NMComplexField

jnball.spad line 1385 [edit on github]

NMComplexField implements arbitrary precision ball arithmetic using the Nemo Julia package.

0: %

from SetWithZero

1: %

from MagmaWithUnit

*: (%, %) -> %

from Magma

*: (%, Fraction Integer) -> %

from RightModule Fraction Integer

*: (%, Integer) -> %

x*i is the multiplication by an integer.

*: (%, NMRealField) -> %

from RightModule NMRealField

*: (Fraction Integer, %) -> %

from LeftModule Fraction Integer

*: (Integer, %) -> %

from AbelianGroup

*: (NMInteger, %) -> JLObject

from JLObjectRing

*: (NMRealField, %) -> %

from LeftModule NMRealField

*: (NonNegativeInteger, %) -> %

from AbelianMonoid

*: (PositiveInteger, %) -> %

from AbelianSemiGroup

+: (%, %) -> %

from AbelianSemiGroup

-: % -> %

from AbelianGroup

-: (%, %) -> %

from AbelianGroup

/: (%, %) -> %

from Field

/: (%, Integer) -> %

x/i is the division by an integer.

/: (Integer, %) -> %

i/x is the division of an integer by x.

=: (%, %) -> Boolean

from BasicType

^: (%, %) -> %

from ElementaryFunctionCategory

^: (%, Fraction Integer) -> %

from RadicalCategory

^: (%, Integer) -> %

from DivisionRing

^: (%, NonNegativeInteger) -> %

from MagmaWithUnit

^: (%, PositiveInteger) -> %

from Magma

~=: (%, %) -> Boolean

from BasicType

abs2: % -> NMRealField

abs2(x) returns the square of the absolute value of real part of x and imaginary part of x.

abs: % -> NMRealField

from ComplexCategory NMRealField

accuracyBits: % -> JLInt64

accuracyBits(x) returns the relative accuracy of x in bits.

acos: % -> %

from ArcTrigonometricFunctionCategory

acosh: % -> %

from ArcHyperbolicFunctionCategory

acot: % -> %

from ArcTrigonometricFunctionCategory

acoth: % -> %

from ArcHyperbolicFunctionCategory

acsc: % -> %

from ArcTrigonometricFunctionCategory

acsch: % -> %

from ArcHyperbolicFunctionCategory

agm: % -> %

agm(x) returns the arithmetic-geometric mean of 1 and x.

agm: (%, %) -> %

agm(x,y) returns the arithmetic-geometric mean of x and y.

airyAi: R -> R

airyAi(z) is the Airy function Ai(z).

airyAiPrime: R -> R

airyAiPrime(z) is the derivative of the Airy function Ai(z).

airyBi: R -> R

airyBi(z) is the Airy function Bi(z).

airyBiPrime: R -> R

airyBiPrime(z) is the derivative of the Airy function Bi(z).

annihilate?: (%, %) -> Boolean

from Rng

antiCommutator: (%, %) -> %

from NonAssociativeSemiRng

argument: % -> NMRealField

from ComplexCategory NMRealField

asec: % -> %

from ArcTrigonometricFunctionCategory

asech: % -> %

from ArcHyperbolicFunctionCategory

asin: % -> %

from ArcTrigonometricFunctionCategory

asinh: % -> %

from ArcHyperbolicFunctionCategory

associates?: (%, %) -> Boolean

from EntireRing

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

from NonAssociativeRng

atan: % -> %

from ArcTrigonometricFunctionCategory

atanh: % -> %

from ArcHyperbolicFunctionCategory

barnesG: % -> %

barnesG(x) is the Barnes G-function evaluated at x.

basis: () -> Vector %

from FramedModule NMRealField

besselI: (%, %) -> %

besselI(nu,x) is the Bessel I function.

besselJ: (%, %) -> %

besselJ(nu,x) is the Bessel J function.

besselK: (%, %) -> %

besselK(nu,x) is the Bessel K function.

besselY: (%, %) -> %

besselY(nu,x) is the Bessel Y function.

characteristic: () -> NonNegativeInteger

from NonAssociativeRing

characteristicPolynomial: % -> SparseUnivariatePolynomial NMRealField

from FiniteRankAlgebra(NMRealField, SparseUnivariatePolynomial NMRealField)

charthRoot: % -> % if NMRealField has FiniteFieldCategory

from FiniteFieldCategory

charthRoot: % -> Union(%, failed) if % has CharacteristicNonZero and NMRealField has PolynomialFactorizationExplicit or NMRealField has FiniteFieldCategory

from PolynomialFactorizationExplicit

Chi: % -> %

Chi(x) returns the hyperbolic cosine integral of x.

