1 ! Copyright (C) 2009 Jason W. Merrill.
2 ! See http://factorcode.org/license.txt for BSD license.
3 USING: help.markup help.syntax kernel words math math.functions math.derivatives.syntax ;
8 { "ordinary-part" real } { "epsilon-part" real }
11 { $description "Creates a dual number from its ordinary and epsilon parts." } ;
15 { "x" dual } { "y" dual }
18 { $description "Multiply dual numbers." } ;
22 { "x" dual } { "y" dual }
25 { $description "Add dual numbers." } ;
29 { "x" dual } { "y" dual }
32 { $description "Subtract dual numbers." } ;
36 { "x" dual } { "y" dual }
39 { $description "Divide dual numbers." }
40 { $errors "Throws an error if the ordinary part of " { $snippet "x" } " is zero." } ;
44 { "x" dual } { "y" dual }
47 { $description "Raise a dual number to a (possibly dual) power" } ;
54 { $description "Absolute value of a dual number." } ;
61 { $description "Inverse hyberbolic cosine of a dual number." } ;
68 { $description "Inverse hyberbolic sine of a dual number." } ;
75 { $description "Inverse hyberbolic tangent of a dual number." } ;
82 { $description "Negative of a dual number." } ;
89 { $description "Reciprocal of a dual number." } ;
95 { $description "Defines a word " { $snippet "d[word]" } " in the " { $vocab-link "math.dual" } " vocabulary that operates on dual numbers." }
96 { $notes "Uses the derivative word-prop, which holds a list of quotations giving the partial derivatives of the word with respect to each of its arguments. This can be set using " { $link POSTPONE: DERIVATIVE: } "." } ;
98 { define-dual dual-op POSTPONE: DERIVATIVE: } related-words
101 { $class-description "The class of dual numbers with non-zero epsilon part." } ;
107 { $description "Similar to " { $link execute } ", but promotes word to operate on duals." }
108 { $notes "Uses the derivative word-prop, which holds a list of quotations giving the partial derivatives of the word with respect to each of its arguments. This can be set using " { $link POSTPONE: DERIVATIVE: } ". Once a derivative has been defined for a word, dual-op makes it easy to extend the definition to dual numbers." }
110 { $unchecked-example "USING: math math.dual math.derivatives.syntax math.functions ;"
111 "DERIVATIVE: sin [ cos * ]"
112 "M: dual sin \\sin dual-op ;" "" }
113 { $unchecked-example "USING: math math.dual math.derivatives.syntax ;"
114 "DERIVATIVE: * [ drop ] [ nip ]"
115 ": d* ( x y -- x*y ) \ * dual-op ;" "" }
121 { "ordinary-part" number } { "epsilon-part" number }
123 { $description "Extracts the ordinary and epsilon part of a dual number." } ;
125 ARTICLE: "math.dual" "Dual Numbers"
126 "The " { $vocab-link "math.dual" } " vocabulary implements dual numbers, along with arithmetic methods for working with them. Many of the functions in " { $vocab-link "math.functions" } " are extended to work with dual numbers."
128 "Dual numbers are ordered pairs " { $snippet "<o,e>"} "--an ordinary part and an epsilon part--with component-wise addition and multiplication defined by "{ $snippet "<o1,e1>*<o2,e2> = <o1*o2,e1*o2 + e2*o1>" } ". They are analagous to complex numbers with " { $snippet "i^2 = 0" } "instead of " { $snippet "i^2 = -1" } ". For well-behaved functions " { $snippet "f" } ", " { $snippet "f(<o1,e1>) = f(o1) + e1*f'(o1)" } ", where " { $snippet "f'"} " is the derivative of " { $snippet "f" } "."