Numerical Calculations in Clifford Algebra: A Practical Guide for Engineers and Scientists

ISBN-13: 9781394173242 / Angielski / Inna / 2023

Seagar, Andrew

Spis treści

List of Figures xvList of Tables xixPreface xxiPart I Entities and Operations 11 Introduction 31.1 Operations 31.2 History 41.3 Alternative Forms 51.4 Naming 61.5 Structure 71.5.1 Algebraic 71.5.2 Numeric 81.6 Entities 11References 122 Input 132.1 Syntax 132.2 Constants 142.2.1 Specific Types 142.2.2 General 162.3 Variables 192.3.1 Checking and Converting 19Reference 233 Output 253.1 Tree Format 263.2 Numeric Formats 293.2.1 Default Format 293.2.2 Defined Format 313.3 Extended Formats 323.3.1 Rounding 323.3.2 Parts of Coefficients 333.4 Selected Components 353.5 Primitive Formats 363.6 Recovered Values 384 Unary Operations 414.1 Theory 414.1.1 Negation 414.1.2 Involution 414.1.3 Pair Exchange 424.1.4 Reversion 434.1.5 Clifford Conjugation 444.1.6 Supplementation and Pseudo-scalar 444.2 Practice 454.2.1 Example Code 454.2.2 Example Output 475 Binary Operations 495.1 Geometric Origins 495.1.1 Outer Multiplication 495.1.2 Orthogonal Components 525.1.3 Inner Multiplication 535.1.4 Names 545.2 Multiplication of Units 555.2.1 Progressive and Regressive Multiplication 555.2.2 Outer, Inner, and Central Multiplication 575.2.3 Multiplication By Scalars 585.3 Central Multiplication 595.3.1 Primal Units 605.3.2 Evolved and Other Units 615.3.3 Numbers 625.4 Practice 635.4.1 Example Code 635.4.2 Example Output 655.4.3 Multiplication Tables 65References 706 Vectors and Geometry 716.1 Theory 716.1.1 Magnitude 716.1.2 Inverse 726.1.3 Reflection 726.1.4 Projection 736.1.5 Rotation 736.2 Practice 746.2.1 Example Code 746.2.2 Example Output 767 Quaternions 797.1 Theory 797.1.1 Magnitude 807.1.2 Inverse 807.1.3 Reflection and Projection 807.1.4 Rotation 817.1.5 Intersection 827.1.6 Factorisation 827.2 Practice 837.2.1 Example Code 837.2.2 Example Output 86References 878 Pauli Matrices 898.1 Theory 898.1.1 Recovery of Components 908.1.2 Magnitude 908.1.3 Inverse 918.1.4 Reflection, Projection, and Rotation 918.2 Practice 918.2.1 Example Code 918.2.2 Example Output 94Reference 959 Bicomplex Numbers 979.1 Theory 979.1.1 Conjugate 989.1.2 Magnitude 989.1.3 Inverse 989.1.4 Reflection, Projection, and Rotation 999.2 Practice 999.2.1 Example Code 999.2.2 Example Output 101Reference 10210 Electromagnetic Fields 10310.1 Theory 10310.1.1 Time and Frequency 10310.1.2 Electromagnetic Entities 10410.1.3 Dirac Operators 10510.1.4 Maxwell's Equations 10510.1.5 Simplified Notation 10510.1.6 Magnitude 10610.1.7 Inverse 10610.1.8 Reflection 10710.1.9 Projection 10710.1.10 Rotation 10710.2 Practice 10710.2.1 Example Code 10710.2.2 Example Output 11010.3 Field Arithmetic 11210.3.1 Extensions Based on Quaternions 11210.3.2 Inverses 11310.3.3 Example Code 11510.3.4 Example Output 117References 11811 Arrays of Clifford Numbers 11911.1 Theory 11911.2 Practice 12011.2.1 Example Code 12011.2.2 Example Output 123Reference 12512 Power Series 12712.1 Theory 12712.1.1 User Defined 12712.1.2 Predefined 12812.1.3 Convergence 12912.1.4 Factorisation 13012.1.5 Squaring 13112.2 Practice 13112.2.1 User Defined 13112.2.2 Predefined 13312.2.2.1 Standard Convergence 13612.2.2.2 Extended Convergence 14112.2.2.3 Doubly Extended Convergence 146References 14813 Matrices of Clifford Numbers 14913.1 Background 14913.2 Inversion 