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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 domain LO Localize
++ Author: Dave Barton, Barry Trager
++ Date Created:
++ Date Last Updated:
++ Basic Functions: + - / numer denom
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords: localization
++ References:
++ Description: Localize(M,R,S) produces fractions with numerators
++ from an R module M and denominators from some multiplicative subset
++ D of R.
Localize(M:Module R,
R:CommutativeRing,
S:SubsetCategory(Monoid, R)): Module R with
if M has OrderedAbelianGroup then OrderedAbelianGroup
/ :(%,S) -> %
++ x / d divides the element x by d.
/ :(M,S) -> %
++ m / d divides the element m by d.
numer: % -> M
++ numer x returns the numerator of x.
denom: % -> S
++ denom x returns the denominator of x.
==
add
Rep == Record(num:M,den:S)
x,y: %
n: Integer
m: M
r: R
d: S
0 == per [0,1]
zero? x == zero? rep(x).num
-x ==
per [-rep(x).num,rep(x).den]
x=y ==
rep(y).den*rep(x).num = rep(x).den*rep(y).num
before?(x,y) ==
before?(rep(y).den*rep(x).num, rep(x).den*rep(y).num)
numer x == rep(x).num
denom x == rep(x).den
if M has OrderedAbelianGroup then
x < y ==
rep(y).den*rep(x).num < rep(x).den*rep(y).num
x+y ==
per [rep(y).den*rep(x).num+rep(x).den*rep(y).num, rep(x).den*rep(y).den]
n*x ==
per [n*rep(x).num,rep(x).den]
r*x ==
r=rep(x).den => per [rep(x).num,1]
per [r*rep(x).num,rep(x).den]
x/d ==
u: S := d*rep(x).den
zero? u => error "division by zero"
per [rep(x).num,u]
m/d ==
zero? d => error "division by zero"
per [m,d]
coerce(x:%):OutputForm ==
one?(xd:=rep(x).den) => rep(x).num::OutputForm
rep(x).num::OutputForm / (xd::OutputForm)
latex(x:%): String ==
one?(xd:=rep(x).den) => latex rep(x).num
nl : String := concat("{", concat(latex rep(x).num, "}")$String)$String
dl : String := concat("{", concat(latex rep(x).den, "}")$String)$String
concat("{ ", concat(nl, concat(" \over ", concat(dl, " }")$String)$String)$String)$String
)abbrev domain LA LocalAlgebra
++ Author: Dave Barton, Barry Trager
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description: LocalAlgebra produces the localization of an algebra, i.e.
++ fractions whose numerators come from some R algebra.
LocalAlgebra(A: Algebra R,
R: CommutativeRing,
S: SubsetCategory(Monoid, R)): Algebra R with
if A has OrderedRing then OrderedRing
/ : (%,S) -> %
++ x / d divides the element x by d.
/ : (A,S) -> %
++ a / d divides the element \spad{a} by d.
numer: % -> A
++ numer x returns the numerator of x.
denom: % -> S
++ denom x returns the denominator of x.
== Localize(A, R, S) add
1 == 1$A / 1$S
x:% * y:% == (numer(x) * numer(y)) / (denom(x) * denom(y))
characteristic == characteristic$A
)abbrev category QFCAT QuotientFieldCategory
++ Author:
++ Date Created:
++ Date Last Updated: 5th March 1996
++ Basic Functions: + - * / numer denom
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description: QuotientField(S) is the
++ category of fractions of an Integral Domain S.
QuotientFieldCategory(S: IntegralDomain): Category ==
Join(Field, Algebra S, RetractableTo S, FullyEvalableOver S,
DifferentialExtension S, FullyLinearlyExplicitRingOver S,
Patternable S, FullyPatternMatchable S) with
/ : (S, S) -> %
++ d1 / d2 returns the fraction d1 divided by d2.
numer : % -> S
++ numer(x) returns the numerator of the fraction x.
denom : % -> S
++ denom(x) returns the denominator of the fraction x.
numerator : % -> %
++ numerator(x) is the numerator of the fraction x converted to %.
denominator : % -> %
++ denominator(x) is the denominator of the fraction x converted to %.
