1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
|
\documentclass{article}
\usepackage{axiom}
\begin{document}
\title{\$SPAD/src/input fr.input}
\author{The Axiom Team}
\maketitle
\begin{abstract}
\end{abstract}
\eject
\tableofcontents
\eject
\section{License}
<<license>>=
--Copyright The Numerical Algorithms Group Limited 1991.
@
<<*>>=
<<license>>
-- Manipulation of factored integers
)clear all
(x,y,z,w): FR INT
-- automatic coercion of integers to factored integers
x := 2**8 * 78**7 * 111**3 * 74534
y := 2**4 * 45**3 * 162**6 * 774325
-- computation of 50!
z1 := factorial 50
z := z1 :: (FR INT)
-- examine the structure if a factor
nthFactor(z,1)
nthFlag(z,1)
nthExponent(z,1)
-- extract the factors in another form
factorList z
-- construct an object that has the factors to multiplicity one
r:=reduce(*,[(nthFactor(z,i) :: (FR INT)) for i in 1..(numberOfFactors z)])
-- some arithmetic
exquo(z,r)
x*y
y*x
(x*y = y*x) :: BOOLEAN
gcd(x,z)
x+y
-- this is how you multiply the terms together
expand(x+y)
-- now look at quotients
f := x/y
g := (x ** 9) / y
f * g
(f * g) / (g * primeFactor(2,200))
(f * g) / (g * primeFactor(2,200)) * z
--% Manipulation of factored polynomials
)clear all
)set history on
(u,v,w): FR POLY INT
-- coercion to FR POLY INT involves factoring
u := (x**4 - y**4) :: POLY INT
-- primeFactor creates factors that are asserted to be prime
v := primeFactor(x-y,2) * primeFactor(x+y,2) * primeFactor(x**2 + y**2,1)
w := factor(x**2 + 2*x*y + 2*x + 2*y + y**2 + 1) * primeFactor(x-y,2)
unit w
-- some ways of looking at the components of an elements of FR P I
l := factorList u
factorList v
factorList w
l.1.fctr
l.1.xpnt
nthFactor(u,1)
nthFactor(u,2)
nthFactor(u,3)
nthExponent(u,3)
nthFlag(u,3)
nthFactor(u,4)
-- this computes a factored object that is similar to v except that
-- each factor occurs with multiplicity 1
s:=reduce(*,[(nthFactor(v,i) :: FR POLY INT) for i in 1..(numberOfFactors v)])
-- some arithmetic
exquo(v,s)
gcd(u,v)
u + v
lcm(v,w)
u * v * w
-- "expand" multiplies the factors together
expand(u * v * w)
-- some quotients
u/w
w/(u*v)
-- %%(-1) is the last result, %%(-2) is the one before that
w/(u*v) * u/w
w/(u*v) + u/w
differentiate(w,x)
differentiate(w,y)
associates?(x,-x)
characteristic()$FR POLY INT
1$FR POLY INT
0$FR POLY INT
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}
|