Ci: % -> %

Ci(x) returns the cosine integral of x.

cispi: % -> %

cispi(x) returns exp(πix).

coerce: % -> %

from Algebra %

coerce: % -> JLObject

from JLObjectType

coerce: % -> OutputForm

from CoercibleTo OutputForm

coerce: Complex Integer -> %

coerce(z) coerces z. Convenience function.

coerce: Float -> %

coerce(x) coerces x.

coerce: Fraction Integer -> %

from Algebra Fraction Integer

coerce: Integer -> %

from NonAssociativeRing

coerce: JLFloat64 -> %

coerce(x) coerces x.

coerce: NMRealField -> %

from CoercibleFrom NMRealField

coerce: String -> %

coerce(str) returns a NMComplexField element from evaluation in Julia of str.

commutator: (%, %) -> %

from NonAssociativeRng

complex: (NMRealField, NMRealField) -> %

from ComplexCategory NMRealField

conditionP: Matrix % -> Union(Vector %, failed) if % has CharacteristicNonZero and NMRealField has PolynomialFactorizationExplicit or NMRealField has FiniteFieldCategory

from PolynomialFactorizationExplicit

conjugate: % -> %

from ComplexCategory NMRealField

contains?: (%, %) -> Boolean

contains?(x,y) checks whether or not y is contained in x.

contains?: (%, NMFraction NMInteger) -> Boolean

contains?(x,y) checks whether or not y is contained in x.

contains?: (%, NMInteger) -> Boolean

contains?(x,y) checks whether or not y is contained in x.

containsZero?: % -> Boolean

containsZero?(x) checks whether or not 0 is contained in x.

convert: % -> Complex DoubleFloat

from ConvertibleTo Complex DoubleFloat

convert: % -> Complex Float

from ConvertibleTo Complex Float

convert: % -> InputForm if NMRealField has ConvertibleTo InputForm

from ConvertibleTo InputForm

convert: % -> Pattern Float

from ConvertibleTo Pattern Float

convert: % -> Pattern Integer if NMRealField has ConvertibleTo Pattern Integer

from ConvertibleTo Pattern Integer

convert: % -> SparseUnivariatePolynomial NMRealField

from ConvertibleTo SparseUnivariatePolynomial NMRealField

convert: % -> String

from ConvertibleTo String

convert: % -> Vector NMRealField

from FramedModule NMRealField

convert: SparseUnivariatePolynomial NMRealField -> %

from MonogenicAlgebra(NMRealField, SparseUnivariatePolynomial NMRealField)

convert: Vector NMRealField -> %

from FramedModule NMRealField

coordinates: % -> Vector NMRealField

from FramedModule NMRealField

coordinates: (%, Vector %) -> Vector NMRealField

from FiniteRankAlgebra(NMRealField, SparseUnivariatePolynomial NMRealField)

coordinates: (Vector %, Vector %) -> Matrix NMRealField

from FiniteRankAlgebra(NMRealField, SparseUnivariatePolynomial NMRealField)

coordinates: Vector % -> Matrix NMRealField

from FramedModule NMRealField

cos: % -> %

from TrigonometricFunctionCategory

cosh: % -> %

from HyperbolicFunctionCategory

cot: % -> %

from TrigonometricFunctionCategory

coth: % -> %

from HyperbolicFunctionCategory

createPrimitiveElement: () -> % if NMRealField has FiniteFieldCategory

from FiniteFieldCategory

csc: % -> %

from TrigonometricFunctionCategory

csch: % -> %

from HyperbolicFunctionCategory

D: % -> %

from DifferentialRing

D: (%, List Symbol) -> % if NMRealField has PartialDifferentialRing Symbol

from PartialDifferentialRing Symbol

D: (%, List Symbol, List NonNegativeInteger) -> % if NMRealField has PartialDifferentialRing Symbol

from PartialDifferentialRing Symbol

D: (%, NMRealField -> NMRealField) -> %

from DifferentialExtension NMRealField

D: (%, NMRealField -> NMRealField, NonNegativeInteger) -> %

from DifferentialExtension NMRealField

D: (%, NonNegativeInteger) -> %

from DifferentialRing

D: (%, Symbol) -> % if NMRealField has PartialDifferentialRing Symbol

from PartialDifferentialRing Symbol

D: (%, Symbol, NonNegativeInteger) -> % if NMRealField has PartialDifferentialRing Symbol

from PartialDifferentialRing Symbol

dedekindEta: % -> %

dedekindEta(z) returns the Dedekind eta function evaluated at z.

