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constant.spad
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constant.spad
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)abbrev domain AN AlgebraicNumber
++ Author: Manuel Bronstein
++ Date Created: 22 March 1988
++ Description: Algebraic closure of the rational numbers.
++ Keywords: algebraic, number.
AlgebraicNumber() : Exports == Implementation where
Z ==> Integer
FE ==> Expression Z
K ==> Kernel %
P ==> SparseMultivariatePolynomial(Z, K)
SUP ==> SparseUnivariatePolynomial
Exports ==> Join(ExpressionSpace, AlgebraicallyClosedField,
RetractableTo Z, RetractableTo Fraction Z,
LinearlyExplicitOver Z, RealConstant,
LinearlyExplicitOver Fraction Z,
PolynomialFactorizationExplicit,
CharacteristicZero,
ConvertibleTo Complex Float,
ConvertibleTo InputForm,
DifferentialRing) with
coerce : P -> %
++ coerce(p) returns p viewed as an algebraic number.
numer : % -> P
++ numer(f) returns the numerator of f viewed as a
++ polynomial in the kernels over Z.
denom : % -> P
++ denom(f) returns the denominator of f viewed as a
++ polynomial in the kernels over Z.
reduce : % -> %
++ reduce(f) simplifies all the unreduced algebraic numbers
++ present in f by applying their defining relations.
trueEqual : (%, %) -> Boolean
++ trueEqual(x, y) tries to determine if the two numbers are equal
norm : (SUP(%), Kernel %) -> SUP(%)
++ norm(p, k) computes the norm of the polynomial p
++ with respect to the extension generated by kernel k
norm : (SUP(%), List Kernel %) -> SUP(%)
++ norm(p, l) computes the norm of the polynomial p
++ with respect to the extension generated by kernels l
norm : (%, Kernel %) -> %
++ norm(f, k) computes the norm of the algebraic number f
++ with respect to the extension generated by kernel k
norm : (%, List Kernel %) -> %
++ norm(f, l) computes the norm of the algebraic number f
++ with respect to the extension generated by kernels l
Implementation ==> FE add
Rep := FE
ALGOP := '%alg
reducedSystem(m : Matrix(%)) : Matrix(Fraction(Z)) ==
mz : Matrix(Z) := reducedSystem(m)$Rep
map((c : Z) : Fraction(Z) +-> c::Fraction(Z), mz
)$MatrixCategoryFunctions2(Z, Vector(Z), Vector(Z), Matrix(Z),
Fraction(Z), Vector(Fraction(Z)), Vector(Fraction(Z)),
Matrix(Fraction(Z)))
reducedSystem(m : Matrix(%), v : Vector(%)
) : Record(mat : Matrix(Fraction(Z)), vec : Vector(Fraction(Z))) ==
mzr : Record(mat : Matrix(Z), vec : Vector(Z)) :=
reducedSystem(m, v)$Rep
mq := map((c : Z) : Fraction(Z) +-> c::Fraction(Z), mzr.mat
)$MatrixCategoryFunctions2(Z, Vector(Z), Vector(Z), Matrix(Z),
Fraction(Z), Vector(Fraction(Z)), Vector(Fraction(Z)),
Matrix(Fraction(Z)))
vq := map((c : Z) : Fraction(Z) +-> c::Fraction(Z), mzr.vec
)$VectorFunctions2(Z, Fraction(Z))
[mq, vq]
-- private
mainRatDenom(f : %) : % ==
ratDenom(f::Rep::FE)$AlgebraicManipulations(Integer, FE)::Rep::%
-- mv := mainVariable denom f
-- mv case "failed" => f
-- algv := mv::K
-- q := univariate(f, algv, minPoly(algv))$PolynomialCategoryQuotientFunctions(IndexedExponents K, K, Integer, P, %)
-- q(algv::%)
findDenominator(z : SUP %) : Record(num : SUP %, den : %) ==
zz := z
while not(zz = 0) repeat
dd := (denom leadingCoefficient zz)::%
not(dd = 1) =>
rec := findDenominator(dd*z)
return [rec.num, rec.den*dd]
zz := reductum zz
[z, 1]
makeUnivariate(p : P, k : Kernel %) : SUP % ==
map(x +-> x::%,
univariate(p, k))$SparseUnivariatePolynomialFunctions2(P, %)
-- public
a, b : %
differentiate(x : %) : % == 0
zero? a == zero? numer a
one? a == (numer a = 1) and (denom a = 1)
x : % / y : % == mainRatDenom(x /$Rep y)
x : % ^ n : Integer ==
n < 0 => mainRatDenom (x ^$Rep n)
x ^$Rep n
trueEqual(a, b) ==
-- if two algebraic numbers have the same norm (after deleting repeated
-- roots, then they are certainly conjugates. Note that we start with a
-- monic polynomial, so don't have to check for constant factors.
-- this will be fooled by sqrt(2) and -sqrt(2), but the = in
-- AlgebraicNumber knows what to do about this.
ka := reverse tower a
kb := reverse tower b
empty? ka and empty? kb => retract(a)@Fraction Z = retract(b)@Fraction Z
pa, pb : SparseUnivariatePolynomial %
pa := monomial(1, 1)-monomial(a, 0)
pb := monomial(1, 1)-monomial(b, 0)
na := map(retract, norm(pa, ka))$SparseUnivariatePolynomialFunctions2(%, Fraction Z)
nb := map(retract, norm(pb, kb))$SparseUnivariatePolynomialFunctions2(%, Fraction Z)
(sa := squareFreePart(na)) = (sb := squareFreePart(nb)) => true
g := gcd(sa, sb)
(dg := degree g) = 0 => false
-- of course, if these have a factor in common, then the
-- answer is really ambiguous, so we ought to be using Duval-type
-- technology
dg = degree sa or dg = degree sb => true
false
norm(z : %, k : Kernel %) : % ==
p := minPoly k
n := makeUnivariate(numer z, k)
d := makeUnivariate(denom z, k)
resultant(n, p)/resultant(d, p)
norm(z : %, l : List Kernel %) : % ==
for k in l repeat
z := norm(z, k)
z
norm(z : SUP %, k : Kernel %) : SUP % ==
p := map(x +-> x::SUP %,
minPoly k)$SparseUnivariatePolynomialFunctions2(%, SUP %)
f := findDenominator z
zz := map(x +-> makeUnivariate(numer x, k),
f.num)$SparseUnivariatePolynomialFunctions2(%, SUP %)
zz := swap(zz)$CommuteUnivariatePolynomialCategory(%, SUP %, SUP SUP %)
resultant(p, zz)/norm(f.den, k)
norm(z : SUP %, l : List Kernel %) : SUP % ==
for k in l repeat
z := norm(z, k)
z
belong? op == belong?(op)$ExpressionSpace2_&(%, K) or has?(op, ALGOP)
convert(x : %) : InputForm ==
coerceToType(convert(x pretend Rep)$Rep)$InputFormFunctions1(%)
convert(x : %) : Float ==
retract map(y +-> y::Float, x pretend FE)$ExpressionFunctions2(Z, Float)
convert(x : %) : DoubleFloat ==
retract map(y +-> y::DoubleFloat,
x pretend FE)$ExpressionFunctions2(Z, DoubleFloat)
convert(x : %) : Complex(Float) ==
retract map(y +-> y::Complex(Float),
x pretend FE)$ExpressionFunctions2(Z, Complex Float)
--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.