NMAlgebraicNumber

jnemo.spad line 207 [edit on github]

This domain allows the manipulation of Nemo algebraic numbers, i.e. algebraic closure of rational field, represented by minimal polynomials using the Nemo package for Julia (Calcium based). https://fredrikj.net/calcium/

0: %

from AbelianMonoid

1: %

from MagmaWithUnit

*: (%, %) -> %

from Magma

*: (%, Fraction Integer) -> %

from RightModule Fraction Integer

*: (%, Integer) -> %

x*n is the product of x by the integer n.

*: (%, NMFraction NMInteger) -> %

x*i is the multiplication by an rational number.

*: (%, NMInteger) -> %

x*i is the multiplication by an integer.

*: (Fraction Integer, %) -> %

from LeftModule Fraction Integer

*: (Integer, %) -> %

from AbelianGroup

*: (NMInteger, %) -> %

from JLObjectRing

*: (NonNegativeInteger, %) -> %

from AbelianMonoid

*: (PositiveInteger, %) -> %

from AbelianSemiGroup

+: (%, %) -> %

from AbelianSemiGroup

-: % -> %

from AbelianGroup

-: (%, %) -> %

from AbelianGroup

/: (%, %) -> %

from Field

=: (%, %) -> Boolean

from BasicType

^: (%, %) -> %

a^b returns the value of a raised to power b.

^: (%, Fraction Integer) -> %

from RadicalCategory

^: (%, Integer) -> %

from DivisionRing

^: (%, NonNegativeInteger) -> %

from MagmaWithUnit

^: (%, PositiveInteger) -> %

from Magma

~=: (%, %) -> Boolean

from BasicType

abs2: % -> %

abs2(a) returns the squared absolute value of a.

abs: % -> %

abs(a) returns the absolute value of a.

acospi: % -> %

acospi(x) returns acos(x)/%pi

algebraicInteger?: % -> Boolean

algebraicInteger?(a) tests whether or not a is an algebraic integer.

annihilate?: (%, %) -> Boolean

from Rng

antiCommutator: (%, %) -> %

from NonAssociativeSemiRng

asinpi: % -> %

asinpi(x) returns asin(x)/%pi

associates?: (%, %) -> Boolean

from EntireRing

associator: (%, %, %) -> %

from NonAssociativeRng

atanpi: % -> %

atanpi(x) returns atan(x)/%pi

ceiling: % -> %

ceiling(a) returns the smallest integer above or equal to a.

characteristic: () -> NonNegativeInteger

from NonAssociativeRing

coerce: % -> %

from Algebra %

coerce: % -> AlgebraicNumber

coerce(nan) coerces nan to AlgebraicNumber using the root of the minimal polynomial.

coerce: % -> JLObject

from JLObjectType

coerce: % -> OutputForm

from CoercibleTo OutputForm

coerce: Fraction Integer -> %

from Algebra Fraction Integer

coerce: Integer -> %

from NonAssociativeRing

coerce: NMFraction NMInteger -> %

from CoercibleFrom NMFraction NMInteger

coerce: NMInteger -> %

from CoercibleFrom NMInteger

commutator: (%, %) -> %

from NonAssociativeRng

conjugate: % -> %

conjugate(a) returns the complex conjugate of a.

conjugates: % -> JLVector %

conjugates(a) returns all the roots of a.

convert: % -> NMComplexField

from ConvertibleTo NMComplexField

convert: % -> String

from ConvertibleTo String

cospi: % -> %

cospi(x) returns cos(%pi*x).

crandom: (NonNegativeInteger, NonNegativeInteger) -> %

crandom(deg, bits) returns a random algebraic number (complex) of degree up to deg and coefficients size up to bits. Requires at least degree 2.

csign: % -> %

csign(a) returns an extension of the real sign function equivalent to a/sqrt(a^2).

D: % -> %

from DifferentialRing

D: (%, NonNegativeInteger) -> %

from DifferentialRing

degree: % -> JLInt64

degree(a) returns the degree of the minimal polynomial of a.

denominator: % -> NMInteger

denominator(anum) returns the denominator of anum, i.e. the leading coefficient of the minimal polynomial of a.

differentiate: % -> %

from DifferentialRing

differentiate: (%, NonNegativeInteger) -> %

from DifferentialRing

divide: (%, %) -> Record(quotient: %, remainder: %)

from EuclideanDomain

equal?: (%, %) -> Boolean

from NMRing

euclideanSize: % -> NonNegativeInteger

from EuclideanDomain

exact?: % -> Boolean

from NMRing

exactDivide: (%, %) -> %

from NMRing

expPiI: % -> %

expPiI(a) returns exp(%pi*%i*a).

expressIdealMember: (List %, %) -> Union(List %, failed)

from PrincipalIdealDomain

exquo: (%, %) -> Union(%, failed)

from EntireRing

extendedEuclidean: (%, %) -> Record(coef1: %, coef2: %, generator: %)

from EuclideanDomain

extendedEuclidean: (%, %, %) -> Union(Record(coef1: %, coef2: %), failed)

from EuclideanDomain

factor: % -> Factored %

from UniqueFactorizationDomain

factorPolynomial: SparseUnivariatePolynomial % -> Factored SparseUnivariatePolynomial %

from PolynomialFactorizationExplicit

factorSquareFreePolynomial: SparseUnivariatePolynomial % -> Factored SparseUnivariatePolynomial %

from PolynomialFactorizationExplicit

floor: % -> %

floor(a) returns the largest integer below or equal ot a.

gcd: (%, %) -> %

from GcdDomain

gcd: List % -> %

from GcdDomain

gcdPolynomial: (SparseUnivariatePolynomial %, SparseUnivariatePolynomial %) -> SparseUnivariatePolynomial %

from GcdDomain

height: % -> NonNegativeInteger

height(a) returns the height of a.

heightBits: % -> NonNegativeInteger

heightBits(a) returns the height of a as a number of bits.

imag: % -> %

imag(x) returns imaginary part of x.

integer?: % -> Boolean

integer?(x) tests whether or not x is an integer.

inv: % -> %

from DivisionRing

jlAbout: % -> Void

from JLObjectType

jlApply: (String, %) -> %

from JLObjectType

jlApply: (String, %, %) -> %

from JLObjectType

jlApply: (String, %, %, %) -> %

from JLObjectType

jlApply: (String, %, %, %, %) -> %

from JLObjectType

jlApply: (String, %, %, %, %, %) -> %

from JLObjectType

jlDisplay: % -> Void

from JLObjectType

jlDump: JLObject -> Void

from JLObjectType

jlId: % -> JLInt64

from JLObjectType

jlNMRing: () -> String

from NMRing

jlObject: () -> String

from JLObjectType

jlRef: % -> SExpression

from JLObjectType

jlref: String -> %

from JLObjectType

jlType: % -> String

from JLObjectType

jnan: Fraction Integer -> %

jnan(q) returns q as a NMAlgebraicNumber.

jnan: Integer -> %

jnan(x) returns x as a NMAlgebraicNumber.

jnan: NMFraction NMInteger -> %

jnan(q) returns q as a NMAlgebraicNumber.

jnan: NMInteger -> %

jnan(x) returns x as a NMAlgebraicNumber.

jnan: String -> %

jnan(str) evaluates str in Julia that returns a NMAlgebraicNumber.

latex: % -> String

from SetCategory

lcm: (%, %) -> %

from GcdDomain

lcm: List % -> %

from GcdDomain

lcmCoef: (%, %) -> Record(llcm_res: %, coeff1: %, coeff2: %)

from LeftOreRing

leftPower: (%, NonNegativeInteger) -> %

from MagmaWithUnit

leftPower: (%, PositiveInteger) -> %

from Magma

leftRecip: % -> Union(%, failed)

from MagmaWithUnit

logPiI: % -> %

logPiI(a) returns log(a)/(%pi*%i).

minimalPolynomial: % -> SparseUnivariatePolynomial %

minimalPolynomial(an) returns the minimal polynomial of an over algebraic numbers.

minimalPolynomial: % -> SparseUnivariatePolynomial Integer

minimalPolynomial(an) returns the minimal polynomial of an.

multiEuclidean: (List %, %) -> Union(List %, failed)

from EuclideanDomain

mutable?: % -> Boolean

from JLObjectType

nothing?: % -> Boolean

from JLObjectType

nthRoot: (%, Integer) -> %

from RadicalCategory

numerator: % -> %

numerator(anum) returns anum multiplied by its denominator i.e. an algebraic integer.

one?: % -> Boolean

from MagmaWithUnit

opposite?: (%, %) -> Boolean

from AbelianMonoid

plenaryPower: (%, PositiveInteger) -> %

from NonAssociativeAlgebra %

prime?: % -> Boolean

from UniqueFactorizationDomain

principalIdeal: List % -> Record(coef: List %, generator: %)

from PrincipalIdealDomain

QQbar: Fraction Integer -> %

jnan(q) returns q as a NMAlgebraicNumber.

QQbar: Integer -> %

jnan(x) returns x as a NMAlgebraicNumber.

QQbar: NMFraction NMInteger -> %

jnan(q) returns q as a NMAlgebraicNumber.

QQbar: NMInteger -> %

jnan(x) returns x as a NMAlgebraicNumber.

quo: (%, %) -> %

from EuclideanDomain

random: (NonNegativeInteger, NonNegativeInteger) -> %

random(deg, bits) returns a random algebraic number (real) of degree up to deg and coefficients size up to bits.

rational?: % -> Boolean

rational?(x) tests whether or not x is a rational number.

real?: % -> Boolean

real?(x) tests whether or not x is a real number.

real: % -> %

real(x) returns real part of x.

