ & = x + \frac{a}{nx^{n−1}} − \frac{x}{n} \\ & = (1 − \frac{1}{n})x + \frac{1}{n}(a/x^{n−1}) \end{aligned}  (28)

We will ensure convergence by computing f′ and plugging in the value s = ^{n} \sqrt{a}.

f′(x) = (1 − 1/n) − [a(n − 1)]/[nx^{n}].  (29)

\begin{aligned}f′( ^{n} \sqrt{a}) & = (1 + \frac{1}{n}) − \frac{n−1}{n} \frac{a}{a} \\ & = 1 − \frac{1}{n} − \frac{n}{n} + \frac{1}{n} \\ & = 0 \end{aligned}  (30)

Theory assured us this would happen; our f′ has been correctly computed.

f′′(x) = [a(n − 1)]/x^{n+1}  (31)

For a > 0,

f′′( ^{n} \sqrt{a}) = \frac{a(n − 1)}{a ^{n} \sqrt{a}} = \frac{n − 1}{\sqrt[n]{a}}.  (32)

n ≠ 1 in any case. Therefore we have quadratic convergence in some (yet to be determined) interval. Again, requiring |f′(x)| < 1:

−1 < (1 − \frac{1}{n}) − \frac{(n − 1)a}{nx^{n}} < 1

−2 + \frac{1}{n} < −\frac{(n − 1)a}{nx^{n}} < \frac{1}{n}

\frac{n}{(n − 1)a} (2 − \frac{1}{n}) > \frac{1}{x^{n}} > \frac{n}{(n − 1)a} \frac{1}{n}

\frac{(n − 1)a}{2n − 1} < x^{n} < (n − 1)a

Therefore,

n \sqrt{\frac{(n − 1)a}{2n − 1}} < x < ^{n} \sqrt{(n − 1)a}  (33)

is an interval in which the iteration algorithm converges quadratically. Finally, we note that x^{2} is just short-hand for x·x or x × x, and x^{3} stands for x·x·x or x × x × x and so on. For a given, arbitrary (but fixed) value of n, the above algorithm can be written without exponentiation, substituting

\underbrace{x · x ⋯ x}_{n times} for x^{n}.

BASIC computer program (fast):

10 REM *****************************************************
20 REM * Harry A. Watson, Jr.
30 REM * Math 219b
40 REM * 27 April 1991
50 REM *****************************************************
60 INPUT "Root n, n = "; N
70 INPUT "a, x 0 = "; A, X0
80 FOR I = 1 TO 100
90 X1 = (1-1/N)*X0 +A/(N*X0^(N-1))
100 IF ABS(X1-X0) < .0001 THEN PRINT X1,A^(1/N),I:STOP
110 X0 = X1
120 NEXT I
130 REM ***************************************************
140 REM * Recall that the cube root of 20 is approximately

150 REM * e = 2.7182818 …

160 REM * Plug in n = 3, a = 20,
170 REM * and x_{0} = 3
180 REM * Answer = 2.714418 2.714418 4 (steps)
190 REM ***************************************************

2.3 The Algorithm. The algorithm associated with Newton’s method may be seen graphically as follows:

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FIGURE 12

tan β = f′(x_{0}) = \frac{f(x_{0})}{x_{0} − x_{1}}

x_{1} = x_{0} − \frac{f(x_{0})}{f′(x_{0})}

Solve the equation f(x) = 0. We assume that f has a continuous first derivative, that is f ∈ C^{1}, where

C^{1} = { functions with continuous f′ }.

We will construct the following feedback diagram:

Last estimate x_{n}

x_{n+1} = x_{n} − \frac{f(x_{n})}{f′(x_{n})}

Next estimate x_{n+1}

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n := n + 1

x_{0} ⟶

FIGURE 13

We will see that each error is essentially proportional to the square of the previous error. This means that the number of correct decimal places roughly doubles with each successive approximation. This is called quadratic convergence.

2.4 A Second Opinion. We say a derivation of Newton’s method for finding roots in paragraph 2.1. There was an implicit assumption made that the function in question, f(x), had a Taylor series representation. This is a very strong assumption and we may find examples easily which do not satisfy this condition. One very common engineering example can be constructed as follows:

R′(x) = \{ \begin{matrix}1 & if 0 ≤ x < 1; \\ 0 & otherwise. \end{matrix}

R(x) = \{ \begin{matrix}0 & if x < 0; \\ x & if 0 ≤ x < 1; \\ 1 & if x ≥ 1; \end{matrix}

The function R(x) is the so-called Ramp function so highly favored by electrical engineers. Clearly the derivative is discontinuous at 0 and 1. We can construct f(x) = R(x) − \frac{1}{2} and this function has a root, x_{0} = \frac{1}{2}, f(1/2) = R(1/2) − 1/2 = 1/2 − 1/2 = 0. We can weaken the assumption that f(x) has a Taylor series representation and keep the same basic proof if we employ the so-called Extended Law of the Mean (Mean Value Theorem).^{25}

Theorem. Extended Law of the Mean (Mean Value Theorem). If each of f(x), f′(x), …, f^{(n−1)}(x) is continuous on a (closed) number interval [a, b], and if f^{(n)}(x) exists in the (open) number interval (a, b), then there exists a number ξ ∈ (a, b) such that

f(b) = f(a) + f′(a)(b − a) + \frac{f′′(x)}{2!}(b − a)^{2} + ⋯ + \frac{f^{(n−1)}(a)}{(n − 1)!}(b − a)^{n−1} + \frac{f^{(n)}(ξ)}{n!}(b − a)^{n}. (34)

We let n = 2 and write

f(x) = f(x_{0}) + f′(x_{0})(x − x_{0}) + \frac{f′′(ξ)}{2}(x − x_{0})^{2}.  (35)

We are looking for an x_{1} such that f(x_{1}) = 0. This being the case, we have

x_{1} = x_{0} − \frac{f(x_{0})}{f′(x_{0})} − \frac{f′′(ξ)}{2f′(x_{0})}(x_{1} − x_{0})^{2}.  (36)

We will examine other example of convergence, graphing the steps. In this case the function is not a polynomial, we will use

f(x) = e^{−(x−1/4)} − 1/4.  (37)

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x_{0}  x_{1}  x_{2} ↙(x_{0}, f(x_{0}))  ↙(x_{1}, f(x_{1})) ↙(x_{2}, f(x_{2}))^{•}  ^{•}  ^{•}_{•}  _{•}  _{•}  x → y = f(x)

FIGURE 14

25 John M. H. Olmsted, Advanced Calculus, (NY: Appleton-Century-Crofts, Inc., 1956), pages 75-85.

2.5 The Quasi-Newton Method. The quasi-Newton method, also known as the method of regula falsi, does not require the function for the derivative. It relies on approximation of the derivative by the ratio

\frac{f(x_{n−1}) − f(x_{n−2})}{x_{n−1} − x_{n−2}}.

Making the substitution

f′(x_{n−1}) ≈ \frac{f(x_{n−1}) − f(x_{n−2})}{x_{n−1} − x_{n−2}},

we obtain

x_{n} = x_{n−1} − \frac{(x_{n−1} − x_{n−2}) f(x_{n−1})}{f(x_{n−1}) − f(x_{n−2})}.  (38)

We will examine a problem from a standard textbook^{26} on quantum physics which will employ this technique. In order to derive the Wien displacement law, λmaxT = 0.2014 hc/k,^{27} it is necessary to solve the transcendental equation

e^{−x} + x/5 = 1.  (39)

We know already that the answer is approximately 5, in fact, it is approximately 4.965. We will use the following elementary BASIC computer program to see how well this method works.

5 DEFDBL A-H, O-Z
6 OPEN "92 12 16.txt" FOR OUTPUT AS #1
10 DEF fna (x) = EXP(-x) + .2 * x - 1!
20 INPUT "Initial approximation = ", x0
25 PRINT #1, "Initial approximation = ", x0
26 PRINT #1, "x1 = x0 + .0001 = ", x0 + .0001
30 h = .0001
40 x1 = x0 + h
50 x2 = x1 - (h * fna(x1)) / (fna(x1) - fna(x0))
60 PRINT x2
65 PRINT #1, "Next approximation = ", x2
70 h = x2 - x1
80 IF INKEY$ = "" THEN GOTO 80
90 x0 = x1:  x1 = x2:  GOTO 50

Now, we will look at the output (starting with an initial guess of 5):

TYPE 92 12 16.TXT

Initial approximation = 5
x1 = x0 + .0001 = 5.0001
Next approximation = 4.965135680044722
Next approximation = 4.965114168528612
Next approximation = 4.965114155083961
Next approximation = 4.965114155083955
Next approximation = 4.965114155083955

So, it didn’t take but four steps to converge! Now, let’s draw a graph and see just how this looks graphically. Incidently, in the real world this particular experimental value is rarely calculated with accuracy better than 10%. There is a good approximation to 0.2014, namely,

\frac{1}{2π(1 − cos(e/2))} = \frac{1}{4.963222169} = .20148 …  (40)

(It is accurate to within 0.019%.)

26 Robert Eisberg and Resnick, Robert, Quantum Physics, 2nd Edition, (NY: John Wiley & Sons, 1985), page 24. 27 h = 6.626 × 10^{−34} joule-sec is the Planck constant; c = 2.998 × 10^{8} m/sec is the speed of light in vacuum; k = 1.381 × 10^{−23} joule/^{∘}K is the Boltzmann’s constant.

0  x 1  2  3  4  5  6^{•}^{•} y_{1} = e^{−x} ↙y_{2} = 1 − x/5  _{•}↙y_{1} = 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FIGURE 15

Let’s take a look at another example, this time an iteration for a transcendental equation. We want to find the positive solution for 2 sin(x) = x. The first thing we will want to do is to draw a graph and estimate the root.

0 1 2 3 4 _{0  1  2  3}_{FIGURE 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⟵ y_{1} = 2 · sin(x) ↖y_{2} = x y_{1} = y_{2}↓ /

Of course, we will examine both the quasi-Newton and the Newton method. The Newton method is listed to the right and we see at once that it has converged both faster and to greater precision. The important point to note here is that with a modern (fast) computer both methods converge rapidly. We can use the two in conjunction to check each other’s results. Moreover, with the Simpson’s rule procedure to check the derivative function, we can be assured that the derivative f′(x) of f(x) is the correct function. This comparison of two different methods is similar to the adaptive quadrature routine (AQR), which will be studied in a later section.

Initial approximation = 2
x1 = x0 - .0001 = 1.99990   1.9009955942039090
Next approximation = 1.90099   1.8955116453795950
Next approximation = 1.89580   1.8954942672087130
Next approximation = 1.89550   1.8954952670339810
Next approximation = 1.89549   1.8954952670339810
Next approximation = 1.89549   1.8954925670339810
Next approximation = 1.89549   1.8954925670339810

2.6 Pathological examples. Newton’s method will fail for certain situations. There are an excellent developments of such examples in many elementary textbooks^{28} We won’t dwell on the analytic details of the failure of the Newton method; on the other hand, we will present graphs to illustrate the point. It has already been pointed out that the very first thing to do is to construct a graph and then estimate the root. With all the excellent graphics software available, there is no excuse for not doing so. Without further ado, let’s first look at a graph where Newton’s method fails to converge.

f(x) = \{ \begin{matrix}\sqrt{|x − r|} & if x > r \\ 0 & if x = 0 \\ −\sqrt{|x − r|} & if x < r \end{matrix}  (41)

Successive approximants oscillate back and forth.

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•

• • x_{0} O x_{1} r x

y

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.

FIGURE 17

Now, let’s look at a graph that really gets crazy! This time the successive approximations, x_{0} and x_{1} don’t oscillate, they get worse!

Successive approximants diverge from root.

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•

• • x_{0} O x_{1} r x

y

FIGURE 18

28 George B. Thomas, Jr., Calculus and Analytic Geometry, 3rd Edition, (Reading Massachusetts: Addison-Wesley Publishing Company, 1966), page 455.

Linear Least-squares Line

Chapter 3

3.0 Introduction. One of the best introductions to the so-called methods of least-squares follows from a simple display of a least-squares linear fit graph properly done (with the error bars and all the statistical appendages) and a dissection the components of the graph. The points in the given point set are graphed as central dots (^{•}) in the error bars ( \begin{matrix}⊤ \\ ⊥ \end{matrix} ). The length of the error bar is determined by the standard deviation, σ_{y}. From the central dot to the cross-bar is one unbiased standard deviation.

Missile ‘A’ Warrantied in shaded area

1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991

years ⟶_{FIGURE 19}

85 86 87 88 89 90 91 92 93 94 95

\begin{matrix}↑ \\ R% \end{matrix} •

• •

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Unwarrantied

Region

σ_{y} = 0.180

The underlying principle of the method of least squares is the determination of a curve which minimizes the squares of the deviations of a given point set to that curve. This is another way of saying that we will determine a curve such that the sum of the squares of the distances from the points in a given point set to the curve is a minimum. Since the sought-for curve is a straight line, this case is frequently referred to as a linear least-squares curve fit. The determined line is sometimes called a trend line, sometimes called a regression line, and sometimes called a line of best fit. The technique is known as the maximum likelihood criterion. ^{29} The theory involves an understanding of partial derivatives; it will be covered later in an optional paragraph. The computer programs are straightforward to easy to apply; they will be covered first. 29 Glenn James and James, Robert C., Mathematics Dictionary, (Princeton, NJ: D. Van Nostrand Company, Inc., 1964), page 253.

Linear Least-squares Line

3.1 Dissection of a graph. The first thing to consider in any least-squares construction is the underlying point set. This point set is sometimes called the sample space, sometimes called the sample data, and sometimes called the sample population. Confusion arises because of the various names that are attached to the data points.

Sample Data

abscissa (x-axis) ⟶

FIGURE 20

\begin{matrix}↑ \\ y(x) \end{matrix} • • •

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•

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• •

The next thing to consider are the arithmetic quantities associated with the data points (with the so-called point set). If we simply add up all the x-values (the abscissas) and divide by the count of the number of points, we will get the x-average, also called the x-mean, and usually denoted as

μ_{x} = \frac{∑_{i=1}^{n} x_{i}}{n} .  (42)

In a similar manner, we compute the y-average, μ_{y},

μ_{y} = \frac{∑_{i=1}^{n} y_{i}}{n} .  (43)

The averages μ_{x} and μ_{y} are familiar. Other computations are also necessary to obtain the ingredients needed to calculate the trend line, shown below. Because this line is the best possible fit to the data, it is also known as the line of best fit.

Sample Data

abscissa (x-axis) ⟶

FIGURE 21

\begin{matrix}↑ \\ y(x) \end{matrix} • • •

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↖ Line of Best Fit Now is a good time to introduce the BASIC computer program. Since we have seen several BASIC and “C” programs already, this listing will be provided without comments. The statistical parameters μ_{x} and μ_{y} will be denoted as XMU and YMU, respectively. The statistical parameters for the standard deviations, σ_{x}, σ_{y}, and σ_{xy} (biased and unbiased), will be denoted SIGMAX, SIGMAY, and SIGMAXY, respectively.

