NMExactCalciumField

jnemo.spad line 1172 [edit on github]

NMExactCalciumField implements exact complex field arithmetic using the NM package. Reference: https://nemocas.github.io/NM.jl See https://flintlib.org/doc/introduction_calcium.html for the C library.

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.

*: (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

/: (%, Integer) -> %

/ undocumented

/: (Integer, %) -> %

/ undocumented

<=: (%, %) -> Boolean

from PartialOrder

<: (%, %) -> Boolean

from PartialOrder

=: (%, %) -> Boolean

from BasicType

>=: (%, %) -> Boolean

from PartialOrder

>: (%, %) -> Boolean

from PartialOrder

^: (%, %) -> %

from ElementaryFunctionCategory

^: (%, Fraction Integer) -> %

from RadicalCategory

^: (%, Integer) -> %

from DivisionRing

^: (%, NonNegativeInteger) -> %

from MagmaWithUnit

^: (%, PositiveInteger) -> %

from Magma

~=: (%, %) -> Boolean

from BasicType

acos: % -> %

from ArcTrigonometricFunctionCategory

acos: (%, JLSymbol) -> %

acos(x,repr) returns acos(x) using one of the repr representations: :default (since ‘default’ is a FriCAS keyword use jsym(String) to coerce it) :logarithm :direct

acosh: % -> %

from ArcHyperbolicFunctionCategory

acot: % -> %

from ArcTrigonometricFunctionCategory

acoth: % -> %

from ArcHyperbolicFunctionCategory

acsc: % -> %

from ArcTrigonometricFunctionCategory

acsch: % -> %

from ArcHyperbolicFunctionCategory

algebraic?: % -> Boolean

algebraic?(x) checks whether or not x is algebraic.

annihilate?: (%, %) -> Boolean

from Rng

antiCