RealClosedField

reclos.spad line 258 [edit on github]

RealClosedField provides common access functions for all real closed fields.

* : (%, %) -> %

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

0 : () -> %

from AbelianMonoid

1 : () -> %

from MagmaWithUnit

< : (%, %) -> 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

abs : % -> %

from OrderedRing

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 : Fraction(Integer) -> %

from CoercibleFrom(Fraction(Integer))

coerce : Integer -> %

from NonAssociativeRing

coerce : % -> OutputForm

from CoercibleTo(OutputForm)

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 OrderedRing

nthRoot : (%, Integer) -> %

from RadicalCategory

one? : % -> Boolean

from MagmaWithUnit

opposite? : (%, %) -> Boolean

from AbelianMonoid

plenaryPower : (%, PositiveInteger) -> %

from NonAssociativeAlgebra(Integer)

positive? : % -> Boolean

from OrderedRing

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) gives a new number that prints as name

rename! : (%, OutputForm) -> %

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

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 AbelianMonoid

sign : % -> Integer

from OrderedRing

sizeLess? : (%, %) -> Boolean

from EuclideanDomain

smaller? : (%, %) -> Boolean

from Comparable

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 AbelianMonoid

~= : (%, %) -> Boolean

from BasicType

Module(Fraction(Integer))

PrincipalIdealDomain

PartialOrder

NonAssociativeSemiRing

BiModule(%, %)

Field

canonicalUnitNormal

Rng

CoercibleFrom(Integer)

TwoSidedRecip

OrderedAbelianGroup

SemiRing

EntireRing

NonAssociativeAlgebra(Fraction(Integer))

unitsKnown

RadicalCategory

Algebra(Integer)

LeftModule(Integer)

noZeroDivisors

RetractableTo(Fraction(Integer))

OrderedSet

UniqueFactorizationDomain

SemiGroup

NonAssociativeAlgebra(Integer)

RightModule(Fraction(Integer))

Magma

GcdDomain

LeftModule(%)

NonAssociativeSemiRng

CharacteristicZero

CommutativeRing

CoercibleFrom(Fraction(Integer))

Algebra(%)

OrderedAbelianMonoid

DivisionRing

LeftOreRing

CancellationAbelianMonoid

EuclideanDomain

canonicalsClosed

Comparable

RetractableTo(Integer)

OrderedCancellationAbelianMonoid

OrderedRing

CommutativeStar

AbelianMonoid

MagmaWithUnit

NonAssociativeRing

RightModule(%)

OrderedAbelianSemiGroup

Module(%)

CoercibleTo(OutputForm)

BiModule(Integer, Integer)

Module(Integer)

SemiRng

Monoid

NonAssociativeAlgebra(%)

Algebra(Fraction(Integer))

BasicType

Ring

RightModule(Integer)

LeftModule(Fraction(Integer))

AbelianSemiGroup

IntegralDomain

SetCategory

FullyRetractableTo(Fraction(Integer))

NonAssociativeRng

BiModule(Fraction(Integer), Fraction(Integer))

AbelianGroup