NUMERICAL CALCULATIONS IN CLIFFORD ALGEBRAAn intuitive combination of the theory of Clifford algebra with numerous worked and computed examples and calculationsNumerical Calculations in Clifford Algebra: A Practical Guide for Engineers and Scientists is an accessible and practical introduction to Clifford algebra, with comprehensive coverage of the theory and calculations. The book offers many worked and computed examples at a variety of levels of complexity and over a range of different applications making extensive use of diagrams to maintain clarity. The author introduces and documents the Clifford Numerical Suite, developed to overcome the limitations of existing computational packages and to enable the rapid creation and deployment of sophisticated and efficient code. Applications of the suite include Fourier transforms for arrays of any types of Clifford numbers and the solution of linear systems in which the coefficients are Clifford numbers of particular types, including scalars, bicomplex numbers, quaternions, Pauli matrices, and extended electromagnetic fields. Readers will find: A thorough introduction to Clifford algebra, with a combination of theory and practical implementation in a range of engineering problemsComprehensive explorations of a variety of worked and computed examples at various levels of complexityPractical discussions of the conceptual and computational tools for solving common engineering problemsDetailed documentation on the deployment and application of the Clifford Numerical SuitePerfect for engineers, researchers, and academics with an interest in Clifford algebra, Numerical Calculations in Clifford Algebra: A Practical Guide for Engineers and Scientists will particularly benefit professionals in the areas of antenna design, digital image processing, theoretical physics, and geometry.
Author(s): Andrew Seagar
Publisher: Wiley
Year: 2023
Language: English
Pages: 529
City: Hoboken
Cover
Title Page
Copyright
Contents
List of Figures
List of Tables
Preface
Part I Entities and Operations
Chapter 1 Introduction
1.1 Operations
1.2 History
1.3 Alternative Forms
1.4 Naming
1.5 Structure
1.5.1 Algebraic
1.5.2 Numeric
1.6 Entities
References
Chapter 2 Input
2.1 Syntax
2.2 Constants
2.2.1 Specific Types
2.2.2 General
2.3 Variables
2.3.1 Checking and Converting
Reference
Chapter 3 Output
3.1 Tree Format
3.2 Numeric Formats
3.2.1 Default Format
3.2.2 Defined Format
3.3 Extended Formats
3.3.1 Rounding
3.3.2 Parts of Coefficients
3.4 Selected Components
3.5 Primitive Formats
3.6 Recovered Values
Chapter 4 Unary Operations
4.1 Theory
4.1.1 Negation
4.1.2 Involution
4.1.3 Pair Exchange
4.1.4 Reversion
4.1.5 Clifford Conjugation
4.1.6 Supplementation and Pseudo‐scalar
4.2 Practice
4.2.1 Example Code
4.2.2 Example Output
Chapter 5 Binary Operations
5.1 Geometric Origins
5.1.1 Outer Multiplication
5.1.2 Orthogonal Components
5.1.3 Inner Multiplication
5.1.4 Names
5.2 Multiplication of Units
5.2.1 Progressive and Regressive Multiplication
5.2.2 Outer, Inner, and Central Multiplication
5.2.3 Multiplication By Scalars
5.3 Central Multiplication
5.3.1 Primal Units
5.3.2 Evolved and Other Units
5.3.3 Numbers
5.4 Practice
5.4.1 Example Code
5.4.2 Example Output
5.4.3 Multiplication Tables
References
Chapter 6 Vectors and Geometry
6.1 Theory
6.1.1 Magnitude
6.1.2 Inverse
6.1.3 Reflection
6.1.4 Projection
6.1.5 Rotation
6.2 Practice
6.2.1 Example Code
6.2.2 Example Output
Chapter 7 Quaternions
7.1 Theory
7.1.1 Magnitude
7.1.2 Inverse
7.1.3 Reflection and Projection
7.1.4 Rotation
7.1.5 Intersection
7.1.6 Factorisation
7.2 Practice
7.2.1 Example Code
7.2.2 Example Output
References
Chapter 8 Pauli Matrices
8.1 Theory
8.1.1 Recovery of Components
8.1.2 Magnitude
8.1.3 Inverse
8.1.4 Reflection, Projection, and Rotation
8.2 Practice
8.2.1 Example Code
8.2.2 Example Output
Reference