if S has StepThrough then StepThrough
if S has RetractableTo Integer then
RetractableTo Integer
RetractableTo Fraction Integer
if S has OrderedSet then OrderedSet
if S has OrderedIntegralDomain then OrderedIntegralDomain
if S has RealConstant then RealConstant
if S has ConvertibleTo InputForm then ConvertibleTo InputForm
if S has CharacteristicZero then CharacteristicZero
if S has CharacteristicNonZero then CharacteristicNonZero
if S has RetractableTo Symbol then RetractableTo Symbol
if S has EuclideanDomain then
wholePart: % -> S
++ wholePart(x) returns the whole part of the fraction x
++ i.e. the truncated quotient of the numerator by the denominator.
fractionPart: % -> %
++ fractionPart(x) returns the fractional part of x.
++ x = wholePart(x) + fractionPart(x)
if S has IntegerNumberSystem then
random: () -> %
++ random() returns a random fraction.
ceiling : % -> S
++ ceiling(x) returns the smallest integral element above x.
floor: % -> S
++ floor(x) returns the largest integral element below x.
if S has PolynomialFactorizationExplicit then
PolynomialFactorizationExplicit
add
import MatrixCommonDenominator(S, %)
numerator(x) == numer(x)::%
denominator(x) == denom(x) ::%
if S has StepThrough then
init() == init()$S / 1$S
nextItem(n) ==
m:= nextItem numer n
m case nothing =>
error "We seem to have a Fraction of a finite object"
just(m / 1)
map(fn, x) == (fn numer x) / (fn denom x)
reducedSystem(m:Matrix %):Matrix S == clearDenominator m
characteristic == characteristic$S
differentiate(x:%, deriv:S -> S) ==
n := numer x
d := denom x
(deriv n * d - n * deriv d) / (d**2)
if S has ConvertibleTo InputForm then
convert(x:%):InputForm == (convert numer x) / (convert denom x)
if S has RealConstant then
convert(x:%):Float == (convert numer x) / (convert denom x)
convert(x:%):DoubleFloat == (convert numer x) / (convert denom x)
-- Note that being a Join(OrderedSet,IntegralDomain) is not the same
-- as being an OrderedIntegralDomain.
if S has OrderedIntegralDomain then
if S has canonicalUnitNormal then
x:% < y:% ==
(numer x * denom y) < (numer y * denom x)
else
x:% < y:% ==
if negative? denom(x) then (x,y):=(y,x)
if negative? denom(y) then (x,y):=(y,x)
(numer x * denom y) < (numer y * denom x)
else if S has OrderedSet then
x:% < y:% ==
(numer x * denom y) < (numer y * denom x)
if (S has EuclideanDomain) then
fractionPart x == x - (wholePart(x)::%)
if S has RetractableTo Symbol then
coerce(s:Symbol):% == s::S::%
retract(x:%):Symbol == retract(retract(x)@S)
retractIfCan(x:%):Union(Symbol, "failed") ==
(r := retractIfCan(x)@Union(S,"failed")) case "failed" =>"failed"
retractIfCan(r::S)
if (S has ConvertibleTo Pattern Integer) then
convert(x:%):Pattern(Integer)==(convert numer x)/(convert denom x)
if (S has PatternMatchable Integer) then
patternMatch(x:%, p:Pattern Integer,
l:PatternMatchResult(Integer, %)) ==
patternMatch(x, p,
l)$PatternMatchQuotientFieldCategory(Integer, S, %)
if (S has ConvertibleTo Pattern Float) then
convert(x:%):Pattern(Float) == (convert numer x)/(convert denom x)
if (S has PatternMatchable Float) then
patternMatch(x:%, p:Pattern Float,
l:PatternMatchResult(Float, %)) ==
patternMatch(x, p,
l)$PatternMatchQuotientFieldCategory(Float, S, %)
if S has RetractableTo Integer then
coerce(x:Fraction Integer):% == numer(x)::% / denom(x)::%
if not(S is Integer) then
retract(x:%):Integer == retract(retract(x)@S)
retractIfCan(x:%):Union(Integer, "failed") ==
(u := retractIfCan(x)@Union(S, "failed")) case "failed" =>
"failed"
retractIfCan(u::S)
if S has IntegerNumberSystem then
random():% ==
d : S
while zero?(d:=random()$S) repeat d
random()$S / d
reducedSystem(m:Matrix %, v:Vector %):
Record(mat:Matrix S, vec:Vector S) ==
n := reducedSystem(horizConcat(v::Matrix(%), m))@Matrix(S)
[subMatrix(n, minRowIndex n, maxRowIndex n, 1 + minColIndex n,
maxColIndex n), column(n, minColIndex n)]
)abbrev package QFCAT2 QuotientFieldCategoryFunctions2
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ This package extends a function between integral domains
++ to a mapping between their quotient fields.
