1% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
2% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
3\newcommand{\XPolynomialXmpTitle}{XPolynomial}
4\newcommand{\XPolynomialXmpNumber}{9.93}
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7\begin{page}{XPolynomialXmpPage}{9.93 XPolynomial}
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17
18The \spadtype{XPolynomial} domain constructor implements multivariate polynomials
19whose set of variables is \spadtype{Symbol}.
20These variables do not commute.
21The only parameter of this constructor is
22the coefficient ring which may be non-commutative.
23However, coefficients and variables commute.
24The representation of the polynomials is recursive.
25The abbreviation for \spadtype{XPolynomial} is \spadtype{XPOLY}.
26
27Other constructors like \spadtype{XPolynomialRing},
28\spadtype{XRecursivePolynomial},
29\spadtype{XDistributedPolynomial},
30\spadtype{LiePolynomial} and
31\spadtype{XPBWPolynomial}
32implement multivariate polynomials
33in non-commutative variables.
34
35We illustrate now some of the facilities of the \spadtype{XPOLY} domain constructor.
36
37
38\xtc{
39Define a polynomial ring over the integers.
40}{
41\spadpaste{poly := XPolynomial(Integer) \bound{poly}}
42}
43
44\xtc{
45Define a first polynomial,
46}{
47\spadpaste{pr: poly := 2*x + 3*y-5  \free{poly} \bound{pr}}
48}
49
50
51\xtc{
52and a second one.
53}{
54\spadpaste{pr2: poly := pr*pr  \free{poly} \bound{pr2}}
55}
56
57\xtc{
58Rewrite {\bf pr} in a distributive way,
59}{
60\spadpaste{pd  := expand pr   \free{pr} \bound{pd}}
61}
62
63\xtc{
64compute its square,
65}{
66\spadpaste{pd2 := pd*pd   \free{pd} \bound{pd2}}
67}
68
69\xtc{
70and checks that:
71}{
72\spadpaste{expand(pr2) - pd2 \free{pr2} \free{pd2}}
73}
74
75
76\xtc{
77We define:
78}{
79\spadpaste{qr :=  pr^3 \free{pr} \bound{qr}}
80}
81
82\xtc{
83and:
84}{
85\spadpaste{qd :=  pd^3 \free{pd} \bound{qd}}
86}
87
88\xtc{
89We truncate {\bf qd} at degree {\bf 3}:
90}{
91\spadpaste{trunc(qd,2) \free{qd}}
92}
93
94\xtc{
95The same for {\bf qr}:
96}{
97\spadpaste{trunc(qr,2) \free{qr}}
98}
99
100\xtc{
101We define:
102}{
103\spadpaste{Word := FreeMonoid Symbol \bound{Word}}
104}
105
106\xtc{
107and:
108}{
109\spadpaste{w: Word := x*y^2 \free{Word} \bound{w}}
110}
111
112\xtc{
113The we can compute the right-quotient of {\bf qr} by {\bf r}:
114}{
115\spadpaste{rquo(qr,w) \free{qr} \free{w}}
116}
117
118\xtc{
119and the shuffle-product of {\bf pr} by {\bf r}:
120}{
121\spadpaste{sh(pr,w::poly) \free{pr} \free{w}}
122}
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