Numerical calculations in Clifford algebra : a practical guide for engineers and scientists /: a practical guide for engineers and scientists. (2023)

• Record Type: Book
• Further Information: Note: Andrew Seagar.
• Authors: Seagar, Andrew
• Contents: Preface xix I Entities and Operations 1 Introduction 3 1.1 Operations 3 1.2 History 4 1.3 Alternative Forms 4 1.4 Naming 5 1.5 Structure 6 1.5.1 Algebraic 6 1.5.2 Numeric 6 1.6 Entities 9 2 Input 11 2.1 Syntax 11 2.2 Constants 12 2.2.1 Specific Types 12 2.2.2 General 13 2.3 Variables 15 2.3.1 Checking and Converting 15 3 Output 21 3.1 Tree Format 21 3.2 Numeric Formats 24 3.2.1 Default Format 24 3.2.2 Defined Format 25 3.3 Extended Formats 26 3.3.1 Rounding 26 v vi c Seagar/Wiley, September 16, 2022 Contents 3.3.2 Parts of Coefficients 27 3.4 Selected Components 28 3.5 Primitive Formats 29 3.6 Recovered Values 30 4 Unary Operations 33 4.1 Theory 33 4.1.1 Negation 33 4.1.2 Involution 34 4.1.3 Pair Exchange 34 4.1.4 Reversion 35 4.1.5 Clifford Conjugation 36 4.1.6 Supplementation and Pseudoscalar 36 4.2 Practice 37 4.2.1 Example Code 37 4.2.2 Example Output 38 5 Binary Operations 41 5.1 Geometric Origins 41 5.1.1 Outer Multiplication 41 5.1.2 Orthogonal Components 43 5.1.3 Inner Multiplication 44 5.1.4 Names 45 5.2 Multiplication of Units 46 5.2.1 Progressive and Regressive Multiplication 46 5.2.2 Outer, Inner and Central Multiplication 47 5.2.3 Multiplication by Scalars 49 5.3 Central Multiplication 49 5.3.1 Primal Units 50 5.3.2 Evolved and Other Units 51 5.3.3 Numbers 52 5.4 Practice 53 5.4.1 Example Code 53 5.4.2 Example Output 54 5.4.3 Multiplication Tables 54 Contents c Seagar/Wiley, September 16, 2022 vii 6 Vectors and Geometry 59 6.1 Theory 59 6.1.1 Magnitude 59 6.1.2 Inverse 60 6.1.3 Reflection 60 6.1.4 Projection 60 6.1.5 Rotation 61 6.2 Practice 61 6.2.1 Example Code 61 6.2.2 Example Output 63 7 Quaternions 65 7.1 Theory 65 7.1.1 Magnitude 65 7.1.2 Inverse 66 7.1.3 Reflection and Projection 66 7.1.4 Rotation 66 7.1.5 Intersection 67 7.1.6 Factorisation 67 7.2 Practice 68 7.2.1 Example Code 68 7.2.2 Example Output 70 8 Pauli Matrices 73 8.1 Theory 73 8.1.1 Recovery of Components 73 8.1.2 Magnitude 74 8.1.3 Inverse 74 8.1.4 Reflection, Projection and Rotation 74 8.2 Practice 74 8.2.1 Example Code 74 8.2.2 Example Output 76 9 Bicomplex Numbers 79 9.1 Theory 79 9.1.1 Conjugate 79 9.1.2 Magnitude 80 viii c Seagar/Wiley, September 16, 2022 Contents 9.1.3 Inverse 80 9.1.4 Reflection, Projection, and Rotation 80 9.2 Practice 81 9.2.1 Example Code 81 9.2.2 Example Output 82 10 Electromagnetic Fields 85 10.1 Theory 85 10.1.1 Time and Frequency 85 10.1.2 Electromagnetic Entities 85 10.1.3 Dirac Operators 86 10.1.4 Maxwell’s Equations 86 10.1.5 Simplified Notation 87 10.1.6 Magnitude 87 10.1.7 Inverse 87 10.1.8 Reflection 88 10.1.9 Projection 88 10.1.10 Rotation 88 10.2 Practice 88 10.2.1 Example Code 88 10.2.2 Example Output 90 10.3 Field Arithmetic 92 10.3.1 Extensions based on Quaternions 92 10.3.2 Inverses 92 10.3.3 Example Code 94 10.3.4 Example Output 95 11 Arrays of Clifford Numbers 97 11.1 Theory 97 11.2 Practice 98 11.2.1 Example Code 98 11.2.2 Example