Chapter 7. Dual Numbers and Algebraic Structures

Dual numbers give the cleanest algebraic model of forward mode automatic differentiation. They extend ordinary real numbers with a formal infinitesimal part. Instead of...

9 items

Section	Title
1	Chapter 7. Dual Numbers and Algebraic Structures
2	Nilpotent Elements
3	Geometric Interpretation
4	Truncated Polynomial Algebras
5	Hyper-Dual Numbers
6	Differential Algebras
7	Category-Theoretic View
8	Differential Lambda Calculus
9	Differential Lambda Calculus

Writes › Book › Auto Diff › Chapter 7. Dual Numbers and Algebraic Structures ›

Chapter 7. Dual Numbers and Algebraic Structures

Dual numbers give the cleanest algebraic model of forward mode automatic differentiation. They extend ordinary real numbers with a formal infinitesimal part. Instead of...

Writes › Book › Auto Diff › Chapter 7. Dual Numbers and Algebraic Structures ›

Nilpotent Elements

The defining feature of dual numbers is the existence of a nonzero element whose square vanishes:

Writes › Book › Auto Diff › Chapter 7. Dual Numbers and Algebraic Structures ›

Geometric Interpretation

Dual numbers provide an algebraic mechanism for differentiation, but they also have a precise geometric meaning. A dual number represents a point together with an...

Writes › Book › Auto Diff › Chapter 7. Dual Numbers and Algebraic Structures ›

Truncated Polynomial Algebras

Dual numbers capture first-order derivatives because the infinitesimal element satisfies

Writes › Book › Auto Diff › Chapter 7. Dual Numbers and Algebraic Structures ›

Hyper-Dual Numbers

Dual numbers compute first derivatives exactly. Truncated polynomial algebras extend this to higher-order derivatives, but practical higher-order differentiation introduces an...

Writes › Book › Auto Diff › Chapter 7. Dual Numbers and Algebraic Structures ›

Differential Algebras

Dual numbers and hyper-dual numbers are special cases of a broader algebraic structure called a differential algebra. This framework abstracts differentiation away from...

Writes › Book › Auto Diff › Chapter 7. Dual Numbers and Algebraic Structures ›

Category-Theoretic View

Automatic differentiation can be described operationally through dual numbers and computational graphs. It can also be described abstractly using category theory.

Writes › Book › Auto Diff › Chapter 7. Dual Numbers and Algebraic Structures ›

Differential Lambda Calculus

Automatic differentiation is deeply connected to functional programming and lambda calculus. Programs can be viewed as mathematical functions, and differentiation can be...

Writes › Book › Auto Diff › Chapter 7. Dual Numbers and Algebraic Structures ›

Differential Lambda Calculus

Automatic differentiation is deeply connected to functional programming and lambda calculus. Programs can be viewed as mathematical functions, and differentiation can be...