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--All rights reserved.
--Copyright (C) 2007-2010, Gabriel Dos Reis.
--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.
)abbrev package INTSLPE IntegerSolveLinearPolynomialEquation
++ Author: Davenport
++ Date Created: 1991
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ This package provides the implementation for the
++ \spadfun{solveLinearPolynomialEquation}
++ operation over the integers. It uses a lifting technique
++ from the package GenExEuclid
IntegerSolveLinearPolynomialEquation(): C ==T
where
ZP ==> SparseUnivariatePolynomial Integer
C == with
solveLinearPolynomialEquation: (List ZP,ZP) -> Union(List ZP,"failed")
++ solveLinearPolynomialEquation([f1, ..., fn], g)
++ (where the fi are relatively prime to each other)
++ returns a list of ai such that
++ \spad{g/prod fi = sum ai/fi}
++ or returns "failed" if no such list of ai's exists.
T == add
oldlp:List ZP := []
slpePrime:Integer:=(2::Integer)
oldtable:Vector List ZP := empty()
solveLinearPolynomialEquation(lp,p) ==
if (oldlp ~= lp) then
-- we have to generate a new table
deg:= +/[degree u for u in lp]
ans:Union(Vector List ZP,"failed"):="failed"
slpePrime:=2147483647::Integer -- 2**31 -1 : a prime
-- a good test case for this package is
-- ([x**31-1,x-2],2)
while (ans case "failed") repeat
ans:=tablePow(deg,slpePrime,lp)$GenExEuclid(Integer,ZP)
if (ans case "failed") then
slpePrime:= prevPrime(slpePrime)$IntegerPrimesPackage(Integer)
oldtable:=(ans:: Vector List ZP)
answer:=solveid(p,slpePrime,oldtable)
answer
)abbrev domain INT Integer
++ Author:
++ Date Created:
++ Change History:
++ Basic Operations:
++ Related Constructors:
++ Keywords: integer
++ Description: \spadtype{Integer} provides the domain of arbitrary precision
++ integers.
Integer: IntegerNumberSystem with
canonical
++ mathematical equality is data structure equality.
canonicalsClosed
++ two positives multiply to give positive.
noetherian
++ ascending chain condition on ideals.
== add
ZP ==> SparseUnivariatePolynomial %
ZZP ==> SparseUnivariatePolynomial Integer
import %icst0: % from Foreign Builtin
import %icst1: % from Foreign Builtin
import %ineg: % -> % from Foreign Builtin
import %iabs: % -> % from Foreign Builtin
import %irandom: % -> % from Foreign Builtin
import %iodd?: % -> Boolean from Foreign Builtin
import %ieven?: % -> Boolean from Foreign Builtin
import %hash: % -> SingleInteger from Foreign Builtin
import %iadd: (%,%) -> % from Foreign Builtin
import %isub: (%,%) -> % from Foreign Builtin
import %imul: (%,%) -> % from Foreign Builtin
import %irem: (%,%) -> % from Foreign Builtin
import %iquo: (%,%) -> % from Foreign Builtin
import %imax: (%,%) -> % from Foreign Builtin
import %imin: (%,%) -> % from Foreign Builtin
import %igcd: (%,%) -> % from Foreign Builtin
import %ieq: (%,%) -> Boolean from Foreign Builtin
import %ilt: (%,%) -> Boolean from Foreign Builtin
import %ile: (%,%) -> Boolean from Foreign Builtin
import %igt: (%,%) -> Boolean from Foreign Builtin
import %ige: (%,%) -> Boolean from Foreign Builtin
import %ilength: % -> % from Foreign Builtin
import %i2s: % -> String from Foreign Builtin
import %strconc: (String,String) -> String from Foreign Builtin
x,y: %
n: NonNegativeInteger
zero? x == x = %icst0
one? x == x = %icst1
0 == %icst0
1 == %icst1
base() == 2 pretend %
copy x == x
inc x == x + %icst1
dec x == x - %icst1
hash x == %hash x
negative? x == x < %icst0
positive? x == %icst0 < x
coerce(x):OutputForm == outputForm(x pretend Integer)
coerce(m:Integer):% == m pretend %
convert(x:%):Integer == x pretend Integer
length a == %ilength a
addmod(a, b, p) ==
c := %iadd(a,b)
c >= p => c - p
c
submod(a, b, p) ==
c := %isub(a,b)
negative? c => c + p
c
mulmod(a, b, p) == %imul(a,b) rem p
convert(x:%):Float == coerce(x)$Float
convert(x:%):DoubleFloat == coerce(x)$DoubleFloat
convert(x:%):InputForm == convert(x)$InputForm
latex(x:%):String ==
s := %i2s x
-%icst1 < x and x < 10 => s
%strconc("{", %strconc(s, "}"))
positiveRemainder(a, b) ==
negative?(r := a rem b) =>
negative? b => r - b
r + b
r
reducedSystem(m:Matrix %):Matrix(Integer) ==
m pretend Matrix(Integer)
reducedSystem(m:Matrix %, v:Vector %):
Record(mat:Matrix(Integer), vec:Vector(Integer)) ==
[m pretend Matrix(Integer), v pretend Vector(Integer)]
abs(x) == %iabs x
random() == random()$Lisp
random(x) == %irandom x
x = y == %ieq(x,y)
x < y == %ilt(x,y)
x > y == %igt(x,y)
x <= y == %ile(x,y)
x >= y == %ige(x,y)
- x == %ineg x
x + y == %iadd(x,y)