QuotientFieldCategoryFunctions2(A, B, R, S): Exports == Impl where
A, B: IntegralDomain
R : QuotientFieldCategory(A)
S : QuotientFieldCategory(B)
Exports ==> with
map: (A -> B, R) -> S
++ map(func,frac) applies the function func to the numerator
++ and denominator of frac.
Impl ==> add
map(f, r) == f(numer r) / f(denom r)
)abbrev domain FRAC Fraction
++ Author:
++ Date Created:
++ Date Last Updated: 12 February 1992
++ Basic Functions: Field, numer, denom
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords: fraction, localization
++ References:
++ Description: Fraction takes an IntegralDomain S and produces
++ the domain of Fractions with numerators and denominators from S.
++ If S is also a GcdDomain, then gcd's between numerator and
++ denominator will be cancelled during all operations.
Fraction(S: IntegralDomain): QuotientFieldCategory S with
if S has canonical and S has GcdDomain and S has canonicalUnitNormal
then canonical
++ \spad{canonical} means that equal elements are in fact identical.
== LocalAlgebra(S, S, S) add
Rep:= Record(num:S, den:S)
coerce(d:S):% == [d,1]
zero?(x:%) == zero? x.num
if S has GcdDomain and S has canonicalUnitNormal then
retract(x:%):S ==
one?(x.den) => x.num
error "Denominator not equal to 1"
retractIfCan(x:%):Union(S, "failed") ==
one?(x.den) => x.num
"failed"
else
retract(x:%):S ==
(a:= x.num exquo x.den) case "failed" =>
error "Denominator not equal to 1"
a
retractIfCan(x:%):Union(S,"failed") == x.num exquo x.den
if S has EuclideanDomain then
wholePart x ==
one?(x.den) => x.num
x.num quo x.den
if S has IntegerNumberSystem then
floor x ==
one?(x.den) => x.num
negative? x => -ceiling(-x)
wholePart x
ceiling x ==
one?(x.den) => x.num
negative? x => -floor(-x)
1 + wholePart x
if S has GcdDomain then
cancelGcd: % -> S
normalize: % -> %
normalize x ==
zero?(x.num) => 0
one?(x.den) => x
uca := unitNormal(x.den)
zero?(x.den := uca.canonical) => error "division by zero"
x.num := x.num * uca.associate
x
recip x ==
zero?(x.num) => "failed"
normalize [x.den, x.num]
cancelGcd x ==
one?(x.den) => x.den
d := gcd(x.num, x.den)
xn := x.num exquo d
xn case "failed" =>
error "gcd not gcd in QF cancelGcd (numerator)"
xd := x.den exquo d
xd case "failed" =>
error "gcd not gcd in QF cancelGcd (denominator)"
x.num := xn :: S
x.den := xd :: S
d
nn:S / dd:S ==
zero? dd => error "division by zero"
cancelGcd(z := [nn, dd])
normalize z
x + y ==
zero? y => x
zero? x => y
z := [x.den,y.den]
d := cancelGcd z
g := [z.den * x.num + z.num * y.num, d]
cancelGcd g
g.den := g.den * z.num * z.den
normalize g
-- We can not rely on the defaulting mechanism
-- to supply a definition for -, even though this
-- definition would do, for thefollowing reasons:
-- 1) The user could have defined a subtraction
-- in Localize, which would not work for
-- QuotientField;
-- 2) even if he doesn't, the system currently
-- places a default definition in Localize,
-- which uses Localize's +, which does not
-- cancel gcds
x - y ==
zero? y => x
z := [x.den, y.den]
d := cancelGcd z
g := [z.den * x.num - z.num * y.num, d]