100 CONST N = 11
110 DEFDBL A-H, O-Z: REM Define everything double precision except the indices.
120 DIM X(N), Y(N)
130 DATA 1981.,1982.,1983.,1984.,1985.,1986.,1987.,1988.,1989.,1990.,1991.
140 DATA 90.08,90.57,90.76,91.30,91.57,92.44,92.87,93.68,93.85,94.49,94.88
150 FOR I = 1 TO N: READ X(I): NEXT I
160 FOR I = 1 TO N: READ Y(I): NEXT I
170 SUMX = 0!:  SUMY = 0!:  SUMX2 = 0!:  SUMXY = 0!:  SUMX2 = 0!
180 SIGMAX = 0!:  SIGMAY = 0!:  SIGMAXY = 0!
190 FOR I = 1 TO N
200 SUMX = SUMX + X(I): SUMY = SUMY + Y(I): SUMX2 = SUMX2 + X(I) * X(I)
210 SUMXY = SUMXY + X(I) * Y(I): SUMY2 = SUMY2 + Y(I) * Y(I)
220 NEXT I
230 DELTA = CDBL(N) * SUMX2 - SUMX * SUMX
240 A = (SUMX2 * SUMY - SUMX * SUMXY) / DELTA
250 B = (CDBL(N) * SUMXY - SUMX * SUMY) / DELTA
260 CLS : PRINT "A (x-intercept) = "; A
270 PRINT "B (slope) = "; B
280 XMU = SUMX / CDBL(N): YMU = SUMY / CDBL(N)
290 FOR I = 1 TO N
300 SIGMAX = SIGMAX + (XMU - X(I)) * (XMU - X(I))
310 SIGMAY = SIGMAY + (YMU - Y(I)) * (YMU - Y(I))
320 SIGMAXY = SIGMAXY + (XMU - X(I)) * (YMU - Y(I))
330 NEXT I
340 R = SIGMAXY / SQR(SIGMAX * SIGMAY)
350 PRINT "Correlation Coefficient r = "; R: SIGMAY = 0!
360 FOR I = 1 TO N
370 SIGMAY = SIGMAY + (Y(I) - A - B * X(I)) ^2
380 PRINT USING "#### ##.##### ##.#####"; X(I); Y(I); A + B * X(I)
390 NEXT I
400 SIGMAY = SQR(SIGMAY / (CDBL(N) - 2!))
410 PRINT "sigma y (unbiased) = "; SIGMAY
420 PRINT "sigma x (unbiased) = "; SIGMAY * SQR(SIGMAX / DELTA)
430 END

After the above BASIC computer program was executed, the following data were obtained:

A (x-intercept) = -900.77236
B (slope) = .50009
Correlation Coefficient r = .99475
Sigma y (unbiased) = .17980
Sigma x (unbiased) = .05421

The next important topic is the construction of the error bars. This is often the most forgotten item in the creation of a data presentation. It is the only way properly introduct statistical uncertainty into a display graph. In the hard science of Physics, such information is mandatory. The distance from the data point to the crossbars (top and bottom of the errorbar) must equal one unbiased standard deviation, σ_{y}. There may also be uncertainty in the abscissa. In this case, it is necessary to construct an error bar horizontal to the x-axis of length σ_{x}. The error bars tell us more than just some statistical fact about the individual data points. If we examine closely Fig. 33, we’ll note that the line of best fit passes through most of the error bars. This tells us that there is a good probability that there is an underlying functional relationship between x and f(x), namely f(x) ≈ a · x + b. If, on the other hand, the best straight line missed a high proportion of the error bars, or if it missed any by a large distance (relative to the length of the error bar), then the relationship between x and y ≈ f(x) would probably not be that of a straight line (if there was any relationship at all).

Error bars with uncertainties • \begin{matrix}↑ \\ σ_{y} \\ ↓ \end{matrix}  • \begin{matrix}↑ \\ σ_{y} \\ ↓ \end{matrix} ← σ_{x} →

FIGURE 22

We’ll conclude this long chapter with a another graph showing the line of best fit and the companion “C” program for the BASIC program above. The next graph was suggested by the textbook An Introduction to Error Analysis by John R. Taylor (Mill Valley, CA: University Science Books, 1982), page 26. It indicates the uncertainties found in any physical measurement. The underlying physical law is know as Hooke’s Law, published by Robert Hooke in 1678.

0  500  1000 m (gm) ⟶_{FIGURE 23} 5 \begin{matrix}↑ \\ x \\ (cm) \end{matrix} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} • • • • • • • •

/* Program for mean, standard deviation */
/* Program for least-squares fit to a straight line */
/* Program for computing correlation coefficient "r" */
#include <stdio.h>
#include <math.h>
#include <float.h>
int main()
{
/* Initialize all the parameters and type variables */
  FILE *fp; /* file pointer */
  static double x[] =  1981., 1982., 1983., 1984., 1985.,
    1986., 1987., 1988., 1989., 1990.,
    1991.  ; /* x values (years) */
  static double y[] =  90.08, 90.57, 90.76, 91.30, 91.57,
    92.44, 92.87, 93.68, 93.85, 94.49,
    94.88 ; /* y values (reliabilities) */
  /* The variables sum something are used as accumulators.  */
  /* a, b are the y-intercept and slope, respectively of the
    least-squares line.  */
  /* r is the correlation coefficient (the so-called Pearson's r).  */
double Delta, a, b, r, sum x, sum x2, sum y, sum y2, sum xy;
/* mu something is a mean, sigma something is a standard
deviation (unbiased for least-squares = n-2 degrees of freedom).  */
double mu x, mu y, sigma x, sigma y, sigma xy;
/* The integer i is an index.  (unsigned, short integer) */
int i;
/* The long integer n is the count of the number of data points.  */
long int n;
/* Begin computations */
/* Initialize n, x[n], y[n], and give the output file a name.  Then you
are ready to begin execution.  */
n = 11;
/* Make sure the accumulators are all initialized to zero.  */
sum x = (double) 0.0;
sum y = (double) 0.0;
sum y2 = (double) 0.0;
sum x2 = (double) 0.0;
sum xy = (double) 0.0;
sigma x = (double) 0.0;
sigma y = (double) 0.0;
sigma xy = (double) 0.0;
/* The so-called "for loop" computes the sum somethings */
for (i=0; i < n; ++i){ /* i marches from 0 to n-1.  */
sum y += (double) y[i]; /* All the arrays in "C" */
sum y2 += (double) y[i]*y[i]; /* programs begin with index */
sum x += (double) x[i]; /* zero.  */
sum x2 += (double) x[i]*x[i];
sum xy += (double) x[i]*y[i];
}
/* Display the results of the computation of all the sums, Delta,
  the slope, and the y-intercept for
  y = a + b * x, the least-squares straight line */

printf("\n sum x = %.16lf", sum x);
printf("\n sum y = %.16lf", sum y);
printf("\n sum x2 = %.16lf", sum x2);
printf("\n sum y2 = %.16lf", sum y2);
printf("\n sum xy = %.16lf", sum xy);
Delta = (double) n*sum x2 - sum x*sum x;
printf("\n Delta = %.16lf", Delta);
a = (double) (sum x2*sum y - sum x*sum xy)/Delta;
b = (double) (n*sum xy - sum x*sum y)/Delta;
printf("\n A (y-intercept) = %.16lf", a); /* Use Capital "A" here */
printf("\n B (slope) = %.16lf", b); /* to be consistent with */
mu y = (double) sum y/n; /* the text book (Taylor).  */
mu x = (double) sum x/n;
/* Display (print to screen) the means only if needed.  */
/* printf("\n mu y = %.16lf", mu y); */
/* printf("\n mu x = %.16lf", mu x); */
for (i=0; i < n; ++i){
  /* Biased estimates for sigma x and sigma y */
sigma y += (mu y-y[i])*(mu y-y[i]); /* We are just using the */
sigma x += (mu x-x[i])*(mu x-x[i]); /* sigma somethings here */
sigma xy += (mu x-x[i])*(mu y-y[i]); /* as accumulators.  */
}
/* compute the correlation coefficient from the values in the
accumulators.  This is not the actual formula.  */
r = sigma xy/sqrt(sigma x*sigma y);
printf("\n correlation coefficient = %.16lf", r);
sigma y = 0.0;
for (i=0; i < n; ++i){
sigma y += (y[i] - a - b*x[i])*(y[i] - a - b*x[i]);
/* This is the least-squares data to be plotted.  */
printf("\n %d %.4lf %.4lf %.4lf", i, x[i], y[i],a + b*x[i]);
}
/* Unbiased estimates for sigma x and sigma y */
/* Note that there are n-2 degrees of freedom (not n-1).  */
sigma y = sqrt((double) sigma y/(n-2.0));
printf("\n sigma y (unbiased) = %.16lf", sigma y);
sigma x = sigma y * sqrt((double) sum x2/Delta);
printf("\n sigma x (unbiased) = %.16lf", sigma x);
/* Output to file.  Make sure you have a unique filename.  */
fp = fopen("pgm3.txt","w"); /* Open the file */
fprintf(fp,"\n A (y-intercept) = %.16lf", a);
fprintf(fp,"\n B (slope) = %.16lf", b);
fprintf(fp,"\n correlation coefficient = %.16lf", r);
fprintf(fp,"\n sigma y (unbiased) = %.16lf", sigma y);
fprintf(fp,"\n x[i] y[i] a+b*x[i]");
for (i=0; i < n; ++i){
fprintf(fp,"\n %.4lf %.4lf %.4lf",x[i],y[i], a + b*x[i]);
}
fclose(fp); /* Close the file */
return(0);
}
/* End Of File */

3.2 Formula Derivation. Whenever a mathematicians are faced with the chore of explaining some difficult point, they just make a definition. Then, by choosing a definition to fit their needs, they can create a theorem which employs the definition and declare the problem solved. That’s basically what we’ll do here. There is a formal derivation based on determining a minimum of the formula

P_{a,b}[y_{i}] ∝ \frac{1}{σ_{y}} e^{−(y_{i}−ax_{i}−b)^{2}/2σ_{y}^{2}};  (44)

however, we’ll approach the problem from a simpler tack and arrive at the same result. Now, suppose we have a point set consisting of n points. Let’s call this point set S and observe that it can be written several ways

\begin{aligned}S & = \{ (x_{1}, y_{1}), (x_{2}, y_{2}), …, (x_{n}, y_{n}) \} \\ & = \{ (x_{i}, y_{i}) | i = 1, …, n \}. \end{aligned}

If we call the difference |y_{i} − a · x_{i} − b| = \sqrt{(y_{i} − ax_{i} − b)^{2}} the deviation, or residual, of y_{i} from the value a · x_{i} + b, then we may define

dev^{2} = dev^{2}(a, b) = ∑_{i=1}^{n} (y_{i} − a · x_{i} − b)^{2} .  (45)

There are many different ways to approach this subject matter. We will choose the most straightforward. If one looks in a textbook, the derivation will the much more elaborate. This is because mathematicians love to generalize things. The line y = f(x) = ax + b is also a point set (not finite) and sometimes it will be written as

L = { (x, y) | y = a · x + b, x ∈ R }.

In any case, we simply want to minimize the value for dev^{2}(a, b). In so doing, we will determine a and b in terms of the values of x_{i}, y_{i}, and the integer n. There’s no way out of taking a partial derivative here. Just take it on faith that the minimum occurs exactly when

\frac{∂ dev^{2}(a, b)}{∂a} = \frac{∂ dev^{2}(a, b)}{∂b} = 0.  (46)

We compute

\frac{∂ dev^{2}(a, b)}{∂a} = ∑_{i=1}^{n} −2(y_{i} − ax_{i} − b) · x_{i}  (47)

and

\frac{∂ dev^{2}(a, b)}{∂b} = ∑_{i=1}^{n} −2(y_{i} − ax_{i} − b).  (48)

Assume that equations (46), (47), and (48) are correct.^{30} Let’s go through the algebra (carefully) and see what happens. Setting equation (47) equal to zero and expanding gives

∑_{i=1}^{n} x_{i}y_{i} − a ∑_{i=1}^{n} x_{i}^{2} − b ∑_{i=1}^{n} x_{i} = 0.  (49)

Setting equation (48) equal to zero and expanding gives

∑_{i=1}^{n} y_{i} − a ∑_{i=1}^{n} x_{i} − b ∑_{i−1}^{n} 1 = 0.  (50)

30 for the derivation, see John R. Taylor, An Introduction to Error Analysis, (Mill Valley, CA: University Science Books, 1982), page 156 A very common error is not realizing that

∑_{i=1}^{n} 1 = n.

We have been writing ∑_{i=1}^{n} again and again. Let’s simplify our notation by agreeing that whenever we use ∑_{i=1}^{n} we can just write capital sigma, ∑. Equations (49) and (50) can now we re-written as

∑ x_{i}y_{i} − a ∑ x_{i}^{2} − b ∑ x_{i} = 0  (49′)

and

∑ y_{i} − a ∑ x_{i} − nb = 0.  (50′)

We want to solve for the values of a and b. To solve for a, we eliminate b. That is done by multiplying equation (49’) by n and by multiplying equation (50’) by ∑ x_{i} and subtracting one from the other (clever). Let’s do just that

\begin{aligned}n ∑ x_{i}y_{i} − na ∑ x_{i}^{2} − nb ∑ x_{i} & = 0 \\ ∑ y_{i} ∑ x_{i} − a (∑ x_{i})^{2} − nb ∑ x_{i} & = 0. \\ a ((∑ x_{i})^{2} − n ∑ x_{i}^{2}) & = ∑ x_{i} ∑ y_{i} − n ∑ x_{i}y_{i} \end{aligned}

So,

a = \frac{n ∑ x_{i}y_{i} − ∑ x_{i} ∑ y_{i}}{n ∑ x_{i}^{2} − ( ∑ x_{i})^{2}} .  (51)

Now all we’ll have to do is to plug the value for a into either (49’) or (50’) to solve for b. (50’) is easier.

∑ y_{i} − a ∑ x_{i} − nb = 0,
∑ y_{i} − \frac{n ∑ x_{i}y_{i} − ∑ x_{i} ∑ y_{i}}{n ∑ x_{i}^{2} − (∑ x_{i})^{2}} ∑ x_{i} − nb = 0,
n ∑ y_{i} ∑ x_{i}^{2} − ∑ y_{i} (∑ x_{i})^{2} − n ∑ x_{i} ∑ x_{i}y_{i}
+ ∑ x_{i} ∑ x_{i} ∑ y_{i} − nb (n ∑ x_{i}^{2} − (∑ x_{i})^{2}) = 0,
n ∑ y_{i} ∑ x_{i}^{2} − n ∑ x_{i} ∑ x_{i}y_{i} − nbΔ = 0,

where

Δ = n ∑ x_{i}^{2} − (∑ x_{i})^{2} .

Thus,

b = \frac{∑ y_{i} ∑ x_{i}^{2} − ∑ x_{i} ∑ x_{i}y_{i}}{n ∑ x_{i}^{2} − (∑ x_{i})^{2}} .  (52)

a = \frac{n ∑ x_{i}y_{i} − ∑ x_{i} ∑ x_{i}}{Δ}  (51′)
b = \frac{∑ y_{i} ∑ x_{i}^{2} − ∑ x_{i} ∑ x_{i}y_{i}}{Δ}  (52′)

This is exactly what we saw in the BASIC computer code:

230 DELTA = CDBL(N) * SUMX2 - SUMX * SUMX
240 A = (SUMX2 * SUMY - SUMX * SUMXY) / DELTA
250 B = (CDBL(N) * SUMXY - SUMX * SUMY) / DELTA
260 CLS : PRINT "A (x-intercept) = "; A
270 PRINT "B (slope) = "; B

(Here A and B stand for b and a, respectively! DELTA is still Δ.)

3.3 Goodness of Fit. We saw how to derive the least squares line of best fit, now the question is “just how good of a fit is it?” Two things are needed: (1) a correlation coefficient; and (2) a quantitative measure of the degree of correlation for a given sample size. The formula for the correlation coefficient is simply

r = \frac{∑_{i=1}^{n} (x_{i} − μ_{x})(y_{i} − μ_{y})}{(∑_{i=1}^{n} (x_{i} − μ_{x})^{2} ∑_{i=1}^{n} (y_{i} − μ_{y})^{2})^{1/2}} .  (53)

This correlation coefficient is also called the linear correlation coefficient or Pearson’s r. It turns out that r is a number lying in the interval [−1, 1]. The closer r is to ±1, the more closely the data points lie on a straight line. On the other hand, a value of r near zero implies little or no linear correlation. But this is vague. What does one mean by saying “|r| lies close to 1?” There must be some statistical test to determine goodness of fit quantitatively. There is! We can calculate the probability that a coefficent of linear correlation will be greater than some given value, say r_{0}, by the integral^{31}

P_{N} (|r| ≥ |r_{0}|) = \frac{2Γ [(N − 1)/2]}{\sqrt{π}Γ [(N − 2)/2]} ∫_{|r_{0}|}^{1} (1 − r^{2})^{(N−4)/2} dr.  (54)

But how to evaluate this integral? We might look for a table of values. But relax, if we employ our Simpson’s rule we can solve this integral accurately and quickly. A BASIC program is listed below:

100 DEFDBL A-H, O-Z
110 INPUT "Number of points, Correlation Coefficient "; P, R
130 IF (P < 4) THEN PRINT "Number of points too small":  GOTO 110
140 DEF FNA (x AS DOUBLE) = (1 - x * x) ^((P - 4!)  / 2!)
150 A = 0!:  B = 1!:  N = 256:  SUM = 0!:  H = (B - A) / N
160 FOR I = 1 TO N STEP 2
170 SUM = SUM + FNA(A + (I - 1) * H)
180 SUM = SUM + 4!  * FNA(A + I * H)
190 SUM = SUM + FNA(A + (I + 1) * H)
200 NEXT I: REM N = 256 gives accuracy to 5 decimal places.
220 PSUM = SUM * H / 3!
240 A = R: B = 1!
250 SUM = 0!:  H = (B - A) / N
260 FOR I = 1 TO N STEP 2
270 SUM = SUM + FNA(A + (I - 1) * H)
280 SUM = SUM + 4!  * FNA(A + I * H)
290 SUM = SUM + FNA(A + (I + 1) * H)
300 NEXT I
310 RSUM = 100!  * (SUM * H / 3!)  / PSUM
320 PRINT USING "Probability |r| >= r 0 is ##.#%"; RSUM

But where are all the Γ()’s and the coefficient 2/\sqrt{π}? What we are doing is normalizing the value of the integral by observing that

P_{N}(|r| ≥ 0) = 1.

