RealClosedField

reclos.spad line 252 [edit on github]

RealClosedField provides common access functions for all real closed fields.

0: %

from SetWithZero

1: %

from MagmaWithUnit

*: (%, %) -> %

from Magma

*: (%, Fraction Integer) -> %

from RightModule Fraction Integer

*: (%, Integer) -> %

from RightModule Integer

*: (Fraction Integer, %) -> %

from LeftModule Fraction Integer

*: (Integer, %) -> %

from AbelianGroup

*: (NonNegativeInteger, %) -> %

from AbelianMonoid

*: (PositiveInteger, %) -> %

from AbelianSemiGroup

+: (%, %) -> %

from AbelianSemiGroup

-: % -> %

from AbelianGroup

-: (%, %) -> %

from AbelianGroup

/: (%, %) -> %

from Field

<=: (%, %) -> Boolean

from PartialOrder

<: (%, %) -> Boolean

from PartialOrder

=: (%, %) -> Boolean

from BasicType

>=: (%, %) -> Boolean

from PartialOrder

>: (%, %) -> Boolean

from PartialOrder

^: (%, Fraction Integer) -> %

from RadicalCategory

^: (%, Integer) -> %

from DivisionRing

^: (%, NonNegativeInteger) -> %

from MagmaWithUnit

^: (%, PositiveInteger) -> %

from Magma

~=: (%, %) -> Boolean

from BasicType

abs: % -> %

from OrderedAbelianSemiGroup

allRootsOf: Polynomial % -> List %

allRootsOf(pol) creates all the roots of pol naming each uniquely

allRootsOf: Polynomial Fraction Integer -> List %

allRootsOf(pol) creates all the roots of pol naming each uniquely

allRootsOf: Polynomial Integer -> List %

allRootsOf(pol) creates all the roots of pol naming each uniquely

allRootsOf: SparseUnivariatePolynomial % -> List %

allRootsOf(pol) creates all the roots of pol naming each uniquely

allRootsOf: SparseUnivariatePolynomial Fraction Integer -> List %

allRootsOf(pol) creates all the roots of pol naming each uniquely

allRootsOf: SparseUnivariatePolynomial Integer -> List %

allRootsOf(pol) creates all the roots of pol naming each uniquely

annihilate?: (%, %) -> Boolean

from Rng

antiCommutator: (%, %) -> %

from NonAssociativeSemiRng

approximate: (%, %) -> Fraction Integer

approximate(n, p) gives an approximation of n that has precision p

associates?: (%, %) -> Boolean

from EntireRing

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

from NonAssociativeRng

characteristic: () -> NonNegativeInteger

from NonAssociativeRing

coerce: % -> %

from Algebra %

coerce: % -> OutputForm

from CoercibleTo OutputForm

coerce: Fraction Integer -> %

from Algebra Fraction Integer

coerce: Integer -> %

from NonAssociativeRing

commutator: (%, %) -> %

from NonAssociativeRng

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

from EuclideanDomain

euclideanSize: % -> NonNegativeInteger

from EuclideanDomain

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

gcd: (%, %) -> %

from GcdDomain

gcd: List % -> %

from GcdDomain

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

from GcdDomain

inv: % -> %

from DivisionRing

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

mainDefiningPolynomial: % -> Union(SparseUnivariatePolynomial %, failed)

mainDefiningPolynomial(x) is the defining polynomial for the main algebraic quantity of x

mainForm: % -> Union(OutputForm, failed)

mainForm(x) is the main algebraic quantity name of x

mainValue: % -> Union(SparseUnivariatePolynomial %, failed)

mainValue(x) is the expression of x in terms of SparseUnivariatePolynomial(%)

