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© George W. Collins, II 2003
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/> Table of Contents

List of Figures .....................................................................................................................................vi

List of Tables.......................................................................................................................................ix

Preface
.............................................................................................................................
xi
Notes to the Internet Edition

2.3 Solution of Linear Equations by Iterative Methods ................................................. 39
a. Solution by The Gauss and Gauss-Seidel Iteration Methods ...................... 39
b. The Method of Hotelling and Bodewig ..................................................... 41
c. Relaxation Methods for the Solution of Linear Equations.......................... 44
d. Convergence and Fixed-point Iteration Theory........................................... 46

2.4 The Similarity Transformations and the Eigenvalues and Vectors of a
Matrix ........................................................................................................................ 48

i

Chapter 2 Exercises ............................................................................................................... 53

Chapter 2 References and Supplemental Reading................................................................ 54 3. Polynomial Approximation, Interpolation, and Orthogonal Polynomials................... 55

3.1 Polynomials and Their Roots.................................................................................... 56
a. Some Constraints on the Roots of Polynomials........................................... 57
b. Synthetic Division......................................................................................... 58
c. The Graffe Root-Squaring Process .............................................................. 60
d. Iterative Methods .......................................................................................... 61

3.2 Curve Fitting and Interpolation................................................................................. 64
a. Lagrange Interpolation ................................................................................. 65
b. Hermite Interpolation.................................................................................... 72

4.3 Monte Carlo Integration Schemes and Other Tricks...............................................115
a. Monte Carlo Evaluation of Integrals...........................................................115
b. The General Application of Quadrature Formulae to Integrals .................117

Chapter 4 Exercises .............................................................................................................119

Chapter 4 References and Supplemental Reading...............................................................120 5. Numerical Solution of Differential and Integral Equations ..........................................121

5.1 The Numerical Integration of Differential Equations .............................................122
a. One Step Methods of the Numerical Solution of Differential
Equations......................................................................................................123
b. Error Estimate and Step Size Control .........................................................131
c. Multi-Step and Predictor-Corrector Methods .............................................134
d. Systems of Differential Equations and Boundary Value
Problems.......................................................................................................138
e. Partial Differential Equations ......................................................................146

5.2 The Numerical Solution of Integral Equations........................................................ 147
a. Types of Linear Integral Equations.............................................................148
b. The Numerical Solution of Fredholm Equations........................................148
c. The Numerical Solution of Volterra Equations ..........................................150
d. The Influence of the Kernel on the Solution...............................................154

Chapter 5 Exercises ..............................................................................................................156

a. The Method of Steepest Descent.................................................................183
b. Linear approximation of f(a
j
,x) ...................................................................184
c. Errors of the Least Squares Coefficients.....................................................186

6.5 Other Approximation Norms ...................................................................................187
a. The Chebyschev Norm and Polynomial Approximation ...........................188
b. The Chebyschev Norm, Linear Programming, and the Simplex
Method .........................................................................................................189
c. The Chebyschev Norm and Least Squares .................................................190

Chapter 6 Exercises ..............................................................................................................192

Chapter 6 References and Supplementary Reading.............................................................194 7. Probability Theory and Statistics .....................................................................................197

7.1 Basic Aspects of Probability Theory .......................................................................200
a. The Probability of Combinations of Events................................................201
b. Probabilities and Random Variables...........................................................202
c. Distributions of Random Variables.............................................................203

7.2 Common Distribution Functions .............................................................................204
a. Permutations and Combinations..................................................................204
b. The Binomial Probability Distribution........................................................205
c. The Poisson Distribution .............................................................................206
d. The Normal Curve .......................................................................................207
e. Some Distribution Functions of the Physical World ..................................210

a. The "Students" t-Test...................................................................................232
b. The χ
2
-test ....................................................................................................233
c. The F-test .....................................................................................................234
d. Kolmogorov-Smirnov Tests ........................................................................235

8.3 Linear Regression, and Correlation Analysis..........................................................237
a. The Separation of Variances and the Two-Variable Correlation
Coefficient....................................................................................................238
b. The Meaning and Significance of the Correlation Coefficient ..................240
c. Correlations of Many Variables and Linear Regression ............................242
d Analysis of Variance....................................................................................243