definingPolynomial: () -> SparseUnivariatePolynomial NMRealField

from MonogenicAlgebra(NMRealField, SparseUnivariatePolynomial NMRealField)

derivationCoordinates: (Vector %, NMRealField -> NMRealField) -> Matrix NMRealField

from MonogenicAlgebra(NMRealField, SparseUnivariatePolynomial NMRealField)

differentiate: % -> %

from DifferentialRing

differentiate: (%, List Symbol) -> % if NMRealField has PartialDifferentialRing Symbol

from PartialDifferentialRing Symbol

differentiate: (%, List Symbol, List NonNegativeInteger) -> % if NMRealField has PartialDifferentialRing Symbol

from PartialDifferentialRing Symbol

differentiate: (%, NMRealField -> NMRealField) -> %

from DifferentialExtension NMRealField

differentiate: (%, NMRealField -> NMRealField, NonNegativeInteger) -> %

from DifferentialExtension NMRealField

differentiate: (%, NonNegativeInteger) -> %

from DifferentialRing

differentiate: (%, Symbol) -> % if NMRealField has PartialDifferentialRing Symbol

from PartialDifferentialRing Symbol

differentiate: (%, Symbol, NonNegativeInteger) -> % if NMRealField has PartialDifferentialRing Symbol

from PartialDifferentialRing Symbol

digamma: % -> %

digamma(x) returns logarithmic derivative of the gamma function evaluated at x.

discreteLog: % -> NonNegativeInteger if NMRealField has FiniteFieldCategory

from FiniteFieldCategory

discreteLog: (%, %) -> Union(NonNegativeInteger, failed) if NMRealField has FiniteFieldCategory

from FieldOfPrimeCharacteristic

discriminant: () -> NMRealField

from FramedAlgebra(NMRealField, SparseUnivariatePolynomial NMRealField)

discriminant: Vector % -> NMRealField

from FiniteRankAlgebra(NMRealField, SparseUnivariatePolynomial NMRealField)

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

from EuclideanDomain

Ei: % -> %

Ei(x) returns the exponential integral of x.

Ei: (%, %) -> %

Ei(s,x) returns the generalized exponential integral of x E_s(x).

eisensteinG: (Integer, %) -> %

eisensteinG(k,z) returns the non-normalized Eisenstein series G_k(τ) of SL_2(Z) also defined for τ=i∞.

ellipticE: % -> %

ellipticE(x) is the complete elliptic integral of the second kind.

ellipticK: % -> %

ellipticK(x) is the complete elliptic integral of the first kind.

elt: (%, NMRealField) -> % if NMRealField has Eltable(NMRealField, NMRealField)

from Eltable(NMRealField, %)

enumerate: () -> List % if NMRealField has Finite

from Finite

equal?: (%, %) -> Boolean

from NMRing

erf: % -> %

erf(x) is the error function evaluated at x.

erfi: % -> %

erfi(x) is the imaginary error function evaluated at x.

euclideanSize: % -> NonNegativeInteger

from EuclideanDomain

eval: (%, Equation NMRealField) -> % if NMRealField has Evalable NMRealField

from Evalable NMRealField

eval: (%, List Equation NMRealField) -> % if NMRealField has Evalable NMRealField

from Evalable NMRealField

eval: (%, List NMRealField, List NMRealField) -> % if NMRealField has Evalable NMRealField

from InnerEvalable(NMRealField, NMRealField)

eval: (%, List Symbol, List NMRealField) -> % if NMRealField has InnerEvalable(Symbol, NMRealField)

from InnerEvalable(Symbol, NMRealField)

eval: (%, NMRealField, NMRealField) -> % if NMRealField has Evalable NMRealField

from InnerEvalable(NMRealField, NMRealField)

eval: (%, Symbol, NMRealField) -> % if NMRealField has InnerEvalable(Symbol, NMRealField)

from InnerEvalable(Symbol, NMRealField)

exact?: % -> Boolean

exact?(x) checks whether x is exact i.e. with 0 radius.