This way the coefficient,

\frac{2Γ [(N − 1)/2]}{\sqrt{π}Γ [(N − 2)/2]}

31 E. M. Pugh and G. H. Winslow, The Analysis of Physical Measurement (Addison-Wesley, 1966), Section 12–8. does not have to be computed explicitly.^{32} Working through this program will give us a deeper insight as to why numerical integration is important and how it ties in to Probability Theory. In fact, this whole topic of calculation of probabilities for correlation coefficients is one frequently misunderstood and confused by working scientists and engineers.

The “C” program is listed below. This program has internal documentation in the form of comments. A comment in “C” begins with the character combination (/*) and ends with the reverse character combination (*/).

#include <stdio.h> /* Header for input/output subroutines.  */
#include <math.h> /* Header for math subroutines.  */
#include <float.h> /* Header for floating point subroutines.  */
#define pi 3.141592653589793238462643383279 /* Accurate value for pi.  */
/* Simpson's rule for approximating integrals.
  a:  left endpoint
  b:  right endpoint
  fc:  pointer to function to integrate
  n:  number of subintervals
*/
double fc (double x);
main()
{
double a,b,h,sum,x,y,p,r,psum; /* In 'C' all variables must be assigned */
double p1, p2, p3;
int i, n, o;
printf("\007"); /* Sound bell in computer.  */
a = (double) 0.0; /* The normalizing integral is */
b = (double) 1.0; /* from 0 to 1.  */
n = 256; /* This has been tested and seen to give 5 decimal accuracy.  */
printf("\nEnter number of points ");
scanf("if (o < 5){
  printf("Too few points"); return(o);
}
p = (double) o;
h = (double) (b-a)/n;
for (i=1, sum=0.0; i<=n; i = i+ 2){
  p1 = fc((double) a+(i-1)*h, p);
  p2 = fc((double) a+i*h, p);
  p3 = fc((double) a+(i+1)*h, p);
  sum += p1 + 4.0 * p2 + p3;
}
psum = (double) h*sum/3.0; /* This normalizes the integral */
/* For a given value of N, P N[|r|>=0] (must)) = 1.0 */
/* We assure this (and eliminate the need to compute some */
/* Gamma function) in computing the value in "psum." */
printf("\nEnter correlation coefficient r = ");
scanf("if (a > 1) a = (double) 1.0;
  /* r 0 must be between -1 and 1.  Force a to assume a value */
  /* between 0 and 1.  */
h = (double) (b-a)/n;

32 This is nice because Γ(m + \frac{1}{2}) = \frac{1·3·5···(2m−1)}{2^{m}} \sqrt{π} for m = 1, 2, 3, . . ..

for (i=1, sum=0.0; i<=n; i = i+ 2){
  p1 = fc((double) a+(i-1)*h, p);
  p2 = fc((double) a+i*h, p);
  p3 = fc((double) a+(i+1)*h, p);
  sum += p1 + 4.0 * p2 + p3;
}
y = (double) h*sum/3.0;
printf("\nPercentage probability that N measurements");
printf("\nof two uncorrelated variables would give an r");
printf("\nas large as r 0, printf("\nP N[|r|>=r 0] = printf("\nIf the number
printf("\nthen the correlation is called significant.");
printf("\nIf the number printf("\nthen the correlation is call highly signifi-
cant.");
printf("\nGiven this simple program and its speedy calculation");
printf("\nof the significance of a correlation in absolute terms,");
printf("\nwe only have to plug in the number of data points and,");
printf("\nthe value gotten from Pearson's correlation coefficient, r.");
printf("\n");
return(0);
}
double fc (double x, double p)
{
  double y;
  y = (double) pow((double)(1.0-x*x),(double) (p-4.0)/2.0);
  return (y);
}
/* End of file */

Curve f(x) = (1 − x^{2})^{(N−4)/2},
where N = 5 and |r_{0}| = \frac{1}{2}

0.0 0.2 0.4 0.6 0.8 1.0

0.0  0.2  0.4  0.6  0.8  1.0

FIGURE 24

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|r_{0}| 3.4 More Theory. We will examine the theory behind the “method of least squares” as applied

to fitting a straight line

y = a · x + b  (55)

to the data points (x_{1}, y_{1}), (x_{2}, y_{2}), …, (x_{n}, y_{n}). As before, we might observe that associated with each value of the abscissa, that is, associated with each x_{i} (1 ≤ i ≤ n) there are two value of y. There is the y_{i} and the value ỹ_{i} = a · x_{i} + b. If we take the numerical difference of the two, d_{i} = y_{i} − ỹ_{i}, we get the amount the predicted value of y, ỹ, differs from the observed value of y. Let’s write out some of these differences, which are called deviations:

\begin{aligned}d_{1} & = y_{1} − (a · x_{1} + b) \\ d_{2} & = y_{2} − (a · x_{2} + b) \\ ⋮ & ⋮ \\ d_{n} & = y_{n} − (a · x_{n} + b) \end{aligned}

The set of all the d_{i}’s (1 ≤ i ≤ n), which is also written

{ d_{1}, d_{2}, …, d_{n} }

or

{ d_{i} | 1 ≤ i ≤ n }.

If it turns out that d_{i} = 0 for all the integers i, i = 1, 2, …, n, then the straight line y = ax + b passes through each of the n points. If this is not the case, then we have to solve the problem of finding a line of best fit. A line of best fit is a line which best fits the data in some sense. One obvious sense is to minimize the values of the deviations. Let’s look at the situation graphically:

Line of Best Fit_{FIGURE 25} O  x y ^{×}(x_{1}, y_{1})^{×(x_{2}, y_{2})}  ^{×}  ^{×(x_{n}, y_{n})}^{•}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}

If a line comes close to some of the points, the value of the deviation from those points will be small. The values of the deviation for points lying above the line is positive, while the values of the deviation for points lying below the line is negative. To get a realistic picture of what is meant by the best fit, we could take the absolute value; however, it is simpler to take the squares of the deviations. Written as a function, this becomes

dev^{2} = (y_{1} − ax_{1} − b)^{2} + (y_{2} − ax_{2} − b)^{+} ⋯ + (y_{n} − ax_{n} − b)^{2}.  (56)

Recall the “Big-Sigma” notation:

∑_{i=1}^{n}(y_{i} − ax_{i} − b)^{2} = (y_{1} − ax_{1} − b)^{2} + (y_{2} − ax_{2} − b)^{+} ⋯ + (y_{n} − ax_{n} − b)^{2}. (57)

So, we have a function, dev^{2}. But what is it a function of? Clearly, the x_{i}’s and y_{i}’s are fixed data points. There are only two things to vary, the constants a and b. And this is a general concept for much advanced mathematics: “hold the variables constant and vary the constants.” No where is this more often done than in thermodynamics. So, we write

\begin{aligned}dev^{2}(a, b) & = (y_{1} − ax_{1} − b)^{2} + (y_{2} − ax_{2} − b)^{2} + ⋯ + (y_{n} − ax_{n} − b)^{2} \\ & = ∑_{i=1}^{n}(y_{i} − ax_{i} − b)^{2}. \end{aligned}  (58)

Now, by definition, we will say that the line of best fit is that line y = ax + b for which the sum of the squares of the deviations is a minimum. This is a philosophically correct definition as well. The problem is how to determine which values of a and b to choose. The minimum value for equation (56) can be obtained from partial differentiation. But we have already done that in a previous paragraph. We solved the equations

\frac{∂ dev^{2}(a, b)}{∂a} = 0  and \frac{∂ dev^{2}(a, b)}{∂b} = 0.  (59)

Line of Best Fit_{1  2  3  4}_{FIGURE 26}^{1 2 3 4 5}O  x y^{×}  ^{×}^{×}  ^{×}  ^{×} (x_{1}, y_{1}) (x_{2}, y_{2}) (x_{3}, y_{3}) (x_{4}, y_{4})(x_{5}, y_{5})^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}

The data points belong to the set { (0, 1), (2, 2), (1, 3), (3, 4), (4, 5) }. The equation of the line of best fit is y = 0.9x + 1.2. Since the data values are integers and the number of points is small, it is possible to evalute the deviation function directly.^{33} All of this can be generalized to a polynomial p(x) of degree n ≥ 2. We simply write y = a_{0} + a_{1}x + a_{2}x^{2} + ⋯ + a_{n}x^{n} to get

dev^{2}(a_{0}, …, a_{n}) = ∑_{i=1}^{n} (y_{i} − a_{0} − a_{1}x − ⋯ − a_{n}x_{i}^{n}).  (60)

Then it is necessary to minimize dev^{2}(a), where (a = (a_{0}, a_{1}, …, a_{n})), as follows:

\frac{∂ dev^{2}(a_{0}, …, a_{n})}{∂a_{i}} = 0.  (61)

33 George B. Thomas, Calculus and Analytic Geometry, (Reading, Massachusetts: Addison-Wesley Publishing Company, Inc., 1966), page 699.

3.5 Problems. In this paragraph we will work problems.^{34} One very important consideration is the plotting of the data. With the digital computer, the very first thing a scientist or engineer should do is plot the data. This ensures proper scaling and gives an intuitive feeling for the behavior of the line of best fit. Moreover, unusual characteristics (such as logarithmic dependence, non-linear behavior, or sectional continuity) can be determined before the machinery of the least-squares method comes into play.

The first data set: { (5, 10), (10, 8.9), (15, 8.2), (20, 7.0) }. Since the x-values are given as integers, we will assume that σ_{x} = 0 for this problem. The only error-bar needed is parallel to the y-axis. First plot the data. Ten points to the inch look like a good scale for the x-axis and one-fourth inch per unit looks good for the y-axis.

5  10  15  20 FIGURE 27^{5 6 7 8 9 10} ^{•}  ^{•}  ^{•}  ^{•}

Now, we run the program and get the following values for the data on the line of best fit: { (5, 9.98), (10, 9.01), (15, 8.04), (20, 7.07) }. The equation for the line of best fit is

y = −0.194x + 10.949  (62)

and the correlation coefficient is r = −.99546. This looks like it will be a good linear fit. Let’s plot it now.

0  5  10  15  20  25 FIGURE 28^{5 6 7 8 9 10 11 12}  ^{•}  ^{•}  ^{•}  ^{•}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}

This graph looks OK without error bars. They would just clutter up the picture, so let’s not put them in. The correlation coefficient is nearly −1, there is no need to do a goodness of fit analysis. In short, this data set very closely approximates a linear relationship over the region in question. We cannot, however, extrapolate outside of the extreme values of the data; our only conclusions are valid within the specified range of data values. This is a very important point. Notice that the line of best fit has been drawn across the entire graph. This is standard mathematical practice; otherwise, the equation would not represent a function—a graph of a function from the x-axis. But this mathematical practice should not obscure the underlying assumptions.

34 Erwin Kreyszig, Advanced Engineering Mathematics, 5th Edition, (NY: John Wiley & Sons, 1983), pages 820-821.

The next data set has a different appearance. The data points are { (4, −17), (15, −4), (30, −7), (100, 50), (200, 70) }. We will choose one printer’s point per data point.

FIGURE 29 ∘ ∘ ∘  ∘  ∘

After running the computer program, the following data along the line of best fit is obtained:

{ (4, −11.67), (15, −6.65), (30, 0.21), (100, 32.20), (200, 77.91) }.  (63)

The equation of the line of best fit and the correlation coefficient r are

y = 0.457x − 13.503 and r = .96074.  (64)

Now, we will plot the data against the curve. It is now easy to scale the data and put in numbered tick marks.

50  100  150  200  250_{FIGURE 30} −20 0 20 40 60 80 100 _{∘} _{∘} _{∘}  ^{∘}  ^{∘}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}

Now we run the computer program to get the value for σ_{y} and see how good our fit is. σ_{y} = 12.47154 and the probability that a random choice of values would give the same correlation coefficient is 0.9%—we have a highly significant correlation. So, let’s put in the error bars to see just how this graph looks in its final form.

50  100  150  200  250_{FIGURE 31} −20 0 20 40 60 80 100 _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} • • •  •  •

Now things are starting to look routine. We will repeat this procedure again and again to ensure that the idea is clear. Again, let’s start with a data set

{ (2.8, 30), (2.9, 26), (3.0, 33), (3.1, 31), (3.2, 33), (3.2, 35), (3.2, 37), (3.3, 36), (3.4, 33) }. (65)

Notice that one x-value has several y-values. This won’t cause the least-square program to have any problems. First the graph

2.8  3.0  3.2  3.4 FIGURE 32^{26 28 30 32 34 36 38} ^{×}  ^{×}  ^{×}  ^{×}  ^{×}^{×}^{×}  ^{×}  ^{×}

We execute the computer program and get a line of best fit with an equation

y = 12.06767x − 5.01128 and r = .69155.  (66)

With nine points and a value of the correlation coefficient of only 0.69 just how significant is our data? —3.9%—significant (under 5%) but not highly significant. Now the finished plot, with error bars:

2.5  3.0  3.5 FIGURE 33^{25 30 35 40}  •  •  •  •  • • •  •  •^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}

Now try your hand at the data set { (0, 100), (3, 130), (5, 140), (8, 170), (10, 190) }, which will give y = 8.8216x + 100.12739 and have a correlation coefficient of r = +0.99666. What is σ_{y}? This correlation is significant; is it highly significant?

Adaptive Quadrature Routine

Chapter 4

4.0 Introduction. Sometimes it’s not easy to tell just how many subdivisions are needed to do a numerical integration. The reasons for this uncertainty are many. Functions exhibiting bizarre behavior at a point or in a subinterval require special attention. Some continuous functions fail to have derivatives. Moreover, a function can be integrable even if it is not be continuous at every point. For these cases and many others it’s nice to have some kind of automatic system for determining just when an approximation is “good enough.” Fast modern computers may still be too slow. In particular, if a function is defined in terms of an integral its numerical evaluation may be time consuming; moreover, functions so defined may be prone to significant error propagation. In general, a function defined over an interval may be easier to integrate over some subintervals than over others. The methods which have been devised to deal with this phenomenon are call adaptive quadrature routines, or AQRs. Adaptive methods make use of error estimates to automatically determine the number of subintervals, n, and the interval size h. These methods may also use smaller values of the interval size h over subintervals where it is needed. There are many systematic ways of doing this. There is one method which is easily programmed (in BASIC), but which is theoretically involved. This method entails comparing Simpson’s rule with the so-called trapezoid rule. Another method, not so easily programmed and requiring recursion, compares two different estimates from Newton-Côtes’ formulas. In the following paragraphs, we’ll develop computer programs for the simplest computers—the programmable calculators—and compare results.

An Adaptive Quadratic Routine, yielding unequal intervals, to integrate the function f(x) = 2x sin x^{2} over the interval [0, π].