15013.3 Practice 15213.3.1 Example Code 15213.3.2 Example Output 155Reference 159Part II Customisation 16114 Memory 16314.1 Memory Usage 16314.2 Examples 16514.2.1 Memory Tree Sparsity 16514.2.2 Memory Expansion 17014.2.3 Memory Recycling 17114.2.3.1 Explicit and Implicit 17114.2.3.2 Implicit and Nested 173Reference 17515 Errors 17715.1 User Errors 17715.1.1 Syntax Errors and Messages 18015.2 System Errors 18115.3 Recovery 18215.4 Beneficial Usage 185Reference 19116 Extension 19316.1 Accumulation 19316.2 Multiplication 19516.3 Transformation 19716.4 Filtration 198Part III Application 20317 Verification 20517.1 Identities 20517.2 Tests 20517.2.1 Example Code 20517.2.2 Example Output 208Reference 21418 Lines Not Parallel 21518.1 Theory 21518.1.1 Common Plane 21518.1.1.1 Inner Product 21618.1.1.2 Outer Product 21718.1.1.3 Geometrical Interpretation 21718.1.2 No Plane in Common 21818.1.2.1 Inner Product 21918.1.2.2 Solution 21918.2 Practice 22018.2.1 Example Code 22018.2.2 Example Output 223Reference 22419 Perspective Projection 22519.1 Theory 22519.2 Practice 22519.2.1 Example Code 22519.2.2 Example Output 229Reference 23020 Linear Systems 23120.1 Theory 23120.2 Practice 23320.2.1 Example Code 23320.2.2 Example Output 235References 23521 Fast Fourier Transform 23721.1 Theory 23721.2 Practice 23821.2.1 Example Code 23821.2.2 Example Output 243References 24422 Hertzian Dipole 24522.1 Theory 24522.2 Practice 24622.2.1 Example Code 24622.2.2 Example Output 251Reference 25323 Finite Difference Time Domain 25523.1 Theory 25523.1.1 Analytical Solution 25523.1.2 Series Solution 25623.1.3 Analytical Example 25723.1.4 Numerical Derivatives 25723.2 Practice 25923.2.1 Example Code 25923.2.2 Example Output 265References 27024 Cauchy Extension 27124.1 Background 27124.2 Theory 27224.2.1 Two Dimensions 27224.2.2 Three Dimensions 27224.2.3 Singularity 27324.2.4 The Taming Function 27324.2.5 Construction 27424.3 Practice 27624.3.1 Example Code 27624.3.2 Example Output 281References 28425 Electromagnetic Scattering 28525.1 Background 28525.2 Theory 28625.3 Practice 28825.3.1 Example Code 28825.3.2 Example Output 289References 293Part IV Programming 29526 Interfaces 29726.1 Configuration and Observation 29726.1.1 Management 29726.1.2 Printing 29826.2 Simple Entities 30026.2.1 Units 30026.2.2 Components 30026.2.3 Numbers 30226.2.3.1 Establishing and Recovering Values 30226.2.3.2 Functions 30326.2.3.3 Addition and Subtraction 30426.2.3.4 Multiplication 30426.2.3.5 Geometric 30526.2.3.6 Filtering 30526.3 Higher Entities 30626.3.1 Vectors 30626.3.2 Bicomplex Numbers 30726.3.3 Quaternions 30726.3.4 Pauli Matrices 30826.3.5 Electromagnetic Fields 30826.4 Multiple Entities 30926.4.1 Arrays 30926.4.2 Fast Fourier Transforms 30926.4.3 Series 31026.4.4 Matrices 310Reference 31127 Descriptions 31327.1 Arguments 31327.2 Data types 31327.3 Formats 31527.4 Manual Pages 31627.4.1 A-e 31627.4.2 F-j 34227.4.3 K-o 36927.4.4 P-t 38727.4.5 U-z 46827.5 Quick Reference 477Reference 487A Key to Example Code and Results 489Index 493

O autorze

Andrew Seagar, PhD, is Director for the Bachelor of Engineering Programs at the Gold Coast Campus of the School of Engineering at Griffith University in Australia. He has experience in a variety of research, commercial development, and academic positions around the world, primarily in the areas of electrical or biomedical engineering.