recip: % -> Union(%, failed)

from MagmaWithUnit

rem: (%, %) -> %

from EuclideanDomain

retract: % -> Fraction Integer

from RetractableTo Fraction Integer

retract: % -> Integer

from RetractableTo Integer

retract: % -> NMFraction NMInteger

from RetractableTo NMFraction NMInteger

retract: % -> NMInteger

from RetractableTo NMInteger

retractIfCan: % -> Union(Fraction Integer, failed)

from RetractableTo Fraction Integer

retractIfCan: % -> Union(Integer, failed)

from RetractableTo Integer

retractIfCan: % -> Union(NMFraction NMInteger, failed)

from RetractableTo NMFraction NMInteger

retractIfCan: % -> Union(NMInteger, failed)

from RetractableTo NMInteger

rightPower: (%, NonNegativeInteger) -> %

from MagmaWithUnit

rightPower: (%, PositiveInteger) -> %

from Magma

rightRecip: % -> Union(%, failed)

from MagmaWithUnit

rootOf: (SparseUnivariatePolynomial %, Symbol) -> %

from AlgebraicallyClosedField

rootOf: Polynomial % -> %

from AlgebraicallyClosedField

rootOf: SparseUnivariatePolynomial % -> %

from AlgebraicallyClosedField

rootOfUnity?: % -> Boolean

rootOfUnity?(x) tests whether or not x is a root of unity.

rootOfUnity: (NonNegativeInteger, Integer) -> %

rootOfUnity(n,k)Return the root of unity exp(2*%pi*%i*k/n).

rootOfUnity: NonNegativeInteger -> %

rootOfUnity(n)Return the root of unity exp(2*%pi*%i/n).

rootsOf: (SparseUnivariatePolynomial %, Symbol) -> List %

from AlgebraicallyClosedField

rootsOf: Polynomial % -> List %

from AlgebraicallyClosedField

rootsOf: SparseUnivariatePolynomial % -> List %

from AlgebraicallyClosedField

sample: %

from AbelianMonoid

sign: % -> %

sign(a) returns the complex sign of a.

signImag: % -> %

signImag(a) returns the sign of the imaginary part.

signReal: % -> %

signReal(a) returns the sign of the real part.

sinpi: % -> %

sinpi(x) returns sin(%pi*x).

sizeLess?: (%, %) -> Boolean

from EuclideanDomain

solveLinearPolynomialEquation: (List SparseUnivariatePolynomial %, SparseUnivariatePolynomial %) -> Union(List SparseUnivariatePolynomial %, failed)

from PolynomialFactorizationExplicit

sqrt: % -> %

from RadicalCategory

squareFree: % -> Factored %

from UniqueFactorizationDomain

squareFreePart: % -> %

from UniqueFactorizationDomain

squareFreePolynomial: SparseUnivariatePolynomial % -> Factored SparseUnivariatePolynomial %

from PolynomialFactorizationExplicit

string: % -> String

from JLType

subtractIfCan: (%, %) -> Union(%, failed)

from CancellationAbelianMonoid

tanpi: % -> %

tanpi(x) returns tan(%pi*x).

unit?: % -> Boolean

from EntireRing

unitCanonical: % -> %

from EntireRing

unitNormal: % -> Record(unit: %, canonical: %, associate: %)

from EntireRing

zero?: % -> Boolean

from AbelianMonoid

zeroOf: (SparseUnivariatePolynomial %, Symbol) -> %

from AlgebraicallyClosedField

zeroOf: Polynomial % -> %

from AlgebraicallyClosedField

zeroOf: SparseUnivariatePolynomial % -> %

from AlgebraicallyClosedField

zerosOf: (SparseUnivariatePolynomial %, Symbol) -> List %

from AlgebraicallyClosedField

zerosOf: Polynomial % -> List %

from AlgebraicallyClosedField

zerosOf: SparseUnivariatePolynomial % -> List %

from AlgebraicallyClosedField

AbelianGroup

AbelianMonoid

AbelianSemiGroup

Algebra %

Algebra Fraction Integer

AlgebraicallyClosedField

BasicType

BiModule(%, %)

BiModule(Fraction Integer, Fraction Integer)

CancellationAbelianMonoid

canonicalsClosed

canonicalUnitNormal

CharacteristicZero

CoercibleFrom Fraction Integer

CoercibleFrom Integer

CoercibleFrom NMFraction NMInteger

CoercibleFrom NMInteger

CoercibleTo OutputForm

CommutativeRing

CommutativeStar

ConvertibleTo NMComplexField

ConvertibleTo String

DifferentialRing

DivisionRing

EntireRing

EuclideanDomain

Field

GcdDomain

IntegralDomain

JLObjectRing

JLObjectType

JLRing

JLType

LeftModule %

LeftModule Fraction Integer

LeftOreRing

Magma

MagmaWithUnit

Module %

Module Fraction Integer

Monoid

NMCommutativeRing

NMRing

NMType

NonAssociativeAlgebra %

NonAssociativeAlgebra Fraction Integer

NonAssociativeRing

NonAssociativeRng

NonAssociativeSemiRing

NonAssociativeSemiRng

noZeroDivisors

PolynomialFactorizationExplicit

PrincipalIdealDomain

RadicalCategory

RetractableTo Fraction Integer

RetractableTo Integer

RetractableTo NMFraction NMInteger

RetractableTo NMInteger

RightModule %

RightModule Fraction Integer

Ring

Rng

SemiGroup

SemiRing

SemiRng

SetCategory

TwoSidedRecip

UniqueFactorizationDomain

unitsKnown