x - y == %isub(x,y)
x * y == %imul(x,y)
(m:Integer) * (y:%) == %imul(m,y) -- for subsumption problem
x ** n == %ipow(x,n)$Foreign(Builtin)
odd? x == %iodd? x
even? x == %ieven? x
max(x,y) == %imax(x,y)
min(x,y) == %imin(x,y)
divide(x,y) == %idivide(x,y)$Foreign(Builtin)
x quo y == %iquo(x,y)
x rem y == %irem(x,y)
shift(x, y) == ASH(x,y)$Lisp
recip(x) == if one? x or x=-1 then x else "failed"
gcd(x,y) == %igcd(x,y)
UCA ==> Record(unit:%,canonical:%,associate:%)
unitNormal x ==
negative? x => [-%icst1,-x,-%icst1]$UCA
[%icst1,x,%icst1]$UCA
unitCanonical x == abs x
solveLinearPolynomialEquation(lp:List ZP,p:ZP):Union(List ZP,"failed") ==
solveLinearPolynomialEquation(lp pretend List ZZP,
p pretend ZZP)$IntegerSolveLinearPolynomialEquation pretend
Union(List ZP,"failed")
squareFreePolynomial(p:ZP):Factored ZP ==
squareFree(p)$UnivariatePolynomialSquareFree(%,ZP)
factorPolynomial(p:ZP):Factored ZP ==
-- GaloisGroupFactorizer doesn't factor the content
-- so we have to do this by hand
pp:=primitivePart p
leadingCoefficient pp = leadingCoefficient p =>
factor(p)$GaloisGroupFactorizer(ZP)
mergeFactors(factor(pp)$GaloisGroupFactorizer(ZP),
map(#1::ZP,
factor((leadingCoefficient p exquo
leadingCoefficient pp)
::%))$FactoredFunctions2(%,ZP)
)$FactoredFunctionUtilities(ZP)
factorSquareFreePolynomial(p:ZP):Factored ZP ==
factorSquareFree(p)$GaloisGroupFactorizer(ZP)
gcdPolynomial(p:ZP, q:ZP):ZP ==
zero? p => unitCanonical q
zero? q => unitCanonical p
gcd([p,q])$HeuGcd(ZP)
-- myNextPrime: (%,NonNegativeInteger) -> %
-- myNextPrime(x,n) ==
-- nextPrime(x)$IntegerPrimesPackage(%)
-- TT:=InnerModularGcd(%,ZP,67108859 pretend %,myNextPrime)
-- gcdPolynomial(p,q) == modularGcd(p,q)$TT
)abbrev domain NNI NonNegativeInteger
++ Author:
++ Date Created:
++ Change History:
++ Basic Operations:
++ Related Constructors:
++ Keywords: integer
++ Description: \spadtype{NonNegativeInteger} provides functions for non
++ negative integers.
NonNegativeInteger: Join(OrderedAbelianMonoidSup,Monoid) with
quo : (%,%) -> %
++ a quo b returns the quotient of \spad{a} and b, forgetting
++ the remainder.
rem : (%,%) -> %
++ a rem b returns the remainder of \spad{a} and b.
gcd : (%,%) -> %
++ gcd(a,b) computes the greatest common divisor of two
++ non negative integers \spad{a} and b.
divide: (%,%) -> Record(quotient:%,remainder:%)
++ divide(a,b) returns a record containing both
++ remainder and quotient.
exquo: (%,%) -> Union(%,"failed")
++ exquo(a,b) returns the quotient of \spad{a} and b, or "failed"
++ if b is zero or \spad{a} rem b is zero.
shift: (%, Integer) -> %
++ shift(a,i) shift \spad{a} by i bits.
random : % -> %
++ random(n) returns a random integer from 0 to \spad{n-1}.
commutative("*")
++ commutative("*") means multiplication is commutative : \spad{x*y = y*x}.
== SubDomain(Integer,#1 >= 0) add
x,y:%
sup(x,y) == %imax(x,y)$Foreign(Builtin)
shift(x:%, n:Integer):% == ASH(x,n)$Lisp
subtractIfCan(x, y) ==
c:Integer := rep x - rep y
negative? c => "failed"
per c
)abbrev domain PI PositiveInteger
++ Author:
++ Date Created:
++ Change History:
++ Basic Operations:
++ Related Constructors:
++ Keywords: positive integer
++ Description: \spadtype{PositiveInteger} provides functions for
++ positive integers.
PositiveInteger: Join(OrderedAbelianSemiGroup,Monoid) with
gcd: (%,%) -> %
++ gcd(a,b) computes the greatest common divisor of two
++ positive integers \spad{a} and b.
commutative("*")
++ commutative("*") means multiplication is commutative : x*y = y*x
== SubDomain(NonNegativeInteger,#1 > 0)
)abbrev domain ROMAN RomanNumeral
++ Author:
++ Date Created:
++ Change History:
++ Basic Operations:
++ convert, roman
++ Related Constructors:
++ Keywords: roman numerals
++ Description: \spadtype{RomanNumeral} provides functions for converting
++ integers to roman numerals.
RomanNumeral(): Join(IntegerNumberSystem,ConvertibleFrom Symbol) with
canonical
++ mathematical equality is data structure equality.
canonicalsClosed
++ two positives multiply to give positive.
noetherian
++ ascending chain condition on ideals.
roman : Symbol -> %
++ roman(n) creates a roman numeral for symbol n.
roman : Integer -> %
++ roman(n) creates a roman numeral for n.
== Integer add
import NumberFormats()
roman(n:Integer) == n::%
roman(sy:Symbol) == convert sy
convert(sy:Symbol):% == ScanRoman(string sy)::%
coerce(r:%):OutputForm ==
n := convert(r)@Integer
-- okay, we stretch it
zero? n => n::OutputForm
negative? n => - ((-r)::OutputForm)
FormatRoman(n::PositiveInteger)::Symbol::OutputForm
|