exactDivide: (%, %) -> %

from NMRing

exp1: () -> %

exp1() returns the NMComplexField ℯ (exp(1)).

exp: % -> %

from ElementaryFunctionCategory

exp: () -> %

exp() returns the NMComplexField ℯ (exp(1)).

expm1: % -> %

expm1(x) computes accurately e^x-1. It avoids the loss of precision involved in the direct evaluation of exp(x)-1 for small values of x.

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

from PrincipalIdealDomain

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

from EntireRing

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

from ComplexCategory NMRealField

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

from EuclideanDomain

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

from EuclideanDomain

factor: % -> Factored %

from UniqueFactorizationDomain

factorPolynomial: SparseUnivariatePolynomial % -> Factored SparseUnivariatePolynomial % if NMRealField has PolynomialFactorizationExplicit

from PolynomialFactorizationExplicit

factorsOfCyclicGroupSize: () -> List Record(factor: Integer, exponent: NonNegativeInteger) if NMRealField has FiniteFieldCategory

from FiniteFieldCategory

factorSquareFreePolynomial: SparseUnivariatePolynomial % -> Factored SparseUnivariatePolynomial % if NMRealField has PolynomialFactorizationExplicit

from PolynomialFactorizationExplicit

finite?: % -> Boolean

finite?(x) checks whether or not x is finite, not an infinity for example.

Gamma: % -> %

Gamma(x) is the Euler Gamma function evaluated at x.

Gamma: (%, %) -> %

Gamma(x,y) is the incomplete Gamma function.

gammaLower: (%, %) -> %

gammaLower(s,x) is the lower incomplete Gamma function.

gammaLowerRegularized: (%, %) -> %

gammaLowerRegularized(s,x) is the regularized lower incomplete Gamma function.

gammaRegularized: (%, %) -> %

gammaRegularized(s,x) is the regularized upper incomplete Gamma function.

gcd: (%, %) -> %

from GcdDomain

gcd: List % -> %

from GcdDomain

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

from GcdDomain

generator: () -> %

from MonogenicAlgebra(NMRealField, SparseUnivariatePolynomial NMRealField)

guess: (%, NonNegativeInteger) -> NMAlgebraicNumber

guess(a, deg) returns the reconstructed algebraic number found if it succeeds. Up to degree deg.

hash: % -> SingleInteger if NMRealField has Hashable

from Hashable

hashUpdate!: (HashState, %) -> HashState if NMRealField has Hashable

from Hashable

hurwitzZeta: (%, %) -> %

hurwitzZeta(s,a) returns the Hurwitz zeta function of s and a.

hypergeometric1F1: (%, %, %) -> %

hypergeometric1F1(a,b,z) is the confluent hypergeometric function 1F1.

hypergeometric1F1Regularized: (%, %, %) -> %

hypergeometric1F1Regularized(a,b,z) is the regularized confluent hypergeometric function 1F1.

hypergeometric2F1: (%, %, %, %) -> %

hypergeometric2F1(a,b,c,z) is the Gauss hypergeometric function 2F1.

hypergeometricU: (%, %, %) -> %

hypergeometricU(a,b,x) is the confluent hypergeometric function U.

imag: % -> NMRealField

from ComplexCategory NMRealField

imaginary: () -> %

from ComplexCategory NMRealField

index: PositiveInteger -> % if NMRealField has Finite

from Finite

init: % if NMRealField has FiniteFieldCategory

from StepThrough

integer?: % -> Boolean

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

inv: % -> %

from DivisionRing

irreducible?: % -> Boolean

from NMRing

jacobiTheta: (%, %) -> List %

jacobiTheta(z,tau) returns a list with the values of the Jacobi theta function evaluated at z and tau.

jInvariant: % -> %

jInvariant(z) returns the j-invariant function evaluated at z.