−5 −4 −3 −2 −1 1 2 3 4 5 x

y = 2x sin(x^{2})

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FIGURE 34

4.1 A Simple Approach. The best introduction to this subject is to work simple examples and then develop the theory as needed. Excellent numerical results can be obtained using a very simple approach. We are given a function, f(x), an interval [α, β], and an error bound ϵ > 0. First, approximate the integral with two (different) approximations: the trapezoid rule and Simpson’s rule. Begin with n = 10 subintervals. Second, compute the error estimate (relative) as follows

E = |\frac{A_{T} − A_{S}}{A_{T} + A_{S}} | ,  (67)

where A_{T} is the approximation obtained from the trapezoid rule and A_{S} is the value from Simpson’s rule. If E < ϵ, then the results are OK. If not, then set n = 20 (n = 2·n) subintervals and repeat the process. Continue until either E < ϵ or until n gets too large, say n > 1000. This is easy to program and fast to run. The first prototype would look like this:

10  DEF FNA(X) = SQR(1.0-SIN(EXP(1)/2)^2*SIN(X)^2)
20  N=10 :  A=0.0 :  B=2.0*ATN(1.0)
30  H=(B-A)/N : S = 0.0 :  T = 0.0
40  FOR I=1 TO N STEP 2
50  X=A+(I-1)*H : S = S + FNA(X) : T = T + FNA(X)
60  X=A+I*H : T = T + 4.0*FNA(X)
70  X=A+(I+1)*H : S = S + FNA(X) : T = T + FNA(X)
80  NEXT I : S1 = H*S : S2 = H*S/3.0 :  E = ABS((S1-S2)/(S1+S2))
90  IF (E>0.000001) THEN N = 2*N : GOTO 30
100 PRINT "Integral = "; T, N

The answer is given immediately as

Integral = 1.054686     20

From our previous calculations, we know that this is precisely the correct number of subintervals for this precision. It is a good idea to examine an alternative means of coding the same program. It runs faster if we use a GOSUB construction instead of the defined function. The answers will be identical and we can consult the table prepared in paragraph 1.2 for numerical results.

10  D=0.0001 :  INPUT "A,B = ";A,B : N=10
20  N=N+10 :  H=(B-A)/N : X=A : S=0.0 :  T=0.0
30  FOR I=1 TO N STEP 2
40  GOSUB 100 :  S = S + Y*H/3 :  T = T + Y*H/2 :  X = X + H
50  GOSUB 100 :  S = S + 4*Y*H/3 :  T = T + Y*H : X = X + H
60  GOSUB 100 :  S = S + Y*H/3 :  T = T + Y*H/2 :  NEXT I
70  IF ABS((S-T)/(S+T)) > D THEN 20
80  PRINT "Integral = "; T, N
90  PAUSE : INPUT "Error % = ";D : GOTO 20
100 Y = SQR(1.0-SIN(EXP(1)/2)^2*SIN(X)^2) :  RETURN

This program gives one the ability to increase the error estimate and thereby determine closely the exact number of steps required. Using this program with a hand-held calculator will give a very good qualitative insight into the workings of the algorithm. The interested reader will later be invited to experiment with other adaptive quadrature routines to determine which is better.

4.2 An Adaptive Quadrature Routine. Suppose that we have a function f(x) defined over an interval [a, b]. We would like to compute the integral close to the actual value. Let’s assume that we know that the function is bounded, |f(x)| ≤ M for all x ∈ [a, b]. Then, for an answer within say 0.01% of the actual error, we would choose

ϵ = \frac{0.0001}{M + 1}.  (68)

With an overall error of ϵ, we want to divide up the interval [a, b] so that the total error of integration is less than ϵ. There is an error formula which depends on the value of the fourth derivative of f(x) on [a, b], but we will depend on another criterion. If we choose Simpson’s rule

A_{2} = \frac{h}{3}(y_{0} + 4y_{1} + 2y_{2} + 4y_{3} + y_{4}),  (69)

then the doubled interval rule

A_{1} = \frac{2h}{3}(y_{0} + 4y_{2} + y_{4}),  (70)

is a good approximant. The estimate of the error is†

E_{2} ≃ \frac{|A_{2} − A_{1}|}{15}.  (71)

This follows from a comparison of the two error estimates:

E_{1} = −\frac{(b − a)(2h)^{4}y^{(4)}(ξ_{1})}{180}  E_{2} = −\frac{(b − a)h^{4}y^{(4)}(ξ_{2})}{180},  (72)

where E_{1} is the error estimate for Simpson’s rule with an interval length of 2h and E_{2} is the error estimate for Simpson’r rule with interval length of h. We can accept a value of A_{2} whenever |A_{2} − A_{1}| ≤ 15ϵ/2^{k} is reached, for an interval being halved k times. An alternate derivation‡ supposes that A is the actual value of the integral and requires E_{1} = |A − A_{1}|, E_{2} = |A − A_{2}|, where A_{1} is Simpson’s rule and A_{2} is the so-called five-point approximation (Simpson’s rule with interval length of h). From the triangle inequality and the inequality

E_{2} ≤ \frac{1}{16} E_{1},  (73)

the result follows. The programming makes use of a technique known as recursion. Recursion occurs when a computer subroutine calls itself. The situation in analysis occurs when a function, say f(x) = sin(x), is used to define g(x) = sin ( sin(x)) = f(f(x)). Now we will examine a “C” program which uses recursion to do adaptive quadrature:

#include <math.h>
#include <float.h>
#include <stdio.h>
double adaptive (double a, double b, double (*f)(double x), double *err);
double f(double x);
int main() {
 double a,b,*err,int1;
 a = (double) 0.0;
 b = (double) 2.0*atan(1.0);

† Francis Scheid, Schaum’s Outline Series Theory and Problems of Numerical Analysis 2nd edition, (NY: McGraw-Hill Book Company, 1988), page 124.

‡ Mike Stimpson, “Numerical Integration Using Adpative Quadrature,” The C Users Journal, May 1992, page 35.

*err = (double) 0.00000001;
printf("\nleft endpoint  = %.16lf",a);
printf("\nright endpoint = %.16lf",b);
printf("\nerror estimate = %.16lf",*err);
int1 = adaptive(a, b, f, err);
printf("\nIntegral       = %.16lf",int1);
printf("\nTheoretical    = %.16lf",1.0);
return(0);
}
double f(double x) {
double y;
y = sin(x);
return (y);
}
double adaptive (double a, double b, double (*f)(double x), double *err) {
double h, s1, s2, s3, s4, s5, error, err1, err2, t1, t2;
h = b - a;
s1 = (*f)(a); s2 = (*f)(a+0.25*h); s3 = (*f)(a+0.5*h);
s4 = (*f)(b-0.25*h); s5 = (*f)(b);
t1 = h*(s1+4.0*s3+s5)/6.0;
t2 = h*(s1+4.0*s2+2.0*s3+4.0*s4+s5)/12.0;
error = fabs(t1-t2)/15.0;
if (error < *err) {
 *err = error; return(t2);
 }
else {
 err1 = err2 = *err/2.0; t2 = adaptive(a, a+0.5*h, f, &err1);
 t2 += adaptive(a+0.5*h, b, f, &err2); return(t2);
 }
}
/* End of File */

The output of the program consists of the endpoints, the error estimate (maximum), the computed answer and the theoretical answer. Here, we know that ∫_{0}^{π/2} sin(x) dx = 1. The quadrature should return an answer close to 1; we see that it does.

   left endpoint  = 0.0000000000000000
   right endpoint = 1.5707963267948970
   error estimate = 0.0000000100000000
   Integral       = 1.00000000025973520
   Theoretical    = 1.0000000000000000

While the adaptive quadrature routine (AQR) is impressive, it often fails to evaluate improper integrals. If the integrand of an improper integral experiences an anomoly (“blows up”) at a point, it is necessary to compute successive approximations for values approaching that point. There is no “quick and easy” solution to improper integrals. One cardinal rule is to graph the function first. It is frequently possible to determine the correct approach from a graph when so-called adaptive methods fail.

METAFONT

Chapter 5

5.0 Introduction. The creation of fonts is usually something not approached lightly. With the typesetting software of TEX this is especially true. The problem of merging text with graphics has been resolved in the earlier chapters using the macros of PICTEX, with varying degrees of success. Now it is necessary to investigate the more sophisticated means of doing graphics. PICTEX can be compared to assembly language; METAFONT to machine code. PICTEX is straightforward and easy to use; however, it uses a great deal of randon access memory (RAM) and it is slow. METAFONT, on the other hand, creates a font which is read in directly. It uses little memory and is fast. There is a price which must be paid for using METAFONT—it has to be prepared in its own language and pre-processed through its own processor.

The public domain version of METAFONT does not include a user-friendly view program. It is necessary to produce files and then process them through a TEX program named testfont.tex. The program testfont.tex is well-written and clever. It produces excellent output; however, one must have a commercial previewer to use it on-line. Otherwise, the output must be printed on a printer. The public domain previewer, cdvi12.arc, only employs the sixteen basic TEX fonts; METAFONT actually produces new fonts—that is, the graphics from METAFONT are fonts. Each graph is actually a single letter or blocks of letters “tiled” together. Donald E. Knuth is correct in saying, “Geometric design are rather easy; . . ..” METAFONT does allow the user a great deal of freedom in constructing graphs. There are pitfalls, however, and some experimentation is needed to construct the desired output.

In METAFONT, the subscripted variables x_{1} and y_{1} may be written either as x[1], y[1] or as x1, y1; likewise, x_{2}, y_{2} may be written either as y[2], y[2] or x2, y2, and so on. The letter z stands for the ordered pair (x, y) (just as is the case in complex analysis). Thus z_{1} is equivalent to (x_{1}, y_{1}), z_{2} is equivalent to (x_{2}, y_{2}), etc. When we write z3 we mean (x3, y3) or (x[3], y[3]). This shorthand aids in construction. For those who recall FORTRAN (the FORmula TRANSlator of the 1960’s), the symbol ** meant exponentiation. x**y meant x^{y}. This was the first encounter many of us had with a “two character” operator. (The binary operators +, −, ∗ (times), and / were well understood.) In BASIC (Beginner’s All-purpose Symbolic Instruction Code), FORTRAN’s ** has been replaced by the single character operator ^. METAFONT has re-introduced the two character operator. This time z1..z2 means draw a (curved) line from z1 (z[1]) to z2 (z[2]) and z1 − −z2 means draw a straight line from z1 to z2. The symbol “:=” means assignment while “=” means simple equality (replacement). Variables followed by the hash mark (American pound sign) (#) are called “sharped” and have absolute values (values that are resolution-independent) as opposed to other variables whose values are derived and determined programmatically.

5.1 A Metafont File. We will begin by listing a simple METAFONT file that will produce a small Gaussian distribution.

% The statement "mode setup" adapts METAFONT to the current task.
mode setup;
em#:=30pt#; cap#:=10pt#;
% The statement "define pixels( , )" converts the values
%   of its (two) arguments into pixel (picture element) units.
define pixels(cap, cap);
% The "beginchar" operation assigns values to the variables
%   w, h, and d, which represent the width, height,
%   and deptgh of the current character's bounding rectagle.
beginchar("A",em#,em#,0); "A for Gaussian Distribution";
pickup pencircle scaled 0.4pt;   % Use a pen of diameter 0.4 point.
z1=(0,0.39894b);
z2=(0.1a,0.39695b);
z3=(0.2a,0.39104b);
z4=(0.3a,0.38138b);
z5=(0.4a,0.36827b);
z6=(0.5a,0.35206b);
z7=(0.6a,0.33322b);
z8=(0.7a,0.31225b);
z9=(0.8a,0.28969b);
z10=(0.9a,0.26608b);
z11=(a,0.24197b);
z12=(1.2a,0.19418b);
z13=(1.4a,0.14972b);
z14=(1.6a,0.11092b);
z15=(1.8a,0.07895b);
z16=(2a,0.05399b);
z17=(2.5a,0.01752b);
z18=(3a,0.00443b);
z19=(3.5a,0.00087b);
z20=(4a,0.0001338b);
a=w; b=8h;
draw z1..z2..z3..z4..z5..z6..z7..z8..z9..z10
   ..z11..z12..z13..z14..z15..z16..z17..z18..z19..z20;
for k=2 step 1 until 20:
z[20+k]=(-x[k],y[k]); endfor
draw z1..z22..z23..z24..z25..z26..z27..z28..z28..z30
   ..z31..z32..z33..z34..z35..z36..z37..z38..z39..z40;
draw (0,-0.02b)--(0,0);
draw (1a,-0.02b)--(1a,0);
draw (2a,-0.02b)--(2a,0);
draw (3a,-0.02b)--(3a,0);
draw (4a,-0.02b)--(4a,0);
draw (-1a,-0.02b)--(-1a,0);
draw (-2a,-0.02b)--(-2a,0);
draw (-3a,-0.02b)--(-3a,0);
draw (-4a,-0.02b)--(-4a,0);
draw (-4a,0)--(4a,0);
draw (1.5a,0)--(1.5a,0.12952b);
endchar;   % Write character to output file.

It is not possible to display the above graph using the public domain view program CDVI12.arc. CDVI-2.com and the other previewers use only the sixteen basic TEX fonts. Some discipline is needed to use public domain software; it is not as flexible as the commercially procured products. However, there is much to be said for the persons and organizations that devote their time and energy to the creation of public domain instruments. Without such volunteerism, many scholars and scientists would simply be unable to afford needed tools. Certain activities are not able to budget for commercial software; governmental agencies are only able to procure software within the strict limits of their charters—and then only with ADPE (Automatic Data Processing Equipment) approval from their MIS (Management Information System) directors. Public domain materials are an invaluable tool for research and development.

Since a public domain TEX previewer is to be employed and only sixteen fonts are allowed, one cannot view a newly created font. The graphs produced by METAFONT are, essentially, new fonts. In particular, each graph is one letter in a font. Some display of the METAFONT graph is needed; therefore, a PICTEX picture is provided to give the reader an idea of the construction.

The density φ(ζ) = e^{−ζ^{2}/2}/\sqrt{2π} of the

standard normal distribution.

−3  −2  −1  0  1 ζ 2  3

↙ φ(ζ)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

FIGURE 35

An Important Improper Integral

Chapter 6

6.0 Introduction. An integral can be interpreted geometrically as the area under a curve. More specifically, it is the area bounded by the x-axis, the lines x = α and y = β, and the curve. Several things might go “wrong” with this idea: (1) the function might not be continuous;

(2) either α = −∞ or β = +∞, or both; or (3) the function to be integrated, known as the integrand, might “blow up” (the function might increase without bound behaving as 1/x near 0). In the first case, if the function is composed of several continuous pieces and there is only a finite collection of “jump” discontinuities, that is, if the function is piece-wise continuous, then our usual methods apply. For pathological cases of discontinuous functions, there is an entire discipline of mathematics, Real Analysis. The principal thrust of Real Analysis follows trying to answer the question: “under what conditions does ∫_{α}^{β} f(x) dx make sense?” Aside from some graphs and geometric persuasions, we won’t concern ourselves with this matter. This chapter is concerned with the concept of the improper integral. One improper integral of particular interest has as an integrand the function

f(x) = \frac{1}{\sqrt{2π}} e^{−x^{2}/2}.  (6.1)

On first glance, this would seem to be the ideal integral to focus on. However, integral has been tabulated and is available in standard tables books. Another important integral, which will be the motivation for this chaper, is

∫_{0}^{∞} \frac{ln x dx}{x^{2} + a^{2}} .  (6.2)

This integral is important in physics, engineering, and applied mathematics. Its evaluation is found in some tables books; however, it is not treated as a classical example in most texts. For this reason, an evaluation and an approximation will be done. The computed value for this definite integral can be found in Schaum’s Outline Complex Variables†

84. Prove that

∫_{0}^{∞} \frac{ln x dx}{x^{2} + a^{2}} = \frac{π ln a}{2a}.

An Integrable, Discontinuous Function_{α  β}_{FIGURE 36}^{y_{min} y_{k−1} y_{k} y_{k+1} y_{max}}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  x y  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}

† Spiegel, Murray R., Schaum’s Outline Series Theory and Problems of Complex Variables with an application to conformal mapping and its applications, (NY: McGraw-Hill, Inc., 1964), page 197.

6.1 Theoretical Evaluation.

−R  →  −ϵ  ϵ  →  R  x_{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}  _{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} y Contour Γ_{FIGURE 37}  ↖ ↙

Evaluate the (improper) integral

∫_{0}^{∞} \frac{ln x dx}{x^{2} + a^{2}}  (6.3)

using the methods of complex analysis. This is the preferred means of evaluation of such integrals by physicists and engineers. We will consider the above contour, Γ. By a contour, we mean a simple closed curve, in this case Γ, which is traversed in the positive sense. If someone walked around this curve, he would not cross his own path (the curve is simple); he would eventually return to his starting point (the curve is closed); and, he would notice that the area contained within the curve would always be to his left (traversing in the positive sense). The curve is also made up of pieces, each of which is well-behaved in some mathematical sense. Let C_{1} denote the outer semi-circle (of radius R) and let C_{2} denote the inner semi-circle (of radius ϵ). Then Γ is composed of four parts: (1) the interval along the x-axis from −R to −ϵ, which is denoted as [−R, −ϵ]; (2) the semi-circle C_{2}; (3) the interval [ϵ, R]; and, (4) the semi-circle C_{1}. We will use the so-called calculus of residues to compute

∫_{Γ} \frac{ln z dz}{z^{2} + a^{2}} = 2πi Res_{z=ia} [\frac{ln(z)}{z^{2} + a^{2}} ] = \frac{iπ^{2}}{2a} + \frac{π · ln a}{a}.  (6.4)

Res_{z=ia} [\frac{ln(z)}{z^{2} + a^{2}} ] = lim_{z→ia}(z − ia)\frac{ln(z)}{(z − ia)(z + ia)} = \frac{ln(ia)}{2ia} = \frac{ln(i)}{2ia} + \frac{ln(a)}{2ia} = \frac{π}{4a} + \frac{ln a}{2ia}.