Output 100 12 Power Series 103 12.1 Theory 103 12.1.1 User Defined 103 12.1.2 Predefined 104 Contents c Seagar/Wiley, September 16, 2022 ix 12.1.3 Convergence 105 12.1.4 Factorisation 105 12.1.5 Squaring 106 12.2 Practice 106 12.2.1 User Defined 106 12.2.2 Predefined 109 13 Matrices of Clifford Numbers 121 13.1 Background 121 13.2 Inversion 122 13.3 Practice 123 13.3.1 Example Code 123 13.3.2 Example Output 126 II Customisation 14 Memory 133 14.1 Memory Usage 133 14.2 Examples 134 14.2.1 Memory Tree Sparsity 134 14.2.2 Memory Expansion 138 14.2.3 Memory Recycling 139 15 Errors 143 15.1 User Errors 143 15.1.1 Syntax Errors and Messages 146 15.2 System Errors 146 15.3 Recovery 146 15.4 Beneficial Usage 150 16 Extension 155 16.1 Accumulation 155 16.2 Multiplication 156 16.3 Transformation 158 16.4 Filtration 159 x c Seagar/Wiley, September 16, 2022 Contents III Application 17 Verification 165 17.1 Identities 165 17.2 Tests 165 17.2.1 Example Code 165 17.2.2 Example Output 167 18 Lines Not Parallel 173 18.1 Theory 173 18.1.1 Common Plane 173 18.1.2 No Plane in Common 175 18.2 Practice 177 18.2.1 Example Code 177 18.2.2 Example Output 180 19 Perspective Projection 181 19.1 Theory 181 19.2 Practice 182 19.2.1 Example Code 182 19.2.2 Example Output 185 20 Linear Systems 187 20.1 Theory 187 20.2 Practice 188 20.2.1 Example Code 188 20.2.2 Example Output 190 21 Fast Fourier Transform 191 21.1 Theory 191 21.2 Practice 192 21.2.1 Example Code 192 21.2.2 Example Output 196 22 Hertzian Dipole 199 22.1 Theory 199 22.2 Practice 200 22.2.1 Example Code 200 22.2.2 Example Output 203 Contents c Seagar/Wiley, September 16, 2022 xi 23 Finite Difference Time Domain 207 23.1 Theory 207 23.1.1 Analytical Solution 207 23.1.2 Series Solution 207 23.1.3 Analytical Example 208 23.1.4 Numerical Derivatives 209 23.2 Practice 210 23.2.1 Example Code 210 23.2.2 Example Output 215 24 Cauchy Extension 221 24.1 Background 221 24.2 Theory 221 24.2.1 Two Dimensions 221 24.2.2 Three Dimensions 222 24.2.3 Singularity 222 24.2.4 The Taming Function 223 24.2.5 Construction 223 24.3 Practice 225 24.3.1 Example Code 225 24.3.2 Example Output 228 25 Electromagnetic Scattering 233 25.1 Background 233 25.2 Theory 234 25.3 Practice 235 25.3.1 Example Code 235 25.3.2 Example Output 236 IV Programming 26 Interfaces 243 26.1 Configuration and Observation 243 26.1.1 Management 243 26.1.2 Printing 244 26.2 Simple Entities 245 26.2.1 Units 245 xii c Seagar/Wiley, September 16, 2022 Contents 26.2.2 Components 245 26.2.3 Numbers 247 26.3 Higher Entities 249 26.3.1 Vectors 249 26.3.2 Bicomplex Numbers 250 26.3.3 Quatern
• Edition: 1st
• Publisher Details: Hoboken : John Wiley & Sons, Inc
• Publication Date: 2023
• Extent: 1 online resource (496 pages)
• Subjects: 512.57; Clifford algebras; Algebra -- Data processing
• Languages: English
• ISBNs: 9781394173259
• Notes: Note: Description based on CIP data; resource not viewed.
• Access Rights: Legal Deposit; Only available on premises controlled by the deposit library and to one user at any one time; The Legal Deposit Libraries (Non-Print Works) Regulations (UK).
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