max: (%, %) -> %

from OrderedSet

min: (%, %) -> %

from OrderedSet

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

from EuclideanDomain

negative?: % -> Boolean

from OrderedSet

nthRoot: (%, Integer) -> %

from RadicalCategory

one?: % -> Boolean

from MagmaWithUnit

opposite?: (%, %) -> Boolean

from AbelianMonoid

plenaryPower: (%, PositiveInteger) -> %

from NonAssociativeAlgebra Fraction Integer

positive?: % -> Boolean

from OrderedSet

prime?: % -> Boolean

from UniqueFactorizationDomain

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

from PrincipalIdealDomain

quo: (%, %) -> %

from EuclideanDomain

recip: % -> Union(%, failed)

from MagmaWithUnit

rem: (%, %) -> %

from EuclideanDomain

rename!: (%, OutputForm) -> %

rename!(x, name) changes the way x is printed

rename: (%, OutputForm) -> %

rename(x, name) gives a new number that prints as name

retract: % -> Fraction Integer

from RetractableTo Fraction Integer

retract: % -> Integer

from RetractableTo Integer

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

from RetractableTo Fraction Integer

retractIfCan: % -> Union(Integer, failed)

from RetractableTo Integer

rightPower: (%, NonNegativeInteger) -> %

from MagmaWithUnit

rightPower: (%, PositiveInteger) -> %

from Magma

rightRecip: % -> Union(%, failed)

from MagmaWithUnit

rootOf: (SparseUnivariatePolynomial %, PositiveInteger) -> Union(%, failed)

rootOf(pol, n) creates the nth root for the order of pol and gives it unique name

rootOf: (SparseUnivariatePolynomial %, PositiveInteger, OutputForm) -> Union(%, failed)

rootOf(pol, n, name) creates the nth root for the order of pol and names it name

sample: %

from SetWithZero

sign: % -> Integer

from OrderedSet

sizeLess?: (%, %) -> Boolean

from EuclideanDomain

smaller?: (%, %) -> Boolean

from ComparableRepresentation

sqrt: % -> %

sqrt(x) is x ^ (1/2)

sqrt: (%, PositiveInteger) -> %

sqrt(x, n) is x ^ (1/n)

sqrt: Fraction Integer -> %

sqrt(x) is x ^ (1/2)

sqrt: Integer -> %

sqrt(x) is x ^ (1/2)

squareFree: % -> Factored %

from UniqueFactorizationDomain

squareFreePart: % -> %

from UniqueFactorizationDomain

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

from CancellationAbelianMonoid

unit?: % -> Boolean

from EntireRing

unitCanonical: % -> %

from EntireRing

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

from EntireRing

zero?: % -> Boolean

from SetWithZero

AbelianGroup

AbelianMonoid

AbelianSemiGroup

Algebra %

Algebra Fraction Integer

Algebra Integer

BasicType

BiModule(%, %)

BiModule(Fraction Integer, Fraction Integer)

BiModule(Integer, Integer)

CancellationAbelianMonoid

canonicalsClosed

canonicalUnitNormal

CharacteristicZero

CoercibleFrom Fraction Integer

CoercibleFrom Integer

CoercibleTo OutputForm

CommutativeRing

CommutativeStar

Comparable

ComparableRepresentation

DivisionRing

EntireRing

EuclideanDomain

Field

FullyRetractableTo Fraction Integer

GcdDomain

IntegralDomain

LeftModule %

LeftModule Fraction Integer

LeftModule Integer

LeftOreRing

Magma

MagmaWithUnit

MagmaWithZero

Module %

Module Fraction Integer

Module Integer

Monoid

NonAssociativeAlgebra %

NonAssociativeAlgebra Fraction Integer

NonAssociativeAlgebra Integer

NonAssociativeRing

NonAssociativeRng

NonAssociativeSemiRing

NonAssociativeSemiRng

noZeroDivisors

OrderedAbelianSemiGroup

OrderedMagmaWithZero

OrderedRing

OrderedSet

PartialOrder

PrincipalIdealDomain

RadicalCategory

RetractableTo Fraction Integer

RetractableTo Integer

RightModule %

RightModule Fraction Integer

RightModule Integer

Ring

Rng

SemiGroup

SemiRing

SemiRng

SetCategory

SetWithZero

TwoSidedRecip

UniqueFactorizationDomain

unitsKnown