∫_{Γ} \frac{ln z dz}{z^{2} + a^{2}} = ∫_{−R}^{−ϵ} \frac{ln z dz}{z^{2} + a^{2}} + ∫_{C_{1}} \frac{ln z dz}{z^{2} + a^{2}} + ∫_{ϵ}^{R} \frac{ln x dx}{x^{2} + a^{2}} + ∫_{C_{1}} \frac{ln z dz}{z^{2} + a^{2}} .  (6.5)

∫_{−R}^{−ϵ} \frac{ln z dz}{z^{2} + a^{2}} = ∫_{−R}^{−ϵ} \frac{ln |z| + i arg(z)}{z^{2} + a^{2}} dz = ∫_{ϵ}^{R} \frac{ln x dx}{x^{2} + a^{2}} + iπ ∫_{ϵ}^{R} \frac{dx}{x^{2} + a^{2}} ,  (6.6)

where arg z = θ. We apply the well-known formula^{∗}

∫ \frac{dx}{x^{2} + a^{2}} = \frac{1}{a} arctan (\frac{x}{a})  (6.7)
lim_{ϵ→0} lim_{R→∞} ∫_{ϵ}^{R} \frac{dx}{x^{2} + a^{2}} = lim_{R→∞} \frac{1}{a} arctan (\frac{R}{a}) − lim_{ϵ→0} \frac{1}{a} arctan (\frac{ϵ}{a})
= \frac{1}{a} arctan ( lim_{R→∞} \frac{R}{a}) = \frac{π}{2a}  (6.8)

∗ Handbook of Mathematical, Scientific, and Engineering, etc., (Piscataway, NJ: REA, 1980), page 323 Claim

lim_{ϵ→0} ∫_{C_{2}} \frac{ln z dz}{z^{2} + a^{2}} = 0  and  lim_{R→∞} ∫_{C_{1}} \frac{ln z dz}{z^{2} + a^{2}} = 0.  (6.9)

This is an interesting problem because one inequality will suffice for both integral estimates:

|ln a| = | ln |z| + i arg z| = \sqrt{(ln ρ)^{2} + θ^{2}},  (6.10)

where z = ρe^{iθ} and θ = arg z. The variable θ lies in the interval [0, π], that is 0 ≤ θ ≤ π. When ρ = R, z = Re^{iθ}, we choose R > π so that

| ln z| = \sqrt{(ln R)^{2} + θ^{2}} ≤ \sqrt{(ln R)^{2} + π^{2}} ≤ \sqrt{2} ln R.  (6.11)

We also require that R > |a| so that

0 < \frac{1}{R^{2} + |a|^{2}} ≤ \frac{1}{R^{2} − |a|^{2}} ,  (6.12)

which follows from the inequality |R^{2} − |a|^{2}| ≤ |R^{2}e^{2iθ} + a^{2}|. The modulus (absolute value) of the contour integral

∫_{0}^{π} \frac{ln Re^{iθ} dz}{R^{2}e^{2iθ} + a^{2}}  (6.13)

can thus be bounded

|∫_{C_{1}} \frac{ln z dz}{z^{2} + a^{2}} | = |∫_{0}^{π} \frac{ln(z)Re^{iθ} dθ}{R^{2}e^{2iθ} + a^{2}} |

≤ ∫_{0}^{π} \frac{| ln(z)| · R dθ}{R^{2} − |a|^{2}}

≤ ∫_{0}^{π} \frac{\sqrt{2} ln(R) · R dθ}{R^{2} − |a|^{2}}

= \frac{π\sqrt{2} ln(R) · R}{R^{2} − |a|^{2}}.

(6.14)

And, taking the limit

lim_{R→∞} \frac{π\sqrt{2} ln(R) · R}{R^{2} − |a|^{2}} = 0.  (6.15)

In the second contour,

|∫_{C_{2}} \frac{ln z dz}{z^{2} + z^{2}} | = |∫_{0}^{π} \frac{ln(Re^{iθ}) · Re^{iθ} dθ}{(R^{2}e^{2iθ} + a^{2})} | ≤ ∫_{0}^{π} \frac{\sqrt{2} | ln(1/ϵ)| · ϵ dθ}{(|a|^{2} − ϵ^{2})}

= \frac{π\sqrt{2} · ln(1/ϵ) · ϵ}{(|a|^{2} − ϵ^{2})},

(6.16)

by choosing 1 > ϵ > 0 so that |a| > ϵ and ln ϵ < −π. This follows from

z = ϵe^{iθ}, |z| = \sqrt{| ln ϵ|^{2} + θ^{2}}, for 0 ≤ θ ≤ π

and

| ln z | = \sqrt{| ln ϵ|^{2} + θ^{2}} ≤ \sqrt{| ln ϵ|^{2} + π^{2}} ≤ \sqrt{| ln ϵ|^{2} + ϵ^{2}} = \sqrt{2} ln(1/ϵ).  (6.17)

And, in the limit as ϵ → 0,

lim_{ϵ→0} \frac{π\sqrt{2} · ln(1/ϵ) · ϵ}{(|a|^{2} − ϵ^{2})} = 0 ≡ lim_{δ→0} \frac{ln(1/δ)}{1/δ} = 0.  (6.18)

6.2 Numerical Evaluation. We will apply a program using the adaptive quadrature routine.† For the value of a in equation (6.3), let a = 2 so that the theoretical solution will be π ln(a)/(2a) = arctan(1) · ln(2). The output from a computer program is

   left endpoint  = 0.000000500000000
   right endpoint = 10000.00000000000000000
   error estimate = 0.0000000010000000
   Integral       = 0.5442733355868506
   Theoretical    = 0.5443965225759005
   Program returned  (0).   Press any key

We must employ “C” because BASIC does not support recursion. The program listing is as follows:

#include <math.h>
#include <float.h>
#include <stdio.h>
double adaptive (double a, double b, double (*f)(double x), double *err);
double f(double x);
int main() {
double a,b,*err,int1;
a = (double) 0.0000005; b = (double) 100000.0;
*err = (double) 0.00000001;
printf("\nleft endpoint  = %.16lf",a);
printf("\nright endpoint = %.16lf",b);
printf("\nerror estimate = %.16lf",*err);
int1 = adaptive( a, b, f, err);
printf("\nIntegral       = %.16lf",int1);
printf("\nTheoretical    = %.16lf",atan(1.0)*log(2.0));
return(0);
}
double f(double x) {
 double y;
 y = log(x)/(x*x+4.0);
 return (y);
}
double adaptive (double a, double b, double (*f)(double x), double *err) {
double h, s1, s2, s3, s4, s5, error, err1, err2, t1, t2;
h = b - a;
s1 = (*f)(a); s2 = (*f)(a+0.25*h); s3 = (*f)(a+0.5*h);
s4 = (*f)(b-0.25*h); s5 = (*f)(b);
t1 = h*(s1+4.0*s3+s5)/6.0;
t2 = h*(s1+4.0*s2+2.0*s3+4.0*s4+s5)/12.0;
error = fabs(t1-t2)/15.0;
if (error < *err) { *err = error; return(t2); }
else {
 err1 = err2 = *err/2.0; t2 = adaptive(a, a+0.5*h, f, &err1);
 t2 += adaptive(a+0.5*h, b, f, &err2); return(t2); }
}
/* End of File */

† Mike Stimpson, “Numerical Integration Using Adaptive Quadrature,” The C Users Journal, May 1992, pages 31–36.

Mathematical Preliminaries

Appendix A

Sometimes the most advanced concepts in mathematics are most quickly grasped by those with the least formal training. This may be because the tyro has less conditioning and has formed fewer bad habits. This is particularly true as regards variables with subscripts and superscripts. The algebra student who first encounters subscripted and superscripted variables is drilled on the idea that the superscript denotes successive multiplication, that is

x^{n} = \underbrace{x · x ⋯ x}_{n times} .  (A.1)

This follows from the explanation that x^{2} = x·x, x^{3} = x·x·x, and so on. The idea that x^{i}, for i an index and not an integer appears later (in tensor analysis). Then there’s the subscript. If we have a collection of data, where x stands for a single data element, then it is customary to denote the data by subscripting the symbol x. So, the set of data is formed by collecting together x_{1}, x_{2}, . . .. There is a need to group this set, so braces ({}) are used. And, for appearance sake, the braces are typeset slightly larger than the symbols, ({}). The variables and other mathematical entities are usually set in italics and the vectors in bold face. Functions, such as sin and cos, are usually typeset in the standard roman font. All these things are recognized by the student and accepted, usually without needing to be verbalized. As a student progresses from algebra through calculus, she/he develops more and more of a mind-set towards the symbols and their appearance. Then there is the rude awakening of tensor analysis. Here superscript means coordinate and not power and there is a special system for arranging the symbols. The beginning algebra student would have no problem whatsoever with this arrangement; however, the math major suffers no end of frustration. Those familiar with chemistry and physics are aware of the conventions associated with isotopes. There are not only subscripts and superscripts following the element symbol, but preceeding it as well. Something like _{1}^{2}H^{+} means the hydrogen (deuterium) ion in chemistry. The idea that elements can be arranged according to their atomic number via subscripts preceeding the element symbol is easy for the beginner. Later, learning such conventions becomes harder and harder.

1H, _{2}He, _{3}Li, etc.  _{1}^{1}H, _{1}^{2}H, _{1}^{3}H, _{2}^{3}He, _{2}^{4}He, ….  (A.2)

Mathematicians and theoretical physicists sometimes align the the subscripts and superscripts in tensor notation to indicate a priority. The symbol here is S and it’s contravariant parts are indexed j, k while its covariant parts are indexed i, , , ℓ.

S_{i}^{jk}_{ℓ}.

This is very different from

S_{i}^{j}_{kℓ}.

Enough on the topic of subscripts and superscripts. Now let’s consider a more difficult situation. Small children understand so well that a number such as 0.9999999999 is close to 1—but it is not exactly one. Only with the introduction of the ellipsis (…) comes true confusion and misunderstanding. And just what is this ellipsis? An ellipsis is a useful notation for something, clearly understood, that is left out. If we wish to write the alphabet, for example, we all know the letters

                          ABCDEFGHIJKLMNOPQRSTUVWXYZ.

So, we usually just write ABC…XYZ or even ABC…Z. And what permits us to do this? Well, as long as we know exactly what the omitted objects are, there is no problem. Let’s dwell on this matter for a while. If someone writers ABC…Z, then it’s clear that the first letter omitted in the ellipsis is D and the last letter omitted is Y. So, we have the situation that ABC…Z is exactly the same as ABCD…YZ. But this is a giant step. The next big jump in using an ellipsis is leaving out the closing element. Consider ABC. . .. Now ABC… means ABCD…, and so on. But there is no terminal letter present, no Z. Maybe the last letter in the ellipsis is Z, maybe not. There is uncertainty. So we feel uncertain about ABC… but not about ABC…XYZ. And this uncertainty carries over to such things as x_{1}, x_{2}, … as opposed to x_{1}, x_{2}, …, x_{12}, x_{13}. Now let’s apply all of this to numbers. Consider 0.4999 . . .. I claim that this number is exactly the same as 0.5 and 0.5000 . . .. Maybe it’s not correct to say exactly the same, maybe it’s more correct to say “equivalent to.”

\begin{aligned}x & = 0.4999 … \\ 10x & = 4.9999 … \\ 10x − x & = 4.5000 … \\ 9x & = 4.5 … \\ x & = \frac{4.5}{9} = 0.5 \end{aligned}  (A.3)

Is there any difference between 0.5000 … and 0.5? Clearly, there is a difference in the way the two are presented. But, aside from that, is there any real difference? Anyone would say, “no.” In fact, scientific calculators all substitute trailing blanks for trailing zeros for “exact calculations.” So, it’s easy to understand that 0.5000 … and 0.5 are exactly the same—it is not at all transparent that 0.5 and 0.4999 … are the same. There seems to be something else going on here. And that’s just what we’re going to explore. But first, there is never any problem understanding that 0.5, .5, +0.5, and +.5 represent exactly the same number, \frac{1}{2}. The leading plus sign (+) is always understood, in fact it looks funny when it is present—like a printout from some old manual calculator. There is a problem with understanding why 0.5 and 0.4999 … are equivalent. It is deep, subtle, and elusive. The mathematician has a name for this phenomenon—limit. This is some finality in that word also, as in the expression “that’s the limit.” It’s easy to see that 0.5 can also be written \frac{1}{2}. What about 0.4999 …? This will prove to be the key:

\begin{aligned}0.49 & = \frac{1}{2} − \frac{1}{100} \\ 0.499 & = \frac{1}{2} − \frac{1}{1000} \\ 0.4999 & = \frac{1}{2} − \frac{1}{10000} \\ ⋮ & = ⋮ \\ 0.4999 … & = \frac{1}{2} − lim_{n→∞} \frac{1}{10^{n}} \end{aligned}  (A.4)

This all looks very complicated, especially the gaggle of symbols

lim_{n→∞} \frac{1}{10^{n}} .  (A.5)

Here is that funny symbol “∞” which is known as the infinity sign, or lemniscate, or “lazy-eight” in Texas-talk. This lemniscate is also seen in the occult hovering over the head of the magician. And what is one to make of the arrow in the expression n → ∞? Well, if it is true that 0.4999 … ≡ 0.5, then that cluster of symbols better be equal to zero. It turns out that it is:

lim_{n→∞} \frac{1}{10^{n}} = 0.  (A.6)

This looks too deep for mathematical preliminaries. The point is that the innocuous, harmless equivalence 0.4999 … ≡ \frac{1}{2} is more than an oddity. It is a model, or paradigm, of a much deeper mathematical situation. And this reasoning is at the very foundation of Simpson’s Rule and Newton’s Method. As soon as the ellipsis is removed, the answer is no longer exact, it is an approximation. Only in the world of theory does the ellipsis make sense. What has happened is that mathematicians have taken a literary device and extended it’s application into another realm. And it all looked so simple with ABC…!

It’s easy to convince engineers that \frac{1}{3} is 0.333 … and \frac{2}{3} is 0.666 . . .. However, it’s no so easy to convince them that 1 ≡ 0.999 . . .. Even so, consider

\begin{aligned}\frac{1}{3} & = 0.333 … \\ \frac{2}{3} & = 0.666 … \\ 1 = \frac{1}{3} + \frac{2}{3} & = 0.333 … + 0.666 … = 0.999 … \end{aligned}  (A.7)

References

Appendix B References

[1] Eisberg, Robert and Robert Resnick, Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles, Second Edition. NY: John-Wiley & Sons, 1985.

[2] Handbook of Mathematical, Scientific, and Engineering Formulas, Tables, Functions, Graphs, Transforms. Piscataway, NJ: Research & Education Assoc., 1972.

[3] James, Glenn and Robert C. James, Mathematics Dictionary—Multilingual Edition. Princeton, NJ: D. Van Nostrand Company, Inc., 1959.

[4] Knuth, Donald E., The TEXbook. Reading, Massachusetts: Addison-Wesley Publishing Company, 1970.

[5] Kreyszig, Erwin, Advanced Engineering Mathematics, 5th Edition. NY: John Wiley & Sons, 1983

[6] Numerical Analysis, 2nd Edition. Schaum’s Outline Series, edited by Francis Scheid. NY: McGraw-Hill Book Company, 1968.

[7] Olmsted, John M. H., Advanced Calculus. NY: Appleton-Century-Crofts, Inc., 1961.

[8] Stoer, J. and R. Bulirsch, Introduction to Numerical Analysis. NY: Springer-Verlag, 1980.

[9] Taylor, John R., An Introduction to Error Analysis—The Study of Uncertainties in Physical Measurements. Mill Valley, CA: University Science Books, 1982.

[10] Thomas, George B., Calculus and Analytic Geometry, 2nd Edition. Reading, Massachusetts: Addison-Wesley Publishing Company, Inc., 1966.

