First Semester in Numerical Analysis with Julia, 2020a

More documents

Recommendations

Info

CHAPTER 1. INTRODUCTION 29 the circle, call this number n. Thenn/10000 is approximately the area of the circle. (This approach is known as the Monte Carlo method.) Exercise 1.2-3: Consider the following function f(x, n) = n∑ i∏ x n−j+1 . i=1 j=1 a) Compute f(2, 3) by hand. b) Write a Julia code that computes f. Verify f(2, 3) matches your answer above. 1.3 Computer arithmetic The way computers store numbers and perform computations could surprise the beginner. In Julia if you type ( √ 3) 2 the result will be 2.9....96, where 9 is repeated 15 times. Here are two obvious but fundamental differences in the way computers do arithmetic: • only finitely many numbers can be represented in a computer; • a number represented in a computer can only have finitely many digits. Therefore the numbers that can be represented in a computer exactly is only a subset of rational numbers. Anytime the computer performs an operation whose outcome is not a number that can be represented exactly in the computer, an approximation will replace the exact number. This is called the roundoff error: error produced when a computer is used to perform real number calculations. Floating-point representation of real numbers Here is a general model for representing real numbers in a computer: x = s(.a 1 a 2 ...a t ) β × β e (1.1)
CHAPTER 1. INTRODUCTION 30 where s → sign of x = ±1 e → exponent, with bounds L ≤ e ≤ U (.a 1 ...a t ) β = a 1 β + a 2 β + ... + a t ; the mantissa 2 βt β → base t → number of digits; the precision. In the floating-point representation (1.1), if we specify e in such a way that a 1 ≠0, then the representation will be unique. This is called the normalized floating-point representation. For example if β =10, in the normalized floating-point we would write 0.012 as 0.12 × 10 −1 , instead of choices like 0.012 × 10 0 or 0.0012 × 10. In most computers today, the base is β = 2. Bases 8 and 16 were used in old IBM mainframes in the past. Some handheld calculators use base 10. An interesting historical example is a short-lived computer named Setun developed at Moscow State University which used base 3. There are several choices to make in the general floating-point model (1.1) for the values of s, β, t, e. The IEEE 64-bit floating-point representation is the specific model used in most computers today: x =(−1) s (1.a 2 a 3 ...a 53 ) 2 2 e−1023 . (1.2) Some comments: • Notice how s appears in different forms in equations (1.1) and(1.2). In (1.2), s is either 0 or 1. If s =0,thenx is positive. If s =1,xis negative. • Since β =2, in the normalized floating-point representation of x the first (nonzero) digit after the decimal point has to be 1. Then we do not have to store this number. That’s why we write x as a decimal number starting at 1 in (1.2). Even though precision is t =52, we are able to access up to the 53rd digit a 53 . • The bounds for the exponent are: 0 ≤ e ≤ 2047. We will discuss where 2047 comes from shortly. But first, let’s discuss why we have e − 1023 as the exponent in the representation (1.2), as opposed to simply e (which we had in the representation (1.1)). If the smallest exponent possible was e =0, then the smallest positive number the computer can generate would be (1.00...0) 2 =1: certainly we need the computer to
Page 1 and 2: First Semester in Numerical Analysi
Page 3 and 4: First Semester in Numerical Analysi
Page 5 and 6: Contents 1 Introduction 5 1.1 Revie