cancelGcd g
g.den := g.den * z.num * z.den
normalize g
x:% * y:% ==
zero? x or zero? y => 0
one? x => y
one? y => x
(x, y) := ([x.num, y.den], [y.num, x.den])
cancelGcd x; cancelGcd y;
normalize [x.num * y.num, x.den * y.den]
n:Integer * x:% ==
y := [n::S, x.den]
cancelGcd y
normalize [x.num * y.num, y.den]
nn:S * x:% ==
y := [nn, x.den]
cancelGcd y
normalize [x.num * y.num, y.den]
differentiate(x:%, deriv:S -> S) ==
y := [deriv(x.den), x.den]
d := cancelGcd(y)
y.num := deriv(x.num) * y.den - x.num * y.num
(d, y.den) := (y.den, d)
cancelGcd y
y.den := y.den * d * d
normalize y
if S has canonicalUnitNormal then
x = y == (x.num = y.num) and (x.den = y.den)
--x / dd == (cancelGcd (z:=[x.num,dd*x.den]); normalize z)
one? x == one? (x.num) and one? (x.den)
-- again assuming canonical nature of representation
else
nn:S/dd:S == if zero? dd then error "division by zero" else [nn,dd]
recip x ==
zero?(x.num) => "failed"
[x.den, x.num]
if (S has RetractableTo Fraction Integer) then
retract(x:%):Fraction(Integer) == retract(retract(x)@S)
retractIfCan(x:%):Union(Fraction Integer, "failed") ==
(u := retractIfCan(x)@Union(S, "failed")) case "failed" => "failed"
retractIfCan(u::S)
else if (S has RetractableTo Integer) then
retract(x:%):Fraction(Integer) ==
retract(numer x) / retract(denom x)
retractIfCan(x:%):Union(Fraction Integer, "failed") ==
(n := retractIfCan numer x) case "failed" => "failed"
(d := retractIfCan denom x) case "failed" => "failed"
(n::Integer) / (d::Integer)
QFP ==> SparseUnivariatePolynomial %
DP ==> SparseUnivariatePolynomial S
import UnivariatePolynomialCategoryFunctions2(%,QFP,S,DP)
import UnivariatePolynomialCategoryFunctions2(S,DP,%,QFP)
if S has GcdDomain then
gcdPolynomial(pp,qq) ==
zero? pp => qq
zero? qq => pp
zero? degree pp or zero? degree qq => 1
denpp:="lcm"/[denom u for u in coefficients pp]
ppD:DP:=map(retract(#1*denpp),pp)
denqq:="lcm"/[denom u for u in coefficients qq]
qqD:DP:=map(retract(#1*denqq),qq)
g:=gcdPolynomial(ppD,qqD)
zero? degree g => 1
one? (lc:=leadingCoefficient g) => map(#1::%,g)
map(#1 / lc,g)
if (S has PolynomialFactorizationExplicit) then
-- we'll let the solveLinearPolynomialEquations operator
-- default from Field
pp,qq: QFP
lpp: List QFP
import Factored SparseUnivariatePolynomial %
if S has CharacteristicNonZero then
if S has canonicalUnitNormal and S has GcdDomain then
charthRoot x ==
n:= charthRoot x.num
n case nothing => nothing
d:=charthRoot x.den
d case nothing => nothing
just(n/d)
else
charthRoot x ==
-- to find x = p-th root of n/d
-- observe that xd is p-th root of n*d**(p-1)
ans:=charthRoot(x.num *
(x.den)**(characteristic$%-1)::NonNegativeInteger)
ans case nothing => nothing
just(ans / x.den)
clear: List % -> List S
clear l ==
d:="lcm"/[x.den for x in l]
[ x.num * (d exquo x.den)::S for x in l]
mat: Matrix %
conditionP mat ==
matD: Matrix S
matD:= matrix [ clear l for l in listOfLists mat ]
ansD := conditionP matD
ansD case "failed" => "failed"
ansDD:=ansD :: Vector(S)
[ ansDD(i)::% for i in 1..#ansDD]$Vector(%)
factorPolynomial(pp) ==