[11] Webster’s Ninth New Collegiate Dictionary. Springfield, Massachusetts: Merriam-Webster, Inc., 1984.

[12] Wichura, Michael J., The PICTEX Manual, TEXniques, Publication for the TEX Community, Number 6. Providence, RI: TEX User Group, 1987.

A Geometric Persuasion

Appendix C

A Geometric Persuasion

6/28/89 H. Watson Corona, CA

Assuming that mass is volume times density, consider first a sphere of radius r_{0},

m = ρV = ρ \frac{4}{3} πr_{0}^{3}.  (C.1)

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For a charged sphere of radius r_{0}, suppose a standing wave exists on its surface. Then, due to energy absorption the wave “splits” forming two surfaces: an inner surface whose circumference is twice the radius, r_{0}, and an outer whose radius is twice the circumference, 2πr_{0}. Thus

R = 2 · 2πr_{0} = 4πr_{0}  (C.2)

and

r = \frac{2r_{0}}{2π} = \frac{r_{0}}{π} .  (C.3)

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_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} R r

The resulting configuration “collapses” to a figure whose radius is R_{0} = 4πr_{0}− \frac{r_{0}}{π} and containing a “cone” whose base is 4πr^{2} = 4π(r_{0}/π)^{2} = 4r_{0}^{2}/π and height R_{0}. The (area of the) base of the cone equals the surface (area) of the inner sphere: 4πr^{2} = 4π(r_{0}/π)^{2} = 4r_{0}^{2}/π. The volume of the collapsed figure is, then,

V_{1} = \frac{4}{3}πR_{0}^{3} − \frac{1}{3}bh = \frac{4}{3}π [4πr_{0} − \frac{r_{0}}{π} ]^{3} − \frac{4}{3}π [(r_{0}/π)^{2}] · [4πr_{0} − \frac{r_{0}}{π} ] .  (C.4)

V_{1} = \frac{4}{3}π [(4π − \frac{1}{π} )^{3} − (4π − \frac{1}{π} ) (\frac{1}{π} )^{2}] r_{0}^{3}  (C.5)

Define V_{0} = \frac{4}{3}πr_{0}^{3} and look at the ratio

\begin{aligned}\frac{V_{1}}{V_{0}} & = (4π − \frac{1}{π} )^{3} − (4π − \frac{1}{π} ) (\frac{1}{π^{2}} ) \\ & = (64π^{3} − 48π + \frac{12}{π} − \frac{1}{π^{3}} ) − \frac{4}{π} + \frac{1}{π^{3}} \\ = 64π^{3} − 48π + \frac{8}{π} . \end{aligned}  (C.6)

Compute (C.6) numerically:

\frac{V_{1}}{V_{0}} = 1836.151739256348  (C.7)

which lies between the (two) most recent values for the proton-electron ratio:

1836.15152 < \frac{V_{1}}{V_{0}} < 1836.152701.  (C.8)

64π^{3} − 48π + \frac{8}{π} = (4π) · (4π − \frac{1}{π} ) · (4π − \frac{2}{π} )  (C.9)

Looking at A = 4r_{0}^{2}/π as the surface area of the inner sphere and S = 4πR_{0}^{2} as the surface area of the larger sphere, we can also compute V_{1}/V_{0} by assuming that that portion of the solid sphere of radius R_{0} subtended from the center to the area A does not contribute to the total volume. This is to say that the volume element whose base region lying on the surface of the sphere has the same volume as a right circular cone, with equal base area.

\frac{V_{1}}{V_{0}} = (1 − \frac{A}{S}) · R_{0}^{3}.  (C.10)

Direct numerical calculation will reveal that

(1 − \frac{4/π}{4π(4π − 1/π)^{2}} ) · (4π − \frac{1}{π} ) ≈ 1836.151738,  (C.11)

or the same result as equation (C.7).

A Contour Integral

Appendix D

A Contour Integral

20 Feb 92

The residue of f(z) at infinity is defined as follows: If f(z) has an isolated singularity at infinity, and if C is a large circle which encloses all the singularities of f(z) except z = ∞, then the residue at z = ∞ is defined to be

\frac{1}{2πi} ∫_{C} f(z) dz  (D.1)

taken around C in the negative sense (negative with respect to the origin), provided the integral has a definite value. If we apply the transformation z = 1/ζ to the integral it becomes

\frac{1}{2πi} ∫ −f (\frac{1}{ζ}) \frac{dζ}{ζ^{2}}  (D.2)

taken positively around a small circle centered at the origin.

The Beta function ∫_{0}^{1} x^{α}(1 − x)^{1−α} dx.

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↘

↙

C = Complex Plane

f(z) = z(1/z − 1)^{1−α}

∂D(0, R)↘

∫_{∂D(0,R)} z^{α}(1 − z)^{1−α} dz − ∫_{0}^{1} x^{α}(1 − x)^{1−α} dx + e^{2πiα} ∫_{0}^{1} x^{α}(1 − x)^{1−α} dx = 0.  (D.3)

A Contour Integral

Let I = ∫_{0}^{1} x^{α}(1 − x)^{1−α} dx.

\begin{aligned}I − e^{2πiα} I & = ∫_{∂D(0,R)} z^{α}(1 − z)^{1−α} dz \\ & = 2πi Res_{∞}(f). \end{aligned}  (D.4)

Let h(w) = (w − 1)^{β}, where w = 1/z and β = 1 − α.

h(w) = (w − 1)^{β} ; h′(w) = β(w − 1)^{β−1} ; h′′(w) = β(β − 1)(w − 1)^{β−2} ; ⋮ ; h^{(n)}(w) = β(β − 1) ⋯; (β − n + 1)(w − 1)^{β−n} ; ⋮  h(0) = (−1)^{β} ; h′(0) = (−1)^{β−1}β ; h′′(0) = (−1)^{β−2}β(β − 1) ; ⋮ ; h^{(n)}(0) = (−1)^{β−n}β(β − 1) ⋯; (β − n + 1) ; ⋮

h(w) = h(0) + wh′(0) + \frac{w^{2}}{2} h′′(0) + … + \frac{w^{n}}{n!} h^{(n)}(0) + …

\begin{aligned}h(w) & = (−1)^{β} + wβ(−1)^{β−1} + \frac{w^{2}}{2} β(β − 1) + … + (−1)^{β−n} \frac{w^{n}}{n!} β(β − 1) ⋯ (β − n + 1) + … \\ & = (−1)^{β} (1 − wβ + \frac{w^{2}}{2} β(β − 1) + … + (−1)^{n} \frac{w^{n}}{n!} β(β − 1) ⋯ (β − n + 1) + …) \end{aligned}

h(1/z) = (−1)^{β} (1 − \frac{β}{z} + \frac{β(β − 1)}{2z^{2}} + … + (−1)^{n} \frac{β(β − 1) ⋯ (β − n + 1)}{z^{n}n!} + …)

\begin{aligned}f(z) & = zh(1/z) \\ & = (−1)^{β} (z − β + \frac{β(β − 1)}{2} \frac{1}{z} + … + (−1)^{n} \frac{β(β − 1) ⋯ (β − n + 1)}{z^{n−1}n!} + …) \end{aligned}

Therefore,

\begin{aligned}Res_{z=∞} f(z) = Res_{∞}(f) & = −a_{−1} = (−1)^{β−1} \frac{β(β − 1)}{2} \\ & = (−1)^{−α} \frac{α(1 − α)}{2} \\ & = e^{−iπα} \frac{α(1 − α)}{2} \end{aligned}  (D.5)

Now we apply the results of our calculation of the residue in equation (D.5) to equation (D.1):

I · (1 − e^{i2πα}) = 2πi · e^{−iπα} · (\frac{α(1 − α)}{2})

Now we equate the real parts:

I · (1 − cos(2πα)) = π sin(πα)α(1 − α)
1 − cos 2θ = 2 sin^{2} θ
I · (2 sin^{2}(πα)) = π sin(πα)α(1 − α)
I = \frac{πα(1 − α)}{2 sin(πα)}.  (D.6)

Equation (D.6) is the sought-for result.

Malgrange-Ehrenpreis Theorem

Appendix E

The Malgrange-Ehrenpreis Theorem

The Malgrange-Ehrenpreis Theorem. If p(D) is a constant coefficient partial differential operator on R^{n}, then there is a distribution E ∈ D′ such that p(D)E = δ. (See [5], page 48.)

Definitions: We are using the standard multi-index notation. A multi-index

α = ⟨α_{1}, …, α_{n}⟩

is an n-tuple of nonnegative integers. The collection of all such multi-indices will be denoted by I_{+}^{n}. The symbols |α|, x^{α}, D^{α}, and x^{2} are defined thusly:

\begin{aligned}|α| & = ∑_{i=1}^{n} α_{i} \\ x^{α} & = x_{1}^{α_{1}}x_{2}^{α_{2}} ⋯ x_{n}^{α_{n}} \\ D^{α} & = \frac{∂^{|α|}}{∂x_{1}^{α_{1}}∂x_{2}^{α_{2}} ⋯ ∂x_{n}^{α_{n}}} \\ x^{2} & = ∑_{i=1}^{n} x_{i}^{2} \end{aligned}

We define the set S(R^{n}) ⊂ C^{∞}(R^{n}) such that f ∈ S iff

sup_{x∈R^{n}} | x^{β}D^{α}f(x)| < ∞  (x^{β} = ∏_{j=1}^{n} x_{j}^{β_{j}} )

for every α = ⟨α_{1}, α_{2}, …, α_{n}⟩ and β = ⟨β_{1}, β_{2}, …, β_{n}⟩ with nonnegative integers α_{j} and β_{k}. Such functions are called rapidly decreasing (at ∞) or functions of rapid decrease. We call S(R^{n}) the Schwartz space of C^{∞} functions of rapid decrease or simply the Schwartz space.

Definition: Let f ∈ S(R^{n}). The Fourier Transform of f is the function

f̂(λ) = \frac{1}{(2π)^{n/2}} ∫_{R^{n}} e^{−ix·λ}f(x) dx  (E.1)

where x·λ = ∑_{i=1}^{n} x_{i}λ_{i}. On occasion it will be necessary to write f̂ = Ff. The inverse Fourier Transform of f is the function

f̌(λ) = \frac{1}{(2π)^{n/2}} ∫_{R^{n}} e^{ix·λ}f(x) dx.  (E.2)

We will occasionally write f̌ = F^{−1}f.

We define the polynomial p(x) = p(x_{1}, …, x_{n}) to be a polynomial of total degree k in n variables. Recall that the total degree, k, of a polynomial p is the maximum of the degrees, k_{i}, of the factors a_{i}x_{1}^{k_{i1}} ⋯ x_{n}^{k_{in}} such that k_{i} = k_{i1} + … + k_{in} and a_{i} ≠ 0. If x = ⟨x_{1}, …, x_{n}⟩, then p(D) denotes the partial differential operator obtained by substituting ∂/∂x_{i} for x_{i} whenever it occurs in the polynomial p(x).

Definition: Let b ∈ R. The symbol δ_{b} is the linear functional δ_{b}(φ) = φ(b). δ_{b} ∈ S′(R).

δ_{b}(φ) = ∫ φ(x)δ(x − b) dx  (E.3)

δ(x) is not a function; it is a symbolic expression known as the (Dirac) delta function, sometimes written T_{δ_{0}} .

We will now discuss a distribution or generalized function. To do this, some background is necessary. Let Ω be an open connected set in R^{n} and C_{0}^{∞}(Ω) be the family of functions with compact support in Ω. (Recall that if f(x) is a complex-valued function defined on Ω, we call the support of f, written supp(f), to be the closure of { x ∈ Ω | f(x) ≠ 0 } in the space Ω. It may be equivalently defined as the smallest closed set of Ω outside which f vanishes identically.) Let K_{n} be an increasing family of compact sets with ⋃ K_{n}^{o} = Ω, that is

K_{1} ⊆ K_{2} ⊆ K_{3} ⊆ … K_{n} ⊆ …

where K_{n}^{o} is the interior of K_{n} (the largest open subset of K_{n}). It is not hard to show that each of the C_{0}^{∞}(K_{n}) is a linear space, where

(f_{1} + f_{2})(x) = f_{1}(x) + f_{2}(x),  (αf)(x) = αf(x).  (E.4)

For each compact set K_{n} and each multi-index α, we can define the semi-norm ∥ · ∥ by

∥f∥_{α,∞} = sup { | D^{β}f(x)| | |β| ≤ α, f ∈ C_{0}^{∞}, x ∈ K_{n} }.  (E.5)

Define

∥D^{α}f∥_{∞} = ∥f∥_{α,∞}.  (E.6)

Put the Fréchet topology on C_{0}^{∞}(K_{n}) generated by the ∥D^{α}f∥_{∞} norms. The set C_{0}^{∞}(Ω) with the inductive limit topology obtained by C_{0}^{∞}(Ω) = ⋃ C_{0}^{∞}(K_{n}) is denoted D_{Ω} . If Ω is the space R^{n}, then we denote D_{R^{n}} by D. The space of all continuous linear functionals on D_{Ω} is denoted D′_{Ω} , where D′_{R^{n}} = D′. It can be shown that the topology on D_{Ω} is independent of the choice of the K_{n}. It can also be shown that a sequence f_{m} ∈ D converges to f ∈ D iff all the f_{m} are in some K_{n} and f_{m} → f in the topology of K_{n}. We finally have all the preliminaries to define a distribution.

Definition. A distribution or generalized function is a continuous linear functional on D_{Ω} (D when Ω = R^{n}). The space of all continuous linear functionals on D_{Ω} is denoted by D′_{Ω} (D′ when Ω = R^{n}).

Definition. Suppose f, g ∈ S(R^{n}). Then the convolution of f and g, denoted f ∗ g is the function

(f ∗ g)(y) = ∫_{R^{n}} f(y − x)g(x) dx.  (E.7)

The partial differential operator p(D) = ∑_{|α|≤k} a_{α}x^{α} extends to D′ by the formula

(p(D)T )(φ) = T ( ∑_{|α|≤k}(−1)^{|α|}D^{α}(a_{α}φ)) .  (E.8)

Definition. A fundamental solution for the partial differential operator p(D) is a distribution E ∈ D′ such that p(D)E = δ.

The reason for studying fundamental solutions is that if we define u = E∗f, where f ∈ C_{0}^{∞}(R^{n}), then

\begin{aligned}p(D)u & = p(D)(E ∗ f) = p(D)E ∗ f \\ & = δ ∗ f = f. \end{aligned}  (E.9)

Thus, if we can find a fundamental solution, then we have an existence theorem for all the partial differential equations p(D)u = f where f ∈ C_{0}^{∞}(R^{n}). Furthermore, if we can find an expression for the distribution E, then we have an explicit representation of a solution, namely

u = E ∗ f.  (E.10)

The convolution operation, E ∗ f, for distributions and functions can be accomplished on digital computers. Note that, in general, partial differential equations will have many fundamental solutions in D′, and sometimes have more than one in S′(R^{n}). Convolutions frequently occur when one uses Fourier transforms because the Fourier transform takes products into convolutions. In particular, for distributions and functions, if T is a distribution (T ∈ D′) and f ∈ D, let f^{†}(x) = f(−x).

(T ∗ f)(φ) = T (f^{†} ∗ φ)  ∀φ ∈ D.  (E.11)

The Malgrange-Ehrenpreis theorem states that every constant coefficient partial differential operator p(D) has a fundamental solution. The proof of this theorem breaks into two parts: the first part is a complex variables argument and the second is an application of the Hahn-Banach theorem.

Proof of the Malgrange-Ehrenpreis Theorem: Define p^{†}(x) = p(−x) and q(x) = p^{†}(ix), with i = \sqrt{−1}. Let φ ∈ C_{0}^{∞}(R^{n}), where C_{0}^{∞}(R^{n}) is the space of all infinitely differentiable functions with compact support. The first thing we observe is that

F(p^{†}(D)φ(x)) = φ̂(s)q(s).  (E.12)

This beautiful and exciting result follows immediately from the following:

Lemma 1. The maps F and F^{−1} are continuous linear transformations of S(R^{n}) into S(R^{n}). Furthermore, if α and β are multi-indices, then

((iλ)^{α}D^{β}f̂)(λ) = F ( D^{α}((−ix)^{β}f(x)))  (E.13)

If we assume that f ∈ C_{0}^{∞}(R^{n}), then for all ζ = ⟨ζ_{1}, …, ζ_{n}⟩ ∈ C^{n}, the integral

f̂(ζ) = (2π)^{n/2} ∫_{R^{n}} e^{−iζ·x}f(x) dx  (E.14)

is well defined. Furthermore, f̂(ζ) is an entire analytic function of the n complex variables ζ_{1} , ζ_{2} , …, ζ_{n} since we can differentiate under the integral sign. The formal differentiation under the integral sign is permitted since the right hand side is uniformly convergent in ζ. This is contained in the following:

Paley-Weiner theorem. An entire analytic function of n complex variables g(ζ) is the Fourier transform of a C_{0}^{∞}(R^{n}) function with support in the ball { x | |x| ≤ R } if and only if for each N there is a C_{N} such that

|g(ζ)| ≤ \frac{C_{N}e^{R|Im(ζ)|}}{(a + |ζ|)^{N}}  (E.15)

for all ζ ∈ C^{n}. (See [5], page 16.)

We apply the Paley-Weiner theorem to the partial differential operator p^{†}(D) acting on the function φ ∈ C_{0}^{∞}(R^{n}). For each y = ⟨y_{1}, …, y_{n}⟩ ∈ R^{n}, we have

(F(p^{†}(D)φ)) (y + ζ) = φ̂(y + ζ) q(y + ζ)  (E.16)

is an entire (analytic) function of ζ ∈ C^{n}. Now define Q(x) as follows:

Q(x) = ∑_{α} | D^{α}q(x)|;  (E.17)

notice that Q is positive and bounded away from zero. If k is the total degree of p, then there is some multi-index α such that |α| = k. D^{α}q(x) = a_{α}, where a_{α} is the coefficient of x^{α}, hence α_{k} ≠ 0. Therefore,

∑_{α} |D^{α}q(x)| ≥ |a_{α}| > 0.  (E.18)

Having defined Q, we will have to use some results from complex analysis. We will state four lemmas:

Lemma 2 (Malgrange). If f(z) is a holomorphic function of a complex variable z for |z| ≤ 1 and p(z) is a polynomial

p(z) = a_{m}z^{m} + a_{m−1}z^{m−1} + … + a_{1}z + a_{0}  (E.19)

of degree m, then

|a_{m}f(0)| ≤ \frac{1}{2π} ∫_{0}^{2π} |f(e^{iθ})p(e^{iθ})| dθ.  (E.20)

Lemma 3. If f(z) is a holomorphic function of a complex variable z for |z| ≤ 1 and p(z) is a polynomial

p(z) = a_{m}z^{m} + a_{m−1}z^{m−1} + … + a_{1}z + a_{0}  (E.21)

of degree m, then

|f(0) p^{(k)}(0)| ≤ \frac{m!}{(m − k)! 2π} ∫_{0}^{2π} |f(e^{iθ})p(e^{iθ})| dθ.  (E.22)

Lemma 4. Let F (ζ) be an entire function of n complex variables and let P (ζ) be a polynomial of degree m. Suppose that g(ζ) is a nonnegative integrable function with compact support depending only on |ζ_{1}|, |ζ_{2}|, …, |ζ_{n}|.

|F (0)D^{α}P (0)| ∫_{C^{n}} |ζ|^{α}g(ζ) dζ ≤ C_{0} ∫_{C^{n}} |F (ζ)P (ζ)| g(ζ) dζ,  (E.23)

where dζ denotes the Lebesgue measure on C^{n} and C_{0} is a constant depending on α and m. dζ is the measure denoted dξ_{1}dη_{1} ⋯ dξ_{n}dη_{n}, where ζ_{k} = ξ_{k} + iη_{k}; and, m is regarded as the multi-index ⟨m, m, …, m⟩.

Lemma 5. Suppose q̃(x) = ( ∑_{|α|≥k} |q^{(α)}(x)|^{2})^{1/2} , where q^{(α)}(x) = D_{x}^{α}q(x), k is the total degree of q, φ ∈ C_{0}^{∞}(R^{n}), and ϵ > 0. Then

|q̃(x)φ̂(x)| ≤ C_{1} ∫_{|ζ|≤ϵ} |φ̂(x + ζ) q(x + ζ)| dζ.  (E.24)

The first step in the proof uses the Cauchy integral formula to show that

\begin{aligned}|Q(x)φ̂(x)| & ≤ C_{1} ∫_{|ζ|≤ϵ} |φ̂(x + ζ) q(x + ζ)| dζ \\ & = C_{1} ∫_{|ζ|≤ϵ} |F(p^{†}(D)φ)(x + ζ)| dζ \end{aligned}  (E.25)

where C_{1} depends on ϵ but is independent of φ. As defined in Lemma 4, dζ is the Lebesgue measure on C^{n}. To derive equation (E.25), first let u ∈ C_{0}^{∞}(R^{n}) and φ = p(D)u. We will make the following substitutions in equation (E.23):

g(ζ) = \{ \begin{matrix}1, & if |ζ| ≤ ϵ, \\ 0, & otherwise; \end{matrix}  (E.26)
F (ζ) = φ̂(x + ζ) and P (ζ) = q(x + ζ).  (E.27)

Equation (E.23) immediately becomes

|φ̂(x)D^{α}q(x)| ∫_{|ζ|≤ϵ} |ζ|^{α} dζ ≤ C_{0} ∫_{|ζ|≤ϵ} |φ̂(x + ζ)q(x + ζ)| dζ.  (E.28)

Now we may define the constant (scalar) C_{1} as

C_{1} = \frac{C_{0} · |α|}{∫_{|ζ|≤ϵ} |ζ|^{α} dζ} ,  (E.29)

where |α| = ∑_{α} 1 = α_{1} + α_{2} + … + α_{n}. Recall that C_{0} is a constant depending only on α and m; therefore, C_{1} is a constant depending only on α, m, and ϵ. In particular, C_{1} does not depend on the choice of the function φ. Now, re-write equation (E.28):

|φ̂(x)D^{α}q(x)| ≤ \frac{C_{1}}{|α|} ∫_{|ζ|≤ϵ} |φ̂(x + ζ)q(x + ζ)| dζ.  (E.30)

\begin{aligned}|φ̂(x)Q(x)| & ≤ ∑_{α} |φ̂(x)D^{α}q(x)| \\ & ≤ \frac{C_{0}}{∫_{|ζ|≤ϵ} |ζ|^{α} dζ} (∑_{α} 1) ∫_{|ζ|≤ϵ} |φ̂(x + ζ)q(x + ζ)| dζ \\ & = C_{1} ∫_{|ζ|≤ϵ} |φ̂(x + ζ)q(x + ζ)| dζ \end{aligned}  (E.31)

We will substitute formula (E.10) into the right-hand side of equation (E.32) to get

|φ̂(x)Q(x)| ≤ C_{1} ∫_{|ζ|≤ϵ} |F(p^{†}(D)φ)(x + ζ)| dζ.  (E.32)

From Fourier’s (inversion) theorem, we have that

φ(0) = \frac{1}{(2π)^{n/2}} ∫_{R^{n}} φ̂(y) dy  (E.33)

|φ(0)| ≤ \frac{1}{(2π)^{n/2}} ∫_{R^{n}} |φ̂(y)| dy  (E.34)

\begin{aligned}|φ(0)| & ≤ \frac{1}{(2π)^{n/2}} ∫_{R^{n}} |φ̂(y)| dy \\ & ≤ C_{2} ∫_{R^{n}} (∫_{|ζ|≤ϵ} |F(p^{†}(D)φ)(y + ζ)| |Q(y)|^{−1} dζ) dy \\ & = C_{2} ∫_{R^{n}} ∫_{|λ|^{2}+|μ|^{2}≤ϵ^{2}} |F(p^{†}(D)φ)(y + λ + iμ)| |Q(y)|^{−1} dλdμdy \end{aligned}  (E.35)

For |λ| ≤ ϵ, we will demonstrate a constant C_{3} such that Q(y+λ)(Q(y))^{−1} ≤ C_{3} independently of y. Take the generalized Taylor series expansion for D^{α}q(y + λ). λ = ⟨λ_{1}, λ_{2}, …, λ_{n}⟩ ∈ R^{n} and |λ| ≤ ϵ.

D^{α}q(y + λ) = ∑_{β} \frac{λ^{β}}{β!} D^{α+β}q(y),  (E.36)

where q is a polynomial of total degree k; therefore, D^{α+β}q(y) = 0 whenever |α + β| > k. Therefore, for all α, |α| ≤ k

|D^{α}q(y + λ)| ≤ ∑_{|α+β|≤k} |\frac{λ^{β}}{β!} D^{α+β}q(y)|;  (E.37)

λ^{β} = λ_{1}^{β_{1}}λ_{2}^{β_{2}} ⋯ λ_{n}^{β_{n}}.
|λ| ≤ ϵ ⟹ |λ|^{|β|} ≤ ϵ^{|β|}
|β| ≤ k ⟹ |λ|^{|β|} ≤ ϵ^{k} for all multi-indices β.

Therefore,

\begin{aligned}|D^{α}q(y + λ)| & ≤ ∑_{|α+β|≤k} |\frac{λ^{β}}{β!} | |D^{α+β}q(y)| \\ & ≤ ∑_{|α+β|≤k} |\frac{ϵ^{k}}{β!}| |D^{α+β}q(y)| \\ & ≤ ϵ^{k} · ( ∑_{|α+β|≤k} |D^{α+β}q(y)|) \end{aligned}  (E.38)

\begin{aligned}|Q(y + λ)| & = ∑_{|α|≤k} |D^{α}q(y + λ)| \\ & ≤ ∑_{|α|≤k} ∑_{|α+β|≤k} |\frac{λ^{β}}{β!} | · |D^{α+β}q(y)| \\ & ≤ ∑_{|α|≤k} ϵ^{k} ∑_{|α+β|≤k} |D^{α+β}q(y)| \\ & ≤ (ϵ^{k} ∑_{|β|≤k} 1) · ∑_{|α|≤k} |D^{α}q(y)| \\ & = (ϵ^{k} ∑_{|β|≤k} 1) · Q(y). \end{aligned}  (E.39)

Therefore, since |Q(y + λ)| = Q(y + λ),

\frac{|Q(y + λ)|}{|Q(y)|} ≤ ϵ^{k} · ∑_{|β|≤k} 1.

Let C_{3} = ϵ^{k} · ∑_{|β|≤k} 1. Setting C_{4} = C_{2} · C_{3} and substituting,

\begin{aligned}|φ(0)| & ≤ C_{4} ∫_{R^{n}} ∫_{|λ|^{2}+|μ|^{2}≤ϵ^{2}} |F(p^{†}(D)φ)(y + λ + iμ)| (Q(y + λ))^{−1} dλdμdy. \\ & ≤ C_{5} ∫_{R^{n}} ∫_{|μ|^{2}≤ϵ^{2}} |F(p^{†}(D)φ)(y + iμ)| (Q(y))^{−1} dμdy. \end{aligned}  (E.40)

We now define the norm ∥ · ∥_{Q} as follows

∥φ∥_{Q} = ∫_{R^{n}} ∫_{|μ|≤ϵ} |φ̂(y + iμ)(Q(y))^{−1}| dμdy.  (E.41)

We will first show that ∥ · ∥_{Q} is a continuous norm on D. To see that ∥ · ∥_{Q} is a norm, observe that

∥ · ∥_{Q}: C_{0}^{∞}(R^{n}) → [0, ∞)  (E.42)

and that φ ≠ 0 implies that

∥φ∥_{Q} > 0.  (E.43)

This follows immediately from the fact that ∥φ̂∥_{2} = ∥φ∥_{2} . This is a corollary to the Fourier inverson theorem. If each of φ and ψ are in C_{0}^{∞}(R^{n}), then we know (from the Paley-Wiener theorem) that each of the following integral exist:

φ̂(λ) = \frac{1}{2π} ∫_{R^{n}} e^{−iλ·x}φ(x) dx,

ψ̂(λ) = \frac{1}{2π} ∫_{R^{n}} e^{−iλ·x}φ(x) dx,

∥φ∥_{Q} = ∫_{R^{n}} ∫_{|μ|≤ϵ} |φ̂(y + iμ)(Q(y))^{−1}| dμdy,

∥ψ∥_{Q} = ∫_{R^{n}} ∫_{|μ|≤ϵ} |ψ̂(y + iμ)(Q(y))^{−1}| dμdy,

∥φ + ψ∥_{Q} = ∫_{R^{n}} ∫_{|μ|≤ϵ} |(φ̂ + ψ̂)(y + iμ)(Q(y))^{−1}| dμdy.  (E.44)

From the so-called triangle inequality, we have that |φ̂(ζ) + ψ̂(ζ)| ≤ |φ̂(ζ)| + |ψ̂(ζ)|. This implies immediately that

∥φ + ψ∥_{Q} ≤ ∥φ∥_{Q} + ∥ψ∥_{Q}  (E.45)

Next, we have to show that ∥ · ∥_{Q} is continuous. Since Q is bounded from below, there is a positive number C_{ℓ} such that C_{ℓ} < Q(y) for all y ∈ R^{n}.

(Q(y))^{−1} < \frac{1}{C_{ℓ}} = C_{6}  for all y ∈ C_{0}^{∞}(R^{n}).  (E.46)

∥φ∥_{Q} ≤ C_{6} ∫_{R^{n}} ∫_{|μ|≤ϵ} |φ̂(y + iμ)| dμdy

≤ C_{7} sup_{{y∈R^{n} \atop |μ|≤ϵ}} |(1 + y^{2})^{n+1}φ̂(y + iμ)|

≤ C_{7} sup_{|μ|≤ϵ} ∥(I − Δ)^{n+1}e^{μ·x}φ(x)∥_{1}.

(E.47)

If we define p_{1}(D) = 1 − Δ = 1 − ∑_{j=1}^{n} ∂^{2}/∂x_{j}^{2} , then we will have p_{1}(x) = 1 − x^{2} , q_{1}(x) = p_{1}^{†}(ix) = 1 + x^{2} and

F ((1 − Δ)^{n+1}e^{μ·x}φ(x)) = (1 + y^{2})^{n+1}φ̂(y + iμ).  (E.48)

We implicitly made use of the fact that for α ∈ C if h(x) = g(x)e^{iαx} then ĥ(s) = ĝ(t − α). In equation (E.47), we have used the fact that if g ∈ L^{1} , then ĝ ∈ C_{0} and

∥ĝ∥_{∞} ≤ ∥g∥_{1}.  (E.49)

The inequality is a direct consequence of the definition of the Fourier transform. The function space inclusions are is consequence of the Lebesgue dominated convergence theorem. The righthand side is a continuous norm on C_{0}^{∞}(K) for each compact set K ⊂ R^{n}. Since D has the inductive limit topology, ∥·∥_{Q} is a continuous norm on D. The basic estimate (equation (E.34)) shows that the map

Ẽ: p^{†}(D)φ → φ(0)  (E.50)

is well-defined. That is, if p^{†}(D)φ_{1} = p^{†}(D)φ_{2} , then φ_{1}(0) = φ_{2}(0). Ẽ is continuous since ∥ · ∥_{Q} is a continuous norm. We now recall the Hahn-Banach Theorem:

The Hahn-Banach Theorem. If M is a subspace of a normed linear space X and if T̃ is a bounded linear functional on M, then T̃ can be extended to a bounded linear functional T on X such that ∥T ∥ = ∥T̃∥. (See [6], page 104.)

Thus by the Hahn-Banach theorem there is an E in D′ which extends Ẽ. Since

(p(D)E)(φ) = E(p^{†}(D)φ) = φ(0),  (E.51)

we have found a fundamental solution for p(D).

Lemma 1. The maps F and F^{−1} are continuous linear transformations of S(R^{n}) into S(R^{n}). Furthermore, if α and β are multi-indices, then

((iλ)^{α}D^{β}f̂)(λ) = F ( D^{α}((−ix)^{β}f(x)))  (E.52)

Proof: The map F is clearly linear. We directly compute the Fourier transform of (λ^{α}D^{β}f̂)(λ) from the definition.

\begin{aligned}(λ^{α}D^{β}f̂)(λ) & = \frac{1}{(2π)^{n/2}} ∫_{R^{n}} λ_{x}^{α}(−ix)^{β}e^{−iλ·x}f(x) dx \\ & = \frac{1}{(2π)^{n/2}} ∫ \frac{1}{(−i)^{α}} (D_{x}^{α}e^{−iλ·x})(−ix)^{β}f(x) dx \\ & = \frac{(−i)^{α}}{(2π)^{n/2}} ∫_{R^{n}} e^{−iλ·x}D_{x}^{α}((−ix)^{β}f(x)) dx \end{aligned}  (E.53)

The above formulas require some justification. First of all, we observe that for

f̂(λ) = \frac{1}{(2π)^{n/2}} ∫_{R^{n}} e^{−ix·λ}f(x) dx,  (E.54)

formal differentiation under the integral sign is permitted because the right hand side is uniformly convergent.

D^{β}f̂(λ) = \frac{1}{(2π)^{n/2}} ∫_{R^{n}} e^{−ix·λ}(−i)^{|β|}x^{β}f(x) dx.  (E.55)

We now multiply both sides by (iλ)^{α}.

(iλ)^{α}D^{β}f̂(λ) = \frac{1}{(2π)^{n/2}} ∫_{R^{n}}(iλ)^{α} e^{−ix·λ}(−i)^{|β|}x^{β}f(x) dx.  (E.56)

The next operation is integration by parts. We define a function ϕ^{(α)} = (−iλ)^{α}e^{−iλ·x} = D_{x}^{α}e^{−iλ·x}. Again, by uniform convergence we have

\frac{1}{(2π)^{n/2}} ∫_{R^{n}}(iλ)^{α} e^{−ix·λ}(−i)^{|β|}x^{β}f(x) dx

= (−1)^{α} \frac{1}{(2π)^{n/2}} ∫_{R^{n}}(−1)^{−α} e^{−iλ·x} D_{x}^{α}((−ix)^{β}f(x)) dx.  (E.57)

The final result is

\begin{aligned}(iλ)^{α}D^{β}f̂(λ) & = \frac{1}{(2π)^{n/2}} ∫_{R^{n}} e^{−iλ·x} D_{x}^{α}((−ix)^{β}f(x)) dx \\ & = FD_{x}^{α}((−ix)^{β}f(x)). \end{aligned}  (E.58)

This is the sought-for result. Now we observe that

\begin{aligned}∥f̂∥_{α,β} & = sup_{λ} |λ^{α}(D^{β}f̂)(λ)| \\ & ≤ \frac{1}{(2π)^{n/2}} ∫_{R^{n}} |D_{x}^{α}(x^{β}f)| dx < ∞. \end{aligned}  (E.59)

Thus F takes S(R^{n}) into S(R^{n}). There is a positive integer k such that

∫_{R^{n}} (1 + x^{2})^{−k} dx < ∞,  (E.60)

where dx = dx_{1}dx_{2} ⋯ dx_{n}, x = ⟨x_{1}, x_{2}, …, x_{n}⟩.

\begin{aligned}∥f̂∥_{α,β} & ≤ \frac{1}{(2π)^{n/2}} ∫_{R^{n}} \frac{(1 + x^{2})^{−k}}{(1 + x^{2})^{−k}} |D_{x}^{α}(−ix)^{β}f(x)| dx \\ & ≤ \frac{1}{(2π)^{n/2}} (∫_{R^{n}} (1 + x^{2})^{−k} dx) sup_{x} \{(1 + x^{2})^{+k} |D_{x}^{α}(−ix)^{β}f(x)| \} \end{aligned}  (E.61)

Using Leibnitz’s rule we easily conclude that there exist multi-indices α_{j}, β_{j} and constants c_{j} such that

∥f̂∥_{α,β} ≤ ∑_{j=1}^{M} c_{j} · ∥f∥_{α_{j},β_{j}}  (E.62)

Therefore, F is bounded. A bounded linear functional is always continuous. A symmetric argument holds for the inverse operator F^{−1} .

Lemma 2 (Malgrange). If f(z) is a holomorphic function of a complex variable z for |z| ≤ 1 and p(z) is a polynomial

p(z) = a_{m}z^{m} + a_{m−1}z^{m−1} + … + a_{1}z + a_{0}  (E.63)

of degree m, then

|a_{m}f(0)| ≤ \frac{1}{2π} ∫_{0}^{2π} |f(e^{iθ})p(e^{iθ})| dθ.  (E.64)

Proof: For j = 1, …, m, let z_{j} be a zero of the polynomial p(z). Order the roots such that for j = 1, …, k, z_{j} is a zero of the polynomial interior to the unit circle, that is |z_{j}| < 1 for j = 1, …, k and |z_{j}| ≥ 1 for k < j ≤ m. Define q(z) as follows

q(z) = p(z) ∏_{j=1}^{k} \frac{z̄_{j}z − 1}{z − z_{j}}.  (E.65)

Clearly, p(z) = p_{1}(z) · (z − z_{1}) ⋯ (z − z_{k}) so that

q(z) = p_{1}(z) · ∏_{j=1}^{k} (z̄_{j}z − 1) = (a_{m} · ∏_{j=k+1}^{m}(z − z_{j})) · ∏_{j=1}^{k}(z̄_{j}z − 1).  (E.66)

Since q(z) is a polynomial in C, q(z) is analytic. Moreover, for 0 ≤ θ < 2π, |e^{iθ}| = 1, and

\begin{aligned}|q(e^{iθ})| & = |p_{1}(e^{iθ}) · (z̄_{1}e^{iθ} − 1)(z̄_{2}e^{iθ} − 1) ⋯ (z̄_{k}e^{iθ} − 1)| \\ & = |p_{1}(e^{iθ})| · ∏_{j=1}^{k} |z̄_{j}e^{iθ} − 1| \\ & = |p_{1}(e^{iθ})| · ∏_{j=1}^{k} |z̄_{j} − e^{−iθ}| \\ & = |p_{1}(e^{iθ})| · ∏_{j=1}^{k} |z_{j} − e^{iθ}| \\ & = |p(e^{iθ})| . \end{aligned}  (E.67)

Thus q(z) is regular in the unit circle and |p(z)| = |q(z)| for |z| = 1. We apply the Cauchy integral formula, ∫_{∂Ω} g(ξ) dξ/(ξ − z) = 2πi · g(z), to the function g(z) = f(z)q(z) to obtain

\begin{aligned}\frac{1}{2π} ∫_{0}^{2π} |f(e^{iθ})| · |p(e^{iθ})| dθ & = \frac{1}{2π} ∫_{0}^{2π} |f(e^{iθ})p(e^{iθ})| dθ \\ & = \frac{1}{2π} ∫_{0}^{2π} |f(e^{iθ})q(e^{iθ})| dθ \\ & ≥ \frac{1}{2π} | ∫_{0}^{2π} f(e^{iθ})q(e^{iθ}) dθ| \\ & = |f(0)q(0)| . \end{aligned}  (E.68)

Now consider q(0). | q(0)| = |p_{1}(0)| = |a_{m}| · ∏_{j=k+1}^{m} |z_{j}| ≥ |a_{m}|.

\frac{1}{2π} ∫_{0}^{2π} |f(e^{iθ})p(e^{iθ})| dθ ≥ |f(0)q(0)| ≥ |a_{m}f(0)| .  (E.69)

This completes the lemma.

p(z) = a_{7}z^{7} + … + a_{1}z + a_{0} a polynomial of degree 7.

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. . . . . . . . . . . . . . . . . . . . . . . . . .

∘ ∘

∘ ∘

∘

∘

∘

z_{1} z_{2}

z_{3} z_{4}

z_{5}

z_{6}

z_{7} 1

∘ zeros
m = 7
k = 4

D = unit disk

q(z) = p(z) · ∏_{j=1}^{k} \frac{z̄_{j}z − 1}{z − z_{j}}

q(z) = p_{1}(z) ∏_{j=1}^{k}(z̄_{j}z − 1)

p_{1}(z) = a_{m} · ∏_{j=k+1}^{m}(z − z_{j}) above p_{1}(z) = a_{7}(z − z_{5})(z − z_{6})(z − z_{7})

p_{1}(0) = a_{m} · ∏_{j=k+1}^{m} z_{j}

|p_{1}(0)| = |a_{m}| · ∏_{j=k+1}^{m} |z_{j}| ≥ |a_{m}| · 1 = |a_{m}|

⟹ |q(0)| ≥ |a_{m}|

\frac{1}{2π} ∫_{0}^{2π} |f(e^{iθ})p(e^{iθ})| dθ ≥ \frac{1}{2π} |∫_{0}^{2π} f(e^{iθ})q(e^{iθ})|dθ| = |f(0)q(0)| ≥ |a_{m}f(0)|

Lemma 3. If f(z) is a holomorphic function of a complex variable z for |z| ≤ 1 and p(z) is a polynomial

p(z) = a_{m}z^{m} + a_{m−1}z^{m−1} + … + a_{1}z + a_{0}  (E.70)

of degree m, then

|f(0) p^{(k)}(0)| ≤ \frac{m!}{(m − k)! · 2π} ∫_{0}^{2π} |f(e^{iθ})p(e^{iθ})| dθ.  (E.71)

Proof: Let p be a monic polynomial of degree m such that

\begin{aligned}p(z) & = z^{m} + a_{m−1}z^{m−1} + … + a_{1}z + a_{0} \\ & = ∏_{j=1}^{m}(z − z_{j}), \end{aligned}  (E.72)

where { z_{j} | j = 1, 2, …, m } are the roots of p. Apply the previous lemma to the polynomial of degree k

∏_{j=1}^{k}(z − z_{j})  (E.73)

and the holomorphic (analytic) function

f(z) · ∏_{j=k+1}^{m}(z − z_{j}),  (E.74)

for some 1 ≤ k ≤ m. This yields the formula

|f(0) ∏_{j=k+1}^{m} z_{j}| ≤ \frac{1}{2π} ∫_{0}^{2π} |f(e^{iθ})p(e^{iθ})| dθ.  (E.75)

We need to consider the derivatives of p(z).

p′(z) = p^{(1)}(z) = (z − z_{2}) ⋯ (z − z_{m}) + (z − z_{1})(z − z_{3}) ⋯ (z − z_{m}) + …
+(z − z_{1})(z − z_{2}) ⋯ (z − z_{m−1});  (E.76)

the first derivative has m terms and each term has m − 1 factors of the form (z − z_{j}). The second derivative can be written

\begin{aligned}p′′(z) & = p^{(2)}(z) \\ & = (z − z_{3}) ⋯ (z − z_{m}) + (z − z_{1})(z − z_{4}) ⋯ (z − z_{m}) \\ & + … + (z − z_{1}) ⋯ (z − z_{m−2}) + (z − z_{2})(z − z_{4}) ⋯ (z − z_{m}) \\ & + (z − z_{1})(z − z_{3})(z − z_{5}) ⋯ (z − z_{m}) + …. \end{aligned}  (E.77)

The second derivative has m(m − 1) terms each of which is a product of (m − 2) factors of the form (z − z_{j}).

p^{(k)}(z) = (z − z_{k+1})(z − z_{k+2}) ⋯ (z − z_{m}) + … + (z − z_{1})(z − z_{2}) ⋯ (z − z_{m−k+1}). (E.78)

The k th derivative has m · (m − 1) ⋯ (m − k + 1) terms and each term has m − k factors. Recall that

m · (m − 1) ⋯ (m − k + 1) = \frac{m!}{(m − k)!}.  (E.79)

Each of the m!/(m − k)! terms has the form

∏_{j=k+1}^{m} (z − w_{j}) ,  (E.80)

where {w_{k+1}, …, w_{m}} is one of the m!/(m − k)! possible subsets of m − k elements from {z_{1}, z_{2}, …, z_{m}}. Thus, we have that p^{(k)}(0) is the sum of L = m!/(m − 1)! terms of the form

(−1)^{m−k} ∏_{j=k+1}^{m} w_{ℓ,j},  (E.81)

for some ℓ ∈ L. This yields

\begin{aligned}|f(0) · p^{(k)}(0)| & = | f(0) · ∑_{ℓ∈L}(−1)^{m−k} ∏_{j=k+1}^{m} w_{ℓ,j}| \\ & ≤ ∑_{ℓ∈L} | f(0) · ∏_{j=k+1}^{m} w_{ℓ,j}| \\ & ≤ \frac{m!}{(m − k)!} | f(0) · ∏_{j=k+1}^{m} w_{j}| \\ & ≤ \frac{m!}{(m − k)! · 2π} ∫_{0}^{2π} |f(e^{iθ})p(e^{iθ})| dθ. \end{aligned}  (E.82)

For a general polynomial p, multiply both sides of the inequality by |a_{m}| > 0. This proves the lemma.

Lemma 5. Suppose q̃(x) = ( ∑_{|α|≤k} |q^{(α)}(x)|^{2})^{1/2} , where q^{(α)}(x) = D_{x}^{α}q(x), k is the total degree of q, φ ∈ C_{0}^{∞}(R^{n}), and ϵ > 0. Then

|q̃(x)φ̂(x)| ≤ C′_{1} ∫_{|ζ|≤ϵ} |φ̂(x + ζ) q(x + ζ)| dζ.  (E.83)

Proof: Put p(D)u = v, where u ∈ C_{0}^{∞}(R^{n}). Then q(ζ)û(ζ) = v̂(ζ). Apply the previous lemma (Lemma 4) with F (ζ) = v̂(ξ + ζ), with p(ζ) replaced by p(ξ + ζ) and with |g(ζ)| = 1 when |ζ| ≤ ϵ and = 0 otherwise. Since q̃(ξ) ≤ ∑_{α} |D^{α}q(ξ)|, we obtain, from the previous lemma,

\begin{aligned}|û(ξ)q̃(ξ)| & ≤ C′_{1} ∫_{|ζ|≤ϵ} |û(ξ + ζ) p(ξ + ζ)| dζ \\ & = C′_{1} ∫_{|ζ|≤ϵ} |v̂(ξ + ξ)| dζ. \end{aligned}  (E.84)

We then apply Fourier’s integral theorem,

\begin{aligned}|u(0)| & ≤ (2π)^{−n/2} ∫_{R^{n}} |û(ξ)| dξ \\ & ≤ C′′_{1} ∫_{|ζ|≤ϵ} (∫_{R^{n}} |v̂(ξ + ζ)|/q̃(ξ) dξ) dζ \\ & ≤ C′′_{1} ∫_{F^{n}} (∫_{ζ^{′2}+η^{′2}≤ϵ^{2}} |v̂(ξ + ξ′ + iη′)|/q̃(ξ) dζ′dη′) dζ. \end{aligned}  (E.85)

On the other hand, we have

q̃(ξ + ξ′)/q̃(ξ) ≤ C′_{2} when |ζ′| ≤ ϵ,  (E.86)

because

D^{α}q(ξ + ξ′) = ∑_{β} \frac{(ξ′)^{β}}{β!} D^{α+β}q(ξ),  (E.87)

so that |D^{α}q(ξ + ξ′)|/q̃(ξ) is bounded when |ξ′| ≤ ϵ. Therefore,

\begin{aligned}|u(0)| & ≤ C′′_{1}C′_{2} ∫_{R^{n}} (∫_{ζ^{′2}+η^{′2}≤ϵ^{2}} |v̂(ξ + ξ′ + iη′)|/q̃(ξ + ξ′)) dξ′dη′) dξ \\ & ≤ C′_{3} ∥v∥′ \end{aligned}  (E.88)

where

∥v∥′ = ∫_{|η|≤ϵ} (∫_{R^{n}} |v̂(ξ + iη)|(q̃(ξ))) dη.  (E.89)

Recall that u ∈ C_{0}^{∞}(R^{n}). Now observe that C′′_{3} is a constant which depends only on ϵ (and, of course, on k and m, the fixed parameters).

Malgrange-Ehrenpreis Theorem

References for Appendix E

[1] Ehrenpreis, Leon; Solutions of some problems of division. Amer. J. Math. 76, 883–903 (1954)

[2] Hörmander, Lars; Local and global properties of fundamental solutions. Math. Scand. 5, 27–39 (1957)

[3] Lewy, Hans; An example of a smooth linear partial differential equation without solutions. Ann. of Math. 66, 155–158 (1957)

[4] Malgrange, B.; Existence et approximation des solutions des équations aux dérivées partielles et des équations de convolution. Ann. Inst. Fourier 6, 271–355 (1955–56).

[5] Reed, Michael and Barry Simon; Methods of Modern Mathematical Physics II: Fourier Analysis, Self-Adjointness (San Diego: Academic Press, Inc., 1975)

[6] Rudin, Walter; Real and Complex Analysis–Third Edition (NY: McGraw-Hill Book Company, 1987)

[7] Yosida, Kôsaku; Functional Analysis—Sixth Edition (Berlin: Springer-Verlag, 1980)