8.4 The Design of Experiments .....................................................................................246
a. The Terminology of Experiment Design ....................................................249
b. Blocked Designs ..........................................................................................250
c. Factorial Designs .........................................................................................252

Chapter 8 Exercises ...........................................................................................................255

Chapter 8 References and Supplemental Reading .............................................................257

Index......................................................................................................................................257 v

schematically shows the curl of a vector field. The direction of the curl is
determined by the "right hand rule" while the magnitude depends on the rate of
change of the x- and y-components of the vector field with respect to y and x. ................. 19

Figure 1.5
schematically shows the gradient of the scalar dot-density in the form of a
number of vectors at randomly chosen points in the scalar field. The direction of
the gradient points in the direction of maximum increase of the dot-density,
while the magnitude of the vector indicates the rate of change of that density. . ................ 20

Figure 3.1
depicts a typical polynomial with real roots. Construct the tangent to the
curve at the point x
k
and extend this tangent to the x-axis. The crossing point
x
k+1
represents an improved value for the root in the Newton-Raphson
algorithm. The point x
k-1
can be used to construct a secant providing a second
method for finding an improved value of x. ......................................................................... 62

Figure 3.2
shows the behavior of the data from Table 3.1. The results of various forms
of interpolation are shown. The approximating polynomials for the linear and
parabolic Lagrangian interpolation are specifically displayed. The specific
results for cubic Lagrangian interpolation, weighted Lagrangian interpolation
and interpolation by rational first degree polynomials are also indicated. ......................... 69


compares the discrete Fourier transform of the function e
-│x│
with the
continuous transform for the full infinite interval. The oscillatory nature of the
discrete transform largely results from the small number of points used to
represent the function and the truncation of the function at t =
±
2. The only
points in the discrete transform that are even defined are denoted by ...............................173

Figure 6.2
shows the parameter space defined by the
φ
j
(x)'s. Each f(a
j
,x
i
) can be
represented as a linear combination of the
φ
j
(x
i
) where the a
j
are the coefficients
of the basis functions. Since the observed variables Y
i
cannot be expressed in


Figure 6.5
shows the parameter space for fitting three points with a straight line under
the Chebyschev norm. The equations of condition denote half-planes which
satisfy the constraint for one particular point.......................................................................189

Figure 7.1
shows a sample space giving rise to events E and F. In the case of the die, E
is the probability of the result being less than three and F is the probability of
the result being even. The intersection of circle E with circle F represents the
probability of E and F [i.e. P(EF)]. The union of circles E and F represents the
probability of E or F. If we were to simply sum the area of circle E and that of
F we would double count the intersection. ..........................................................................202 vii
Figure 7.2
shows the normal curve approximation to the binomial probability
distribution function. We have chosen the coin tosses so that p = 0.5. Here µ
and σ can be seen as the most likely value of the random variable x and the
'width' of the curve respectively. The tail end of the curve represents the region
approximated by the Poisson distribution............................................................................209

Figure 7.3
shows the mean of a function f(x) as <x>. Note this is not the same as the
most likely value of x as was the case in figure 7.2. However, in some real
sense σ is still a measure of the width of the function. The skewness is a


Figure 8.3
shows the probability density distribution function for the F-statistic with
values of N
1
= 3 and N
2
= 5 respectively. Also plotted are the limiting
distribution functions f(χ
2
/N
1
) and f(t
2
). The first of these is obtained from f(F)
in the limit of N
2
→

∞
. The second arises when N
1
≥
1. One can see the tail of
the f(t
2
) distribution approaching that of f(F) as the value of the independent
variable increases. Finally, the normal curve which all distributions approach
for large values of N is shown with a mean equal to F  and a variance equal to the
variance for f(F). ...................................................................................................................220


Table 3.1
Sample Data and Results for Lagrangian Interpolation Formulae .......................... 67

Table 3.2
Parameters for the Polynomials Generated by Neville's Algorithm........................ 71

Table 3.3
A Comparison of Different Types of Interpolation Formulae................................. 79

Table 3.4
Parameters for Quotient Polynomial Interpolation .................................................. 83

Table 3.5
The First Five Members of the Common Orthogonal Polynomials ........................ 90

Table 3.6
Classical Orthogonal Polynomials of the Finite Interval ......................................... 91

Table 4.1
A Typical Finite Difference Table for f(x) = x
2
........................................................99

