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
207
208
209
210
211
212
213
214
|
\documentclass{article}
\usepackage{open-axiom}
\begin{document}
\title{\$SPAD/src/algebra galfactu.spad}
\author{Frederic Lehobey}
\maketitle
\begin{abstract}
\end{abstract}
\eject
\tableofcontents
\eject
\section{package GALFACTU GaloisGroupFactorizationUtilities}
<<package GALFACTU GaloisGroupFactorizationUtilities>>=
)abbrev package GALFACTU GaloisGroupFactorizationUtilities
++ Author: Frederic Lehobey
++ Date Created: 30 June 1994
++ Date Last Updated: 19 October 1995
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ [1] Bernard Beauzamy, Products of polynomials and a priori estimates for
++ coefficients in polynomial decompositions: a sharp result,
++ J. Symbolic Computation (1992) 13, 463-472
++ [2] David W. Boyd, Bounds for the Height of a Factor of a Polynomial in
++ Terms of Bombieri's Norms: I. The Largest Factor,
++ J. Symbolic Computation (1993) 16, 115-130
++ [3] David W. Boyd, Bounds for the Height of a Factor of a Polynomial in
++ Terms of Bombieri's Norms: II. The Smallest Factor,
++ J. Symbolic Computation (1993) 16, 131-145
++ [4] Maurice Mignotte, Some Useful Bounds,
++ Computing, Suppl. 4, 259-263 (1982), Springer-Verlag
++ [5] Donald E. Knuth, The Art of Computer Programming, Vol. 2, (Seminumerical
++ Algorithms) 1st edition, 2nd printing, Addison-Wesley 1971, p. 397-398
++ [6] Bernard Beauzamy, Vilmar Trevisan and Paul S. Wang, Polynomial
++ Factorization: Sharp Bounds, Efficient Algorithms,
++ J. Symbolic Computation (1993) 15, 393-413
++ [7] Augustin-Lux Cauchy, Exercices de Math\'ematiques Quatri\`eme Ann\'ee.
++ De Bure Fr\`eres, Paris 1829 (reprinted Oeuvres, II S\'erie, Tome IX,
++ Gauthier-Villars, Paris, 1891).
++ Description:
++ \spadtype{GaloisGroupFactorizationUtilities} provides functions
++ that will be used by the factorizer.
GaloisGroupFactorizationUtilities(R,UP,F): Exports == Implementation where
R : Ring
UP : UnivariatePolynomialCategory R
F : Join(FloatingPointSystem,RetractableTo(R),Field,
TranscendentalFunctionCategory,ElementaryFunctionCategory)
N ==> NonNegativeInteger
P ==> PositiveInteger
Z ==> Integer
Exports ==> with
beauzamyBound: UP -> Z -- See [1]
++ beauzamyBound(p) returns a bound on the larger coefficient of any
++ factor of p.
bombieriNorm: UP -> F -- See [1]
++ bombieriNorm(p) returns quadratic Bombieri's norm of p.
bombieriNorm: (UP,P) -> F -- See [2] and [3]
++ bombieriNorm(p,n) returns the nth Bombieri's norm of p.
rootBound: UP -> Z -- See [4] and [5]
++ rootBound(p) returns a bound on the largest norm of the complex roots
++ of p.
singleFactorBound: (UP,N) -> Z -- See [6]
++ singleFactorBound(p,r) returns a bound on the infinite norm of
++ the factor of p with smallest Bombieri's norm. r is a lower bound
++ for the number of factors of p. p shall be of degree higher or equal
++ to 2.
singleFactorBound: UP -> Z -- See [6]
++ singleFactorBound(p,r) returns a bound on the infinite norm of
++ the factor of p with smallest Bombieri's norm. p shall be of degree
++ higher or equal to 2.
norm: (UP,P) -> F
++ norm(f,p) returns the lp norm of the polynomial f.
quadraticNorm: UP -> F
++ quadraticNorm(f) returns the l2 norm of the polynomial f.
infinityNorm: UP -> F
++ infinityNorm(f) returns the maximal absolute value of the coefficients
++ of the polynomial f.
height: UP -> F
++ height(p) returns the maximal absolute value of the coefficients of
++ the polynomial p.
length: UP -> F
++ length(p) returns the sum of the absolute values of the coefficients
++ of the polynomial p.
Implementation ==> add
import GaloisGroupUtilities(F)
height(p:UP):F == infinityNorm(p)
length(p:UP):F == norm(p,1)
norm(f:UP,p:P):F ==
n : F := 0
for c in coefficients f repeat
n := n+abs(c::F)**p
nthRoot(n,p::N)
quadraticNorm(f:UP):F == norm(f,2)
infinityNorm(f:UP):F ==
n : F := 0
for c in coefficients f repeat
n := max(n,c::F)
n
singleFactorBound(p:UP,r:N):Z == -- See [6]
n : N := degree p
r := max(2,r)
n < r => error "singleFactorBound: Bad arguments."
nf : F := n :: F
num : F := nthRoot(bombieriNorm(p),r)
if F has Gamma: F -> F then
num := num*nthRoot(Gamma(nf+1$F),2*r)
den : F := Gamma(nf/((2*r)::F)+1$F)
else
num := num*(2::F)**(5/8+n/2)*exp(1$F/(4*nf))
den : F := (pi()$F*nf)**(3/8)
safeFloor( num/den )
singleFactorBound(p:UP):Z == singleFactorBound(p,2) -- See [6]
rootBound(p:UP):Z == -- See [4] and [5]
n := degree p
zero? n => 0
lc := abs(leadingCoefficient(p)::F)
b1 : F := 0 -- Mignotte
b2 : F := 0 -- Knuth
b3 : F := 0 -- Zassenhaus in [5]
b4 : F := 0 -- Cauchy in [7]
c : F := 0
cl : F := 0
for i in 1..n repeat
c := abs(coefficient(p,(n-i)::N)::F)
b1 := max(b1,c)
cl := c/lc
b2 := max(b2,nthRoot(cl,i))
b3 := max(b3,nthRoot(cl/pascalTriangle(n,i),i))
b4 := max(b4,nthRoot(n*cl,i))
min(1+safeCeiling(b1/lc),min(safeCeiling(2*b2),min(safeCeiling(b3/
(nthRoot(2::F,n)-1)),safeCeiling(b4))))
beauzamyBound(f:UP):Z == -- See [1]
d := degree f
zero? d => safeFloor bombieriNorm f
safeFloor( (bombieriNorm(f)*(3::F)**(3/4+d/2))/
(2*sqrt(pi()$F*(d::F))) )
bombieriNorm(f:UP,p:P):F == -- See [2] and [3]
d := degree f
b := abs(coefficient(f,0)::F)
if zero? d then return b
else b := b**p
b := b+abs(leadingCoefficient(f)::F)**p
dd := (d-1) quo 2
for i in 1..dd repeat
b := b+(abs(coefficient(f,i)::F)**p+abs(coefficient(f,(d-i)::N)::F)**p)
/pascalTriangle(d,i)
if even? d then
dd := dd+1
b := b+abs(coefficient(f, dd::N)::F)**p/pascalTriangle(d,dd)
nthRoot(b,p::N)
bombieriNorm(f:UP):F == bombieriNorm(f,2) -- See [1]
@
\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 GALFACTU GaloisGroupFactorizationUtilities>>
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}
|