jlAbout: % -> Void

from JLObjectType

jlApply: (String, %) -> JLObject

from JLObjectType

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

from JLObjectType

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

from JLObjectType

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

from JLObjectType

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

from JLObjectType

jlDisplay: % -> Void

from JLObjectType

jlDump: JLObject -> Void

from JLObjectType

jlFieldNames: % -> JLObject

from JLObjectType

jlGetField: (%, JLSymbol) -> JLObject

from JLObjectType

jlGetJuliaIndex: % -> String

from JLObjectType

jlGetProperty: (%, JLSymbol) -> JLObject

from JLObjectType

jlId: % -> Integer

from JLObjectType

jlimref: String -> %

from JLObjectType

jlNMRing: () -> String

from NMRing

jlObject: () -> String

from JLObjectType

jlPropertyNames: % -> JLObject

from JLObjectType

jlRef: % -> SExpression

from JLObjectType

jlref: String -> %

from JLObjectType

jlText: (%, String) -> List String

from JLObjectType

jlType: % -> Symbol

from JLObjectType

jncf: (Float, Float) -> %

jncf(r,i) returns r as real part and i as imaginary part as a NMComplexField element.

jncf: (Integer, Integer) -> %

jncf(r,i) returns r as real part and i as imaginary part as a NMComplexField element.

jncf: (String, String) -> %

jncf(r,i) returns a NMComplexField element from evaluation in Julia of r and i. r is the real part, i the imaginary one.

jncf: Float -> %

jncf(x) returns x as a NMComplexField element.

jncf: Integer -> %

jncf(i)returns i as NMComplexField element.

jncf: String -> %

jncf(str) returns a NMComplexField element from evaluation in Julia of str.

latex: % -> String

from SetCategory

lcm: (%, %) -> %

from GcdDomain

lcm: List % -> %

from GcdDomain

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

from LeftOreRing

ldexp: (%, NMInteger) -> %

ldexp(x, n) returns x * 2^n.

leftPower: (%, NonNegativeInteger) -> %

from MagmaWithUnit

leftPower: (%, PositiveInteger) -> %

from Magma

leftRecip: % -> Union(%, failed)

from MagmaWithUnit

lift: % -> SparseUnivariatePolynomial NMRealField

from MonogenicAlgebra(NMRealField, SparseUnivariatePolynomial NMRealField)

linearDependence: (JLMatrix %, PositiveInteger) -> JLVector NMInteger

linearDependence(mat, prec) finds a linear dependence of the rows of mat. Heuristic algorithm.

linearDependence: (JLVector %, PositiveInteger) -> JLVector NMInteger

linearDependence(vec, prec) finds a linear dependence of the entries of vec. Heuristic algorithm.

log1p: % -> %

log1p(x) logarithm of 1+x computed accurately.

log: % -> %

from ElementaryFunctionCategory

logBarnesG: % -> %

logBarnesG(x) is the logarithm of the Barnes G-function evaluated at x.

logGamma: % -> %

logGamma(x) returns the logarithm of the gamma function evaluated at x.

logSinPi: % -> %

logSinPi(x) logarithm of sin(pi*x) computed accurately without branch cuts the real line.

lookup: % -> PositiveInteger if NMRealField has Finite

from Finite

map: (NMRealField -> NMRealField, %) -> %

from FullyEvalableOver NMRealField

minimalPolynomial: % -> SparseUnivariatePolynomial NMRealField

from FiniteRankAlgebra(NMRealField, SparseUnivariatePolynomial NMRealField)

missing?: % -> Boolean

from JLObjectType

modularDelta: % -> %

modularDelta(z) returns the modular delta function evaluated at z.

modularLambda: % -> %

modularLambda(z) returns the modular lambda function evaluated at z.

modularWeberF1: % -> %

modularWeberF1(q) returns the Weber F1 function of q in the complex upper half plane.

modularWeberF2: % -> %

modularWeberF2(q) returns the Weber F2 function of q in the complex upper half plane.

modularWeberF: % -> %

modularWeberF(q) returns the Weber F function of q in the complex upper half plane.