Table 4.2
Types of Polynomials for Gaussian Quadrature .....................................................110

Table 4.3
Sample Results for Romberg Quadrature................................................................112


ix
Table 7.2
Examination Statistics for the Sample Test.............................................................215

Table 8.1
Sample Beach Statistics for Correlation Example ..................................................241

Table 8.2
Factorial Combinations for Two-level Experiments with n=2-4............................253 • • •

The origins of this book can be found years ago when I was
a doctoral candidate working on my thesis and finding that I needed numerical tools that I should have
been taught years before. In the intervening decades, little has changed except for the worse. All fields
of science have undergone an information explosion while the computer revolution has steadily and
irrevocability been changing our lives. Although the crystal ball of the future is at best "seen through a
glass darkly", most would declare that the advent of the digital electronic computer will change
civilization to an extent not seen since the coming of the steam engine. Computers with the power that
could be offered only by large institutions a decade ago now sit on the desks of individuals. Methods of
analysis that were only dreamed of three decades ago are now used by students to do homework
exercises. Entirely new methods of analysis have appeared that take advantage of computers to perform
logical and arithmetic operations at great speed. Perhaps students of the future may regard the
multiplication of two two-digit numbers without the aid of a calculator in the same vein that we regard
the formal extraction of a square root. The whole approach to scientific analysis may change with the
advent of machines that communicate orally. However, I hope the day never arrives when the
investigator no longer understands the nature of the analysis done by the machine.

Unfortunately instruction in the uses and applicability of new methods of analysis rarely
appears in the curriculum. This is no surprise as such courses in any discipline always are the last to be
developed. In rapidly changing disciplines this means that active students must fend for themselves.
With numerical analysis this has meant that many simply take the tools developed by others and apply
them to problems with little knowledge as to the applicability or accuracy of the methods. Numerical

expression. The ability to use mathematics in such a fashion is largely what I mean by "mathematical
sophistication". However, this book is primarily intended for scientists and engineers so while there is a
certain familiarity with mathematics that is assumed, the rigor that one expects with a formal
mathematical presentation is lacking. Very little is proved in the traditional mathematical sense of the
word. Indeed, derivations are resorted to mainly to emphasize the assumptions that underlie the results.
However, when derivations are called for, I will often write several forms of the same expression on the
same line. This is done simply to guide the reader in the direction of a mathematical development. I will
often give "rules of thumb" for which there is no formal proof. However, experience has shown that
these "rules of thumb" almost always apply. This is done in the spirit of providing the researcher with
practical ways to evaluate the validity of his or her results. The basic premise of this book is that it can serve as the basis for a wide range of courses that
discuss numerical methods used in science. It is meant to support a series of lectures, not replace them.
To reflect this, the subject matter is wide ranging and perhaps too broad for a single course. It is
expected that the instructor will neglect some sections and expand on others. For example, the social
scientist may choose to emphasize the chapters on interpolation, curve-fitting and statistics, while the
physical scientist would stress those chapters dealing with numerical quadrature and the solution of
differential and integral equations. Others might choose to spend a large amount of time on the principle
of least squares and its ramifications. All these approaches are valid and I hope all will be served by this
book. While it is customary to direct a book of this sort at a specific pedagogic audience, I find that task
somewhat difficult. Certainly advanced undergraduate science and engineering students will have no
difficulty dealing with the concepts and level of this book. However, it is not at all obvious that second
year students couldn't cope with the material. Some might suggest that they have not yet had a formal
course in differential equations at that point in their career and are therefore not adequately prepared.
However, it is far from obvious to me that a student’s first encounter with differential equations should
be in a formal mathematics course. Indeed, since most equations they are liable to encounter will require
a numerical solution, I feel the case can be made that it is more practical for them to be introduced to the
subject from a graphical and numerical point of view. Thus, if the instructor exercises some care in the
presentation of material, I see no real barrier to using this text at the second year level in some areas. In

general collections compiled by users should be preferred for they have at least been screened initially
for efficacy.