zero? pp => 0
denpp:="lcm"/[denom u for u in coefficients pp]
ppD:DP:=map(retract(#1*denpp),pp)
ff:=factorPolynomial ppD
den1:%:=denpp::%
lfact:List Record(flg:Union("nil", "sqfr", "irred", "prime"),
fctr:QFP, xpnt:Integer)
lfact:= [[w.flg,
if leadingCoefficient w.fctr =1 then map(#1::%,w.fctr)
else (lc:=(leadingCoefficient w.fctr)::%;
den1:=den1/lc**w.xpnt;
map(#1::%/lc,w.fctr)),
w.xpnt] for w in factorList ff]
makeFR(map(#1::%/den1,unit(ff)),lfact)
factorSquareFreePolynomial(pp) ==
zero? pp => 0
zero? degree pp => makeFR(pp,empty())
lcpp:=leadingCoefficient pp
pp:=pp/lcpp
denpp:="lcm"/[denom u for u in coefficients pp]
ppD:DP:=map(retract(#1*denpp),pp)
ff:=factorSquareFreePolynomial ppD
den1:%:=denpp::%/lcpp
lfact:List Record(flg:Union("nil", "sqfr", "irred", "prime"),
fctr:QFP, xpnt:Integer)
lfact:= [[w.flg,
if leadingCoefficient w.fctr =1 then map(#1::%,w.fctr)
else (lc:=(leadingCoefficient w.fctr)::%;
den1:=den1/lc**w.xpnt;
map(#1::%/lc,w.fctr)),
w.xpnt] for w in factorList ff]
makeFR(map(#1::%/den1,unit(ff)),lfact)
)abbrev package LPEFRAC LinearPolynomialEquationByFractions
++ Author: James Davenport
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ Given a PolynomialFactorizationExplicit ring, this package
++ provides a defaulting rule for the \spad{solveLinearPolynomialEquation}
++ operation, by moving into the field of fractions, and solving it there
++ via the \spad{multiEuclidean} operation.
LinearPolynomialEquationByFractions(R:PolynomialFactorizationExplicit): with
solveLinearPolynomialEquationByFractions: ( _
List SparseUnivariatePolynomial R, _
SparseUnivariatePolynomial R) -> _
Union(List SparseUnivariatePolynomial R, "failed")
++ solveLinearPolynomialEquationByFractions([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 exists.
== add
SupR ==> SparseUnivariatePolynomial R
F ==> Fraction R
SupF ==> SparseUnivariatePolynomial F
import UnivariatePolynomialCategoryFunctions2(R,SupR,F,SupF)
lp : List SupR
pp: SupR
pF: SupF
pullback : SupF -> Union(SupR,"failed")
pullback(pF) ==
pF = 0 => 0
c:=retractIfCan leadingCoefficient pF
c case "failed" => "failed"
r:=pullback reductum pF
r case "failed" => "failed"
monomial(c,degree pF) + r
solveLinearPolynomialEquationByFractions(lp,pp) ==
lpF:List SupF:=[map(#1@R::F,u) for u in lp]
pF:SupF:=map(#1@R::F,pp)
ans:= solveLinearPolynomialEquation(lpF,pF)$F
ans case "failed" => "failed"
[(vv:= pullback v;
vv case "failed" => return "failed";
vv)
for v in ans]
)abbrev package FRAC2 FractionFunctions2
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description: This package extends a map between integral domains to
++ a map between Fractions over those domains by applying the map to the
++ numerators and denominators.
FractionFunctions2(A, B): Exports == Impl where
A, B: IntegralDomain
R ==> Fraction A
S ==> Fraction B
Exports ==> with
map: (A -> B, R) -> S
++ map(func,frac) applies the function func to the numerator
++ and denominator of the fraction frac.
Impl ==> add
map(f, r) == map(f, r)$QuotientFieldCategoryFunctions2(A, B, R, S)
|