Contents
PREFACE xv
ACKNOWLEDGMENTS xix
Section I Preliminaries
Chapter 1
Mathematics Review 3
1.1 PRELIMINARIES: NUMBERS AND SETS 3
1.2 VECTOR SPACES 4
1.2.1 Defining Vector Spaces 4
1.2.2 Span, Linear Independence, and Bases 5
1.2.3 Our Focus: R
n
7
1.3 LINEARITY 9
1.3.1 Matrices 10
1.3.2 Scalars, Vectors, and Matrices 12
1.3.3 Matrix Storage and Multiplication Methods 13
1.3.4 Model Problem: A~x =
~
b 14
1.4 NON-LINEARITY: DIFFERENTIAL CALCULUS 15
1.4.1 Differentiation in One Variable 16
1.4.2 Differentiation in Multiple Variables 17
1.4.3 Optimization 20
1.5 EXERCISES 23
Chapter 2
Numerics and Error Analysis 27
2.1 STORING NUMBERS WITH FRACTIONAL PARTS 27
2.1.1 Fixed-Point Representations 28
2.1.2 Floating-Point Representations 29
2.1.3 More Exotic Options 31
2.2 UNDERSTANDING ERROR 32
2.2.1 Classifying Error 33
2.2.2 Conditioning, Stability, and Accuracy 35
2.3 PRACTICAL ASPECTS 36
2.3.1 Computing Vector Norms 37
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