Chapter 6 is a lengthy treatment of the principle of least squares and associated topics. I have
found that algorithms based on least squares are among the most widely used and poorest understood of
all algorithms in the literature. Virtually all students have encountered the concept, but very few see and
understand its relationship to the rest of numerical analysis and statistics. Least squares also provides a
logical bridge to the last chapters of the book. Here the huge field of statistics is surveyed with the hope
of providing a basic understanding of the nature of statistical inference and how to begin to use
statistical analysis correctly and with confidence. The foundation laid in Chapter 7 and the tests
presented in Chapter 8 are not meant to be a substitute for a proper course of study in the subject.
However, it is hoped that the student unable to fit such a course in an already crowded curriculum will
at least be able to avoid the pitfalls that trap so many who use statistical analysis without the appropriate
care.

Throughout the book I have tried to provide examples integrated into the text of the more
difficult algorithms. In testing an earlier version of the book, I found myself spending most of my time
with students giving examples of the various techniques and algorithms. Hopefully this initial
shortcoming has been overcome. It is almost always appropriate to carry out a short numerical example
of a new method so as to test the logic being used for the more general case. The problems at the end of
each chapter are meant to be generic in nature so that the student is not left with the impression that this
algorithm or that is only used in astronomy or biology. It is a fairly simple matter for an instructor to
find examples in diverse disciplines that utilize the techniques discussed in each chapter. Indeed, the
student should be encouraged to undertake problems in disciplines other than his/her own if for no other
reason than to find out about the types of problems that concern those disciplines. xiii
use inefficient methods and still obtain answers in a timely fashion. However, with the avalanche of
data about to descend on more and more fields, it does not seem unreasonable to suppose that
numerical tasks will overtake computing power and there will again be a need for efficient and
accurate algorithms to solve problems. I suspect that many of the techniques described herein will be
rediscovered before the new century concludes. Perhaps efforts such as this will still find favor with
those who wish to know if numerical results can be believed.

George W. Collins, II
January 30, 2001

xiv
xv

A Further Note for the Internet Edition Since I put up a version of this book two years ago, I have found numerous errors which
largely resulted from the generations of word processors through which the text evolved. During the
last effort, not all the fonts used by the text were available in the word processor and PDF translator.
This led to errors that were more wide spread that I realized. Thus, the main force of this effort is to
bring some uniformity to the various software codes required to generate the version that will be

Internet, I encourage anyone who is interested to down load the PDF files as they may be of use to
them. I would only request that they observe the courtesy of proper attribution should they find my
efforts to be of use. George W. Collins, II
April, 2003
Case Western Reserve University 1 Introduction and
Fundamental Concepts
• • •
The numerical expression of a scientific statement has traditionally
been the manner by which scientists have verified a theoretical description of the physical world. During this

The extreme speed of contemporary machines has tremendously expanded the scope of numerical
problems that may be considered as well as the manner in which such computational problems may even be
approached. However, this expansion of the degree and type of problem that may be numerically solved has
removed the scientist from the details of the computation. For this, most would shout "Hooray"! But this
removal of the investigator from the details of computation may permit the propagation of errors of various
types to intrude and remain undetected. Modern computers will almost always produce numbers, but
whether they represent the solution to the problem or the result of error propagation may not be obvious.
This situation is made worse by the presence of programs designed for the solution of broad classes of
problems. Almost every class of problems has its pathological example for which the standard techniques
will fail. Generally little attention is paid to the recognition of these pathological cases which have an
uncomfortable habit of turning up when they are least expected.

Thus the contemporary scientist or engineer should be skeptical of the answers presented by the
modern computer unless he or she is completely familiar with the numerical methods employed in obtaining
that solution. In addition, the solution should always be subjected to various tests for "reasonableness".
There is often a tendency to regard the computer and the programs which they run as "black boxes" from
which come infallible answers. Such an attitude can lead to catastrophic results and belies the attitude of
"healthy skepticism" that should pervade all science. It is necessary to understand, at least at some level,
what the "Black Boxes" do. That understanding is one of the primary aims of this book.

It is not my intention to teach the techniques of programming a computer. There are many excellent
texts on the multitudinous languages that exist for communicating with a computer. I will assume that the
reader has sufficient capability in this area to at least conceptualize the manner by which certain processes
could be communicated to the computer or at least recognize a computer program that does so. However, the
programming of a computer does represent a concept that is not found in most scientific or mathematical
presentations. We will call that concept an algorithm. An algorithm is simply a sequence of mathematical
operations which, when preformed in sequence, lead to the numerical answer to some specified problem.
Much time and effort is devoted to ascertaining the conditions under which a particular algorithm will work.
In general, we will omit the proof and give only the results when they are known. The use of algorithms and
the ability of computers to carry out vastly more operations in a short interval of time than the human

a problem thoroughly before any numerical solution is attempted. Very often a better numerical approach
will suggest itself during the analyses and occasionally one may find that the answer has a closed form
analytic solution and a numerical solution is unnecessary.