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

from EuclideanDomain

mutable?: % -> Boolean

from JLObjectType

nextItem: % -> Union(%, failed) if NMRealField has FiniteFieldCategory

from StepThrough

norm: % -> NMRealField

from ComplexCategory NMRealField

nothing?: % -> Boolean

from JLObjectType

nthRoot: (%, Integer) -> %

from RadicalCategory

one?: % -> Boolean

from MagmaWithUnit

opposite?: (%, %) -> Boolean

from AbelianMonoid

order: % -> OnePointCompletion PositiveInteger if NMRealField has FiniteFieldCategory

from FieldOfPrimeCharacteristic

order: % -> PositiveInteger if NMRealField has FiniteFieldCategory

from FiniteFieldCategory

overlaps?: (%, %) -> Boolean

overlaps?(x,y) checks whether or not any part of x and y balls overlaps.

patternMatch: (%, Pattern Float, PatternMatchResult(Float, %)) -> PatternMatchResult(Float, %)

from PatternMatchable Float

patternMatch: (%, Pattern Integer, PatternMatchResult(Integer, %)) -> PatternMatchResult(Integer, %) if NMRealField has PatternMatchable Integer

from PatternMatchable Integer

pi: () -> %

pi() returns the JLFloat representation of π.

plenaryPower: (%, PositiveInteger) -> %

from NonAssociativeAlgebra %

polarCoordinates: % -> Record(r: NMRealField, phi: NMRealField)

from ComplexCategory NMRealField

polygamma: (%, %) -> %

polygamma(x,y) returns the polygamma function applied to x and y.

polylog: (Integer, %) -> %

polylog(n,x) returns the polylogarithm function Li_n(x).

precision: () -> PositiveInteger

precision() returns the actual default precision used when creating a new element.

precision: PositiveInteger -> PositiveInteger

precision(p) changes the precision used for creating elements.

prime?: % -> Boolean

from UniqueFactorizationDomain

primeFrobenius: % -> % if NMRealField has FiniteFieldCategory

from FieldOfPrimeCharacteristic

primeFrobenius: (%, NonNegativeInteger) -> % if NMRealField has FiniteFieldCategory

from FieldOfPrimeCharacteristic

primitive?: % -> Boolean if NMRealField has FiniteFieldCategory

from FiniteFieldCategory

primitiveElement: () -> % if NMRealField has FiniteFieldCategory

from FiniteFieldCategory

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

from PrincipalIdealDomain

quo: (%, %) -> %

from EuclideanDomain

random: () -> % if NMRealField has Finite

from Finite

randtest: JLSymbol -> %

randtest(randtype) returns a random number depending on the Julia symbol randtype. :urandom an uniformly distributed random number contained in the unit disk. To test corner cases: :randtest with separate real and imaginary parts, :randtest_precise with precise real and imaginary parts, :randtest_special with separate real and imaginary parts but eventually infinities and NaNs, :randtest_param with very high probability of returning integers or half-integers.

rank: () -> PositiveInteger

from FramedModule NMRealField

rational?: % -> Boolean if NMRealField has IntegerNumberSystem

from ComplexCategory NMRealField

rational: % -> Fraction Integer if NMRealField has IntegerNumberSystem

from ComplexCategory NMRealField

rationalIfCan: % -> Union(Fraction Integer, failed) if NMRealField has IntegerNumberSystem

from ComplexCategory NMRealField

real: % -> NMRealField

from ComplexCategory NMRealField

recip: % -> Union(%, failed)

from MagmaWithUnit

reciprocalGamma: % -> %

reciprocalGamma(s,x) is the reciprocal Gamma function i.e. 1/Gamma(x,y).

reduce: Fraction SparseUnivariatePolynomial NMRealField -> Union(%, failed)

from MonogenicAlgebra(NMRealField, SparseUnivariatePolynomial NMRealField)

reduce: SparseUnivariatePolynomial NMRealField -> %

from MonogenicAlgebra(NMRealField, SparseUnivariatePolynomial NMRealField)

reducedSystem: (Matrix %, Vector %) -> Record(mat: Matrix Integer, vec: Vector Integer) if NMRealField has LinearlyExplicitOver Integer

from LinearlyExplicitOver Integer

reducedSystem: (Matrix %, Vector %) -> Record(mat: Matrix NMRealField, vec: Vector NMRealField)

from LinearlyExplicitOver NMRealField

reducedSystem: Matrix % -> Matrix Integer if NMRealField has LinearlyExplicitOver Integer

from LinearlyExplicitOver Integer

reducedSystem: Matrix % -> Matrix NMRealField

from LinearlyExplicitOver NMRealField

regularRepresentation: % -> Matrix NMRealField

from FramedAlgebra(NMRealField, SparseUnivariatePolynomial NMRealField)

