Dual Numbers for Common Lisp

This is a Common Lisp library for working with dual numbers. Dual numbers are a simple extension to the real numbers, of the form a + bε, where a and b are real numbers and ε is a nilpotent element such that ε² = 0.

This library provides a dual-number class and implements various arithmetic and transcendental functions for it. The primary application of this library is automatic differentiation.

Features

• Basic arithmetic operations: +, -, *, /
• Trigonometric functions: sin, cos, tan
• Hyperbolic functions: sinh, cosh, tanh
• Transcendental functions: exp, log, sqrt
• Automatic differentiation via the derivative function.

Dependencies

This library depends on:

• alexandria
• generic-arithmetic

These dependencies are managed by ASDF.

Installation

Clone the repository into a directory known to ASDF (e.g., ~/common-lisp/). You can then load the system using Quicklisp:

(ql:quickload :dual-numbers)

Usage

Creating Dual Numbers

You can create a dual number using the make-dual function:

(use-package :dual-numbers)

(make-dual 2 3)
;; => #<DUAL-NUMBER 2 + 3ε>

;; If the infinitesimal part is zero, a regular number is returned
(make-dual 5 0)
;; => 5

Arithmetic

The library uses generic-arithmetic to overload the standard arithmetic functions.

(let ((d1 (make-dual 2 1))
      (d2 (make-dual 3 4)))
  (+ d1 d2))
;; => #<DUAL-NUMBER 5 + 5ε>

(* d1 d2)
;; => #<DUAL-NUMBER 6 + 11ε>

Automatic Differentiation

The core feature of this library is automatic differentiation. The derivative function is a curried function that takes a function and returns a new function that computes the derivative.

(use-package :dual-numbers)

;; The derivative of x^2 at x=3 is 2*3=6
(funcall (derivative (lambda (x) (* x x))) 3)
;; => 6

;; The derivative of sin(x) at x=pi is cos(pi)=-1
(funcall (derivative #'sin) pi)
;; => -1.0

;; The derivative of exp(x) at x=0 is exp(0)=1
(funcall (derivative #'exp) 0)
;; => 1.0

API Reference

• make-dual standard infinitesimal Creates a dual number.

• standard-part number Returns the standard (real) part of a dual number.

• infinitesimal-part number Returns the infinitesimal (dual) part of a dual number.

• derivative function Returns a new function that computes the derivative of function.

Running Tests

The tests can be run by loading the dual-numbers-tests.lisp file in a Common Lisp implementation.

(load "dual-numbers-tests.lisp")

This will run the test suite and print "All tests passed!" if successful.

License

This library is released under the MIT License.