However, it is too easy to say "I don't have the background for this subject" and thereby never
attempt to learn it. The complete study of mathematics is too vast for anyone to acquire in his or her lifetime.
Scientists simply develop a base and then continue to add to it for the rest of their professional lives. To be a
successful scientist one cannot know too much mathematics. In that spirit, we shall "review" some
mathematical concepts that are useful to understanding numerical methods and analysis. The word review
should be taken to mean a superficial summary of the area mainly done to indicate the relation to other areas.
Virtually every area mentioned has itself been a subject for many books and has occupied the study of some
investigators for a lifetime. This short treatment should not be construed in any sense as being complete.
Some of this material will indeed be viewed as elementary and if thoroughly understood may be skimmed.
However many will find some of these concepts as being far from elementary. Nevertheless they will sooner
or later be useful in understanding numerical methods and providing a basis for the knowledge that
mathematics is "all of a piece".

1.1 Basic Properties of Sets and Groups

Most students are introduced to the notion of a set very early in their educational experience.
However, the concept is often presented in a vacuum without showing its relation to any other area of
mathematics and thus it is promptly forgotten. Basically a set is a collection of elements. The notion of an
element is left deliberately vague so that it may represent anything from cows to the real numbers. The
number of elements in the set is also left unspecified and may or may not be finite. Just over a century ago
Georg Cantor basically founded set theory and in doing so clarified our notion of infinity by showing that
there are different types of infinite sets. He did this by generalizing what we mean when we say that two sets
have the same number of elements. Certainly if we can identify each element in one set with a unique
element in the second set and there are none left over when the identification is completed, then we would be
entitled in saying that the two sets had the same number of elements. Cantor did this formally with the
infinite set composed of the positive integers and the infinite set of the real numbers. He showed that it is not

a‡i = a . (1.1.2)
This suggests another useful constraint, namely that there are elements in the set that can be designated
"inverses". An inverse of an element is one that when combined with its element under the law produces the
unit element or
a
-1
‡a = i . (1.1.3)

Now with one further restriction on the law itself, we will have all the conditions required to
produce a group. The restriction is known as associativity. A law is said to be associative if the order in
which it is applied to three elements does not determine the outcome of the application. Thus

(a‡b)‡c = a‡(b‡c) . (1.1.4)

If a set possess a unit element and inverse elements and is closed under an associative law, that set is called a
group under the law. Therefore the normal integers form a group under addition. The unit is zero and the
inverse operation is clearly subtraction and certainly the addition of any two integers produces another
integer. The law of addition is also associative. However, it is worth noting that the integers do not form a
group under multiplication as the inverse operation (reciprocal) does not produce a member of the group (an
integer). One might think that these very simple constraints would not be sufficient to tell us much that is
new about the set, but the notion of a group is so powerful that an entire area of mathematics known as group
theory has developed. It is said that Eugene Wigner once described all of the essential aspects of the
thermodynamics of heat transfer on one sheet of paper using the results of group theory.

While the restrictions that enable the elements of a set to form a group are useful, they are not the
only restrictions that frequently apply. The notion of commutivity is certainly present for the laws of
addition and scalar multiplication and, if present, may enable us to say even more about the properties of our
set. A law is said to be communitative if
a‡b = b‡a . (1.1.5)
A further restriction that may be applied involves two laws say ‡ and ∧. These laws are said to be

careful not to allow division by zero (often known as the cancellation law) such scalars form not only
groups, but also fields.