regularRepresentation: (%, Vector %) -> Matrix NMRealField

from FiniteRankAlgebra(NMRealField, SparseUnivariatePolynomial NMRealField)

rem: (%, %) -> %

from EuclideanDomain

representationType: () -> Union(prime, polynomial, normal, cyclic) if NMRealField has FiniteFieldCategory

from FiniteFieldCategory

represents: (Vector NMRealField, Vector %) -> %

from FiniteRankAlgebra(NMRealField, SparseUnivariatePolynomial NMRealField)

represents: Vector NMRealField -> %

from FramedModule NMRealField

retract: % -> Fraction Integer

from RetractableTo Fraction Integer

retract: % -> Integer

from RetractableTo Integer

retract: % -> NMRealField

from RetractableTo NMRealField

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

from RetractableTo Fraction Integer

retractIfCan: % -> Union(Integer, failed)

from RetractableTo Integer

retractIfCan: % -> Union(NMRealField, failed)

from RetractableTo NMRealField

rightPower: (%, NonNegativeInteger) -> %

from MagmaWithUnit

rightPower: (%, PositiveInteger) -> %

from Magma

rightRecip: % -> Union(%, failed)

from MagmaWithUnit

risingFactorial2: (%, Integer) -> Record(rf: %, d: %)

risingFactorial2(x, n) returns a record containing the rising factorial x(x+1)…(x+n-1) and its derivative.

risingFactorial: (%, Integer) -> %

risingFactorial(x, n) returns the rising factorial of x, i.e. x(x+1)…(x+n-1).

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).

rsqrt: % -> %

rsqrt(x) returns the reciprocal of the square root of x, i.e. 1/sqrt(x).

sample: %

from SetWithZero

sec: % -> %

from TrigonometricFunctionCategory

sech: % -> %

from HyperbolicFunctionCategory

Shi: % -> %

Shi(x) returns the hyperbolic sine integral of x.

Si: % -> %

Si(x) returns the sine integral of x.

sin: % -> %

from TrigonometricFunctionCategory

sinh: % -> %

from HyperbolicFunctionCategory

size: () -> NonNegativeInteger if NMRealField has Finite

from Finite

sizeLess?: (%, %) -> Boolean

from EuclideanDomain

smaller?: (%, %) -> Boolean

from ComparableRepresentation

solveLinearPolynomialEquation: (List SparseUnivariatePolynomial %, SparseUnivariatePolynomial %) -> Union(List SparseUnivariatePolynomial %, failed) if NMRealField has PolynomialFactorizationExplicit

from PolynomialFactorizationExplicit

sqrt: % -> %

from ComplexCategory NMRealField

squareFree: % -> Factored %

from UniqueFactorizationDomain

squareFreePart: % -> %

from UniqueFactorizationDomain

squareFreePolynomial: SparseUnivariatePolynomial % -> Factored SparseUnivariatePolynomial % if NMRealField has PolynomialFactorizationExplicit

from PolynomialFactorizationExplicit

string: % -> String

from JLType

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

from CancellationAbelianMonoid

tableForDiscreteLogarithm: Integer -> Table(PositiveInteger, NonNegativeInteger) if NMRealField has FiniteFieldCategory

from FiniteFieldCategory

tan: % -> %

from TrigonometricFunctionCategory

tanh: % -> %

from HyperbolicFunctionCategory

trace: % -> NMRealField

from FiniteRankAlgebra(NMRealField, SparseUnivariatePolynomial NMRealField)

traceMatrix: () -> Matrix NMRealField

from FramedAlgebra(NMRealField, SparseUnivariatePolynomial NMRealField)

traceMatrix: Vector % -> Matrix NMRealField

from FiniteRankAlgebra(NMRealField, SparseUnivariatePolynomial NMRealField)

trim: % -> %

trim(x) rounds off insignificant bits from the midpoint.

uniqueInteger: % -> Union(NMInteger, failed)

uniqueInteger(x) returns a NMInteger if there is a unique integer in the interval x, “failed” otherwise.

unit?: % -> Boolean

from NMRing

unitCanonical: % -> %

from EntireRing

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

from EntireRing

urand01: () -> %

urand01() returns an uniformly distributed random number contained in the unit disk.