Chapter 9 Bicomplex Numbers
9.1 Theory
9.1.1 Conjugate
9.1.2 Magnitude
9.1.3 Inverse
9.1.4 Reflection, Projection, and Rotation
9.2 Practice
9.2.1 Example Code
9.2.2 Example Output
Reference
Chapter 10 Electromagnetic Fields
10.1 Theory
10.1.1 Time and Frequency
10.1.2 Electromagnetic Entities
10.1.3 Dirac Operators
10.1.4 Maxwell's Equations
10.1.5 Simplified Notation
10.1.6 Magnitude
10.1.7 Inverse
10.1.8 Reflection
10.1.9 Projection
10.1.10 Rotation
10.2 Practice
10.2.1 Example Code
10.2.2 Example Output
10.3 Field Arithmetic
10.3.1 Extensions Based on Quaternions
10.3.2 Inverses
10.3.3 Example Code
10.3.4 Example Output
References
Chapter 11 Arrays of Clifford Numbers
11.1 Theory
11.2 Practice
11.2.1 Example Code
11.2.2 Example Output
Reference
Chapter 12 Power Series
12.1 Theory
12.1.1 User Defined
12.1.2 Predefined
12.1.3 Convergence
12.1.4 Factorisation
12.1.5 Squaring
12.2 Practice
12.2.1 User Defined
12.2.2 Predefined
12.2.2.1 Standard Convergence
12.2.2.2 Extended Convergence
12.2.2.3 Doubly Extended Convergence
References
Chapter 13 Matrices of Clifford Numbers
13.1 Background
13.2 Inversion
13.3 Practice
13.3.1 Example Code
13.3.2 Example Output
Reference
Part II Customisation
Chapter 14 Memory
14.1 Memory Usage
14.2 Examples
14.2.1 Memory Tree Sparsity
14.2.2 Memory Expansion
14.2.3 Memory Recycling
14.2.3.1 Explicit and Implicit
14.2.3.2 Implicit and Nested
Reference
Chapter 15 Errors
15.1 User Errors
15.1.1 Syntax Errors and Messages
15.2 System Errors
15.3 Recovery
15.4 Beneficial Usage
Reference
Chapter 16 Extension
16.1 Accumulation
16.2 Multiplication
16.3 Transformation
16.4 Filtration
Part III Application
Chapter 17 Verification
17.1 Identities
17.2 Tests
17.2.1 Example Code
17.2.2 Example Output
Reference
Chapter 18 Lines Not Parallel
18.1 Theory
18.1.1 Common Plane
18.1.1.1 Inner Product
18.1.1.2 Outer Product
18.1.1.3 Geometrical Interpretation
18.1.2 No Plane in Common
18.1.2.1 Inner Product
18.1.2.2 Solution
18.2 Practice
18.2.1 Example Code
18.2.2 Example Output
Reference
Chapter 19 Perspective Projection
19.1 Theory
19.2 Practice
19.2.1 Example Code
19.2.2 Example Output
Reference
Chapter 20 Linear Systems
20.1 Theory
20.2 Practice
20.2.1 Example Code
20.2.2 Example Output
References
Chapter 21 Fast Fourier Transform
21.1 Theory
21.2 Practice
21.2.1 Example Code
21.2.2 Example Output
References
Chapter 22 Hertzian Dipole
22.1 Theory
22.2 Practice
22.2.1 Example Code
22.2.2 Example Output
Reference
Chapter 23 Finite Difference Time Domain
23.1 Theory
23.1.1 Analytical Solution
23.1.2 Series Solution
23.1.3 Analytical Example
23.1.4 Numerical Derivatives
23.2 Practice
23.2.1 Example Code
23.2.2 Example Output
References
Chapter 24 Cauchy Extension
24.1 Background
24.2 Theory
24.2.1 Two Dimensions
24.2.2 Three Dimensions
24.2.3 Singularity
24.2.4 The Taming Function
24.2.5 Construction
24.3 Practice
24.3.1 Example Code
24.3.2 Example Output
References
Chapter 25 Electromagnetic Scattering
25.1 Background
25.2 Theory
25.3 Practice
25.3.1 Example Code
25.3.2 Example Output
References
Part IV Programming
Chapter 26 Interfaces
26.1 Configuration and Observation
26.1.1 Management
26.1.2 Printing
26.2 Simple Entities
26.2.1 Units
26.2.2 Components
26.2.3 Numbers
26.2.3.1 Establishing and Recovering Values
26.2.3.2 Functions
26.2.3.3 Addition and Subtraction
26.2.3.4 Multiplication
26.2.3.5 Geometric
26.2.3.6 Filtering
26.3 Higher Entities
26.3.1 Vectors
26.3.2 Bicomplex Numbers
26.3.3 Quaternions
26.3.4 Pauli Matrices
26.3.5 Electromagnetic Fields
26.4 Multiple Entities
26.4.1 Arrays
26.4.2 Fast Fourier Transforms
26.4.3 Series
26.4.4 Matrices
Reference
Chapter 27 Descriptions
27.1 Arguments
27.2 Data types
27.3 Formats
27.4 Manual Pages
27.4.1 A–E
27.4.2 F–J
27.4.3 K–O
27.4.4 P–T
27.4.5 U–Z
27.5 Quick Reference
Reference
A Key to Example Code and Results
Index
EULA