Page 7 and 8: Preface This book is based on my le
Page 9 and 10: Chapter 1 Introduction 1.1 Review o
Page 11 and 12: CHAPTER 1. INTRODUCTION 8 Solution.
Page 13 and 14: CHAPTER 1. INTRODUCTION 10 Let’s
Page 15 and 16: CHAPTER 1. INTRODUCTION 12 Out[8]:
Page 17 and 18: CHAPTER 1. INTRODUCTION 14 Press ]
Page 19 and 20: CHAPTER 1. INTRODUCTION 16 Matrix o
Page 21 and 22: CHAPTER 1. INTRODUCTION 18 stdlib/v
Page 23 and 24: CHAPTER 1. INTRODUCTION 20 In [37]:
Page 27 and 28: CHAPTER 1. INTRODUCTION 24 -0.62988
Page 29 and 30: CHAPTER 1. INTRODUCTION 26 5 7 9 He
Page 31: CHAPTER 1. INTRODUCTION 28 Sometime
Page 35 and 36: CHAPTER 1. INTRODUCTION 32 Notice t
Page 37 and 38: CHAPTER 1. INTRODUCTION 34 and real
Page 41 and 42: CHAPTER 1. INTRODUCTION 38 Chopping
Page 43 and 44: CHAPTER 1. INTRODUCTION 40 Machine
Page 45 and 46: CHAPTER 1. INTRODUCTION 42 The abso
Page 47 and 48: CHAPTER 1. INTRODUCTION 44 Next wil
Page 51 and 52: CHAPTER 1. INTRODUCTION 48 Exercise
Page 53 and 54: CHAPTER 1. INTRODUCTION 50 Sources
Page 55 and 56: CHAPTER 2. SOLUTIONS OF EQUATIONS:
Page 83 and 84:
CHAPTER 2. SOLUTIONS OF EQUATIONS:
Page 85 and 86:
Page 87 and 88:
Page 89 and 90:
Page 91 and 92:
Chapter 3 Interpolation In this cha
Page 93 and 94:
CHAPTER 3. INTERPOLATION 90 Here is
Page 95 and 96:
CHAPTER 3. INTERPOLATION 92 basis f
Page 97 and 98:
CHAPTER 3. INTERPOLATION 94 therefo
Page 99 and 100:
CHAPTER 3. INTERPOLATION 96 Summary
Page 101 and 102:
CHAPTER 3. INTERPOLATION 98 and thu
Page 103 and 104:
CHAPTER 3. INTERPOLATION 100 With t
Page 105 and 106:
CHAPTER 3. INTERPOLATION 102 Exampl
Page 107 and 108:
CHAPTER 3. INTERPOLATION 104 Exerci
Page 109 and 110:
CHAPTER 3. INTERPOLATION 106 end en
Page 111 and 112:
CHAPTER 3. INTERPOLATION 108 increa
Page 113 and 114:
CHAPTER 3. INTERPOLATION 110 The ne
Page 115 and 116:
CHAPTER 3. INTERPOLATION 112 there
Page 117 and 118:
CHAPTER 3. INTERPOLATION 114 Figure
Page 119 and 120:
CHAPTER 3. INTERPOLATION 116 Then t
Page 121 and 122:
Page 123 and 124:
CHAPTER 3. INTERPOLATION 120 Exerci
Page 125 and 126:
CHAPTER 3. INTERPOLATION 122 yi yi+
Page 127 and 128:
CHAPTER 3. INTERPOLATION 124 • Cl
Page 129 and 130:
CHAPTER 3. INTERPOLATION 126 In [2]
Page 131 and 132:
CHAPTER 3. INTERPOLATION 128 end fo
Page 133 and 134:
Page 135 and 136:
CHAPTER 3. INTERPOLATION 132 end u=
Page 137 and 138:
CHAPTER 3. INTERPOLATION 134 Especi
Page 139 and 140:
CHAPTER 3. INTERPOLATION 136 Arya a
Page 141 and 142:
CHAPTER 3. INTERPOLATION 138 This l
Page 143 and 144:
CHAPTER 3. INTERPOLATION 140 Recrea
Page 145 and 146:
CHAPTER 4. NUMERICAL QUADRATURE AND
Page 147 and 148:
Page 149 and 150:
Page 151 and 152:
Page 153 and 154:
Page 155 and 156:
Page 157 and 158:
Page 159 and 160:
Page 161 and 162:
Page 163 and 164:
Page 165 and 166:
Page 167 and 168:
Page 169 and 170:
Page 171 and 172:
Page 173 and 174:
Page 175 and 176:
Page 177 and 178:
Page 179 and 180:
Page 181 and 182:
Chapter 5 Approximation Theory 5.1
Page 183 and 184:
CHAPTER 5. APPROXIMATION THEORY 180
Page 185 and 186:
Page 187 and 188:
Page 189 and 190:
Page 191 and 192:
Page 193 and 194:
Page 195 and 196:
Page 197 and 198:
Page 199 and 200:
Page 201 and 202:
Page 203 and 204:
Page 205 and 206:
Page 207 and 208:
Page 209 and 210:
Page 211 and 212:
Page 213 and 214:
Page 215 and 216:
Page 217 and 218:
Page 219 and 220:
Page 221 and 222:
Index Absolute error, 38 Beasley-Sp
Page 223 and 224:
INDEX 220 Chebyshev, 203 Julia code
show all

First Semester in Numerical Analysis with Julia, 2020a

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?