weierstrassP: (%, %) -> %

weierstrassP(z,tau) returns the elliptic Weierstrass P function.

zero?: % -> Boolean

from SetWithZero

AbelianGroup

AbelianMonoid

AbelianSemiGroup

Algebra %

Algebra Fraction Integer

Algebra NMRealField

arbitraryPrecision

ArcHyperbolicFunctionCategory

ArcTrigonometricFunctionCategory

BasicType

BiModule(%, %)

BiModule(Fraction Integer, Fraction Integer)

BiModule(NMRealField, NMRealField)

CancellationAbelianMonoid

canonicalsClosed

canonicalUnitNormal

CharacteristicNonZero if NMRealField has CharacteristicNonZero

CharacteristicZero

CoercibleFrom Fraction Integer

CoercibleFrom Integer

CoercibleFrom NMRealField

CoercibleTo OutputForm

CommutativeRing

CommutativeStar

Comparable

ComparableRepresentation

ComplexCategory NMRealField

ConvertibleTo Complex DoubleFloat

ConvertibleTo Complex Float

ConvertibleTo InputForm if NMRealField has ConvertibleTo InputForm

ConvertibleTo Pattern Float

ConvertibleTo Pattern Integer if NMRealField has ConvertibleTo Pattern Integer

ConvertibleTo SparseUnivariatePolynomial NMRealField

ConvertibleTo String

DifferentialExtension NMRealField

DifferentialRing

DivisionRing

ElementaryFunctionCategory

Eltable(NMRealField, %) if NMRealField has Eltable(NMRealField, NMRealField)

EntireRing

EuclideanDomain

Evalable NMRealField if NMRealField has Evalable NMRealField

Field

FieldOfPrimeCharacteristic if NMRealField has FiniteFieldCategory

Finite if NMRealField has Finite

FiniteFieldCategory if NMRealField has FiniteFieldCategory

FiniteRankAlgebra(NMRealField, SparseUnivariatePolynomial NMRealField)

FramedAlgebra(NMRealField, SparseUnivariatePolynomial NMRealField)

FramedModule NMRealField

FullyEvalableOver NMRealField

FullyLinearlyExplicitOver NMRealField

FullyPatternMatchable NMRealField

FullyRetractableTo NMRealField

GcdDomain

Hashable if NMRealField has Hashable

HyperbolicFunctionCategory

InnerEvalable(NMRealField, NMRealField) if NMRealField has Evalable NMRealField

InnerEvalable(Symbol, NMRealField) if NMRealField has InnerEvalable(Symbol, NMRealField)

IntegralDomain

JLObjectRing

JLObjectType

JLRing

JLType

LeftModule %

LeftModule Fraction Integer

LeftModule NMRealField

LeftOreRing

LinearlyExplicitOver Integer if NMRealField has LinearlyExplicitOver Integer

LinearlyExplicitOver NMRealField

Magma

MagmaWithUnit

Module %

Module Fraction Integer

Module NMRealField

MonogenicAlgebra(NMRealField, SparseUnivariatePolynomial NMRealField)

Monoid

multiplicativeValuation if NMRealField has IntegerNumberSystem

NMCommutativeRing

NMField

NMRing

NMType

NonAssociativeAlgebra %

NonAssociativeAlgebra Fraction Integer

NonAssociativeAlgebra NMRealField

NonAssociativeRing

NonAssociativeRng

NonAssociativeSemiRing

NonAssociativeSemiRng

noZeroDivisors

PartialDifferentialRing Symbol if NMRealField has PartialDifferentialRing Symbol

Patternable NMRealField

PatternMatchable Float

PatternMatchable Integer if NMRealField has PatternMatchable Integer

PolynomialFactorizationExplicit if NMRealField has PolynomialFactorizationExplicit

PrincipalIdealDomain

RadicalCategory

RetractableTo Fraction Integer

RetractableTo Integer

RetractableTo NMRealField

RightModule %

RightModule Fraction Integer

RightModule Integer if NMRealField has LinearlyExplicitOver Integer

RightModule NMRealField

Ring

Rng

SemiGroup

SemiRing

SemiRng

SetCategory

SetWithZero

StepThrough if NMRealField has FiniteFieldCategory

TranscendentalFunctionCategory

TrigonometricFunctionCategory

TwoSidedRecip

UniqueFactorizationDomain

unitsKnown