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
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
|
\documentclass{article}
\usepackage{open-axiom}
\begin{document}
\title{src/algebra weier.spad}
\author{William H. Burge}
\maketitle
\begin{abstract}
\end{abstract}
\tableofcontents
\eject
\section{package WEIER WeierstrassPreparation}
<<package WEIER WeierstrassPreparation>>=
import Field
import NonNegativeInteger
)abbrev package WEIER WeierstrassPreparation
++ Author:William H. Burge
++ Date Created:Sept 1988
++ Date Last Updated:Feb 15 1992
++ Basic Operations:
++ Related Domains:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ Examples:
++ References:
++ Description: This package implements the Weierstrass preparation
++ theorem f or multivariate power series.
++ weierstrass(v,p) where v is a variable, and p is a
++ TaylorSeries(R) in which the terms
++ of lowest degree s must include c*v**s where c is a constant,s>0,
++ is a list of TaylorSeries coefficients A[i] of the
++ equivalent polynomial
++ A = A[0] + A[1]*v + A[2]*v**2 + ... + A[s-1]*v**(s-1) + v**s
++ such that p=A*B , B being a TaylorSeries of minimum degree 0
WeierstrassPreparation(R): Defn == Impl where
R : Field
VarSet==>Symbol
SMP ==> Polynomial R
PS ==> InnerTaylorSeries SMP
NNI ==> NonNegativeInteger
ST ==> Stream
StS ==> Stream SMP
STPS==>StreamTaylorSeriesOperations
STTAYLOR==>StreamTaylorSeriesOperations
SUP==> SparseUnivariatePolynomial(SMP)
ST2==>StreamFunctions2
SMPS==> TaylorSeries(R)
L==>List
null ==> empty?
likeUniv ==> univariate
coef ==> coefficient$SUP
nil ==> empty
Defn ==> with
crest:(NNI->( StS-> StS))
++\spad{crest n} is used internally.
cfirst:(NNI->( StS-> StS))
++\spad{cfirst n} is used internally.
sts2stst:(VarSet,StS)->ST StS
++\spad{sts2stst(v,s)} is used internally.
clikeUniv:VarSet->(SMP->SUP)
++\spad{clikeUniv(v)} is used internally.
weierstrass:(VarSet,SMPS)->L SMPS
++\spad{weierstrass(v,ts)} where v is a variable and ts is
++ a TaylorSeries, impements the Weierstrass Preparation
++ Theorem. The result is a list of TaylorSeries that
++ are the coefficients of the equivalent series.
qqq:(NNI,SMPS,ST SMPS)->((ST SMPS)->ST SMPS)
++\spad{qqq(n,s,st)} is used internally.
Impl ==> add
import TaylorSeries(R)
import StreamTaylorSeriesOperations SMP
import StreamTaylorSeriesOperations SMPS
map1==>map$(ST2(SMP,SUP))
map2==>map$(ST2(StS,SMP))
map3==>map$(ST2(StS,StS))
transback:ST SMPS->L SMPS
transback smps==
if null smps
then nil()$(L SMPS)
else
if null first (smps:(ST StS))
then nil()$(L SMPS)
else
cons(map2(first,smps:ST StS):SMPS,
transback(map3(rest,smps:ST StS):(ST SMPS)))$(L SMPS)
clikeUniv(var)==likeUniv(#1,var)
mind:(NNI,StS)->NNI
mind(n, sts)==
if null sts
then error "no mindegree"
else if first sts=0
then mind(n+1,rest sts)
else n
mindegree (sts:StS):NNI== mind(0,sts)
streamlikeUniv:(SUP,NNI)->StS
streamlikeUniv(p:SUP,n:NNI): StS ==
if n=0
then cons(coef (p,0),nil()$StS)
else cons(coef (p,n),streamlikeUniv(p,(n-1):NNI))
transpose:ST StS->ST StS
transpose(s:ST StS)==delay(
if null s
then nil()$(ST StS)
else cons(map2(first,s),transpose(map3(rest,rst s))))
zp==>map$StreamFunctions3(SUP,NNI,StS)
sts2stst(var, sts)==
zp(streamlikeUniv(#1,#2),
map1(clikeUniv var, sts),(integers 0):(ST NNI))
tp:(VarSet,StS)->ST StS
tp(v,sts)==transpose sts2stst(v,sts)
map4==>map$(ST2 (StS,StS))
maptake:(NNI,ST StS)->ST SMPS
maptake(n,p)== map4(cfirst n,p) pretend ST SMPS
mapdrop:(NNI,ST StS)->ST SMPS
mapdrop(n,p)== map4(crest n,p) pretend ST SMPS
YSS==>Y$ParadoxicalCombinatorsForStreams(SMPS)
weier:(VarSet,StS)->ST SMPS
weier(v,sts)==
a:=mindegree sts
if a=0
then error "has constant term"
else
p:=tp(v,sts) pretend (ST SMPS)
b:StS:=rest(((first p pretend StS)),a::NNI)
c:=retractIfCan first b
c case "failed"=>_
error "the coefficient of the lowest degree of the variable should _
be a constant"
e:=recip b
f:= if e case "failed"
then error "no reciprocal"
else e::StS
q:=(YSS qqq(a,f:SMPS,rest p))
maptake(a,(p*q) pretend ST StS)
cfirst n== first(#1,n)$StS
crest n== rest(#1,n)$StS
qq:(NNI,SMPS,ST SMPS,ST SMPS)->ST SMPS
qq(a,e,p,c)==
cons(e,(-e)*mapdrop(a,(p*c)pretend(ST StS)))
qqq(a,e,p)== qq(a,e,p,#1)
wei:(VarSet,SMPS)->ST SMPS
wei(v:VarSet,s:SMPS)==weier(v,s:StS)
weierstrass(v,smps)== transback wei (v,smps)
@
\section{License}
<<license>>=
--Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd.
--All rights reserved.
--
--Redistribution and use in source and binary forms, with or without
--modification, are permitted provided that the following conditions are
--met:
--
-- - Redistributions of source code must retain the above copyright
-- notice, this list of conditions and the following disclaimer.
--
-- - Redistributions in binary form must reproduce the above copyright
-- notice, this list of conditions and the following disclaimer in
-- the documentation and/or other materials provided with the
-- distribution.
--
-- - Neither the name of The Numerical ALgorithms Group Ltd. nor the
-- names of its contributors may be used to endorse or promote products
-- derived from this software without specific prior written permission.
--
--THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS
--IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED
--TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A
--PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER
--OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
--EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
--PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
--PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF
--LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
--NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
--SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
@
<<*>>=
<<license>>
<<package WEIER WeierstrassPreparation>>
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}
|