Although one can generally describe the condition of the atmosphere locally in terms of scalar
fields, the location itself requires more than a single scalar for its specification. Now we need two (three if
we include altitude) numbers, say the latitude and longitude, which locate that part of the atmosphere for
further description by scalar fields. A quantity that requires more than one number for its specification may
be called a vector. Indeed, some have defined a vector as an "ordered n-tuple of numbers". While many may
not find this too helpful, it is essentially a correct statement, which emphasizes the multi-component side of
the notion of a vector. The number of components that are required for the vector's specification is usually
called the dimensionality of the vector. We most commonly think of vectors in terms of spatial vectors, that
is, vectors that locate things in some coordinate system. However, as suggested in the previous section,
vectors may represent such things as an electric or magnetic field where the quantity not only has a
magnitude or scalar length associated with it at every point in space, but also has a direction. As long as such
quantities obey laws of addition and some sort of multiplication, they may indeed be said to form vector
fields. Indeed, there are various types of products that are associated with vectors. The most common of
these and the one used to establish the field nature of most physical vector fields is called the "scalar
product" or inner product, or sometimes simply the dot product from the manner in which it is usually
written. Here the result is a scalar and we can operationally define what we mean by such a product by
G
G
∑
==•
i
ii
BAcBA
. (1.2.1)
One might say that as the result of the operation is a scalar not a vector, but that would be to put to restrictive
an interpretation on what we mean by a vector. Specifically, any scalar can be viewed as vector having only
one component (i.e. a 1-dimensional vector). Thus scalars become a subgroup of vectors and since the vector

kji
kji
−+−−−==×
G
G
. (1.2.2)
The result of this operation is a vector, but we shall see later that it will be useful to sharpen our definition of
vectors so that this result is a special kind of vector.

Finally, there is the "tensor product" or vector outer product that is defined as
G
G





=
=
jiij
BAC
BA C
. (1.2.3)
Here the result of applying the "law" is an ordered array of (n×m) numbers where n and m are the
dimensions of the vectors
A
G
and
B
G







=δ
10
01
ij
. (1.2.5)
The quantity δ
ij
is called the Kronecker delta and may be generalized to n-dimensions.

Thus the inverse elements of the group will have to satisfy the relation

AA
-1
= 1 , (1.2.6)

and we shall spend some time in the next chapter discussing how these members of the group may be
calculated. Since matrix addition can simply be defined as the scalar addition of the elements of the matrix,
1
@
Fundamental Concepts


ji
. If, in addition, the elements are themselves complex numbers,
then should the elements of the transpose be the complex conjugates of the original matrix, the matrix is said
to be Hermitian or self-adjoint. The conjugate transpose of a matrix A is usually denoted by A
†
. If the
Hermitian conjugate of A is also A
-1
, then the matrix is said to be unitary. Should the matrix A commute
with it Hermitian conjugate so that
AA
†
= A
†
A , (1.2.9)
then the matrix is said to be normal. For matrices with only real elements, Hermitian is the same as
symmetric, unitary means the same as orthonormal and both classes would be considered to be normal.

Finally, a most important characteristic of a matrix is its determinant. It may be calculated by
expansion of the matrix by "minors" so that
)aaaa(a)aaaa(a)aaaa(a
aaa
aaa
aaa
A det
132232211331233321123223332211
332313
232221
131211
−+−−−==

column is added to another.

8. The determinant of the product of two matrices is the product of the determinants of
the two matrices.

One of the important aspects of the determinant is that it is a single parameter that can be used to
characterize the matrix. Any such single parameter (i.e. the sum of the absolute value of the elements) can be
so used and is often called a matrix norm. We shall see that various matrix norms are useful in determining
which numerical procedures will be useful in operating on the matrix. Let us now consider a broader class of
objects that include scalars, vectors, and to some extent matrices. 1.3 Coordinate Systems and Coordinate Transformations

There is an area of mathematics known as topology, which deals with the description of spaces. To
most students the notion of a space is intuitively obvious and is restricted to the three dimensional Euclidian
space of every day experience. A little reflection might persuade that student to include the flat plane as an
allowed space. However, a little further generalization would suggest that any time one has several
independent variables that they could be used to form a space for the description of some phenomena. In the
area of topology the notion of a space is far more general than that and many of the more exotic spaces have
no known counterpart in the physical world.

We shall restrict ourselves to spaces of independent variables, which generally have some physical
interpretation. These variables can be said to constitute a coordinate frame, which describes the space and are
fairly high up in the hierarchy of spaces catalogued by topology. To understand what is meant by a
coordinate frame, imagine a set of rigid rods or vectors all connected at a point. We shall call such a
collection of rods a reference frame. If every point in space can be projected onto the rods so that a unique
set of rod-points represent the space point, the vectors are said to span the space.

If the vectors that define the space are locally perpendicular, they are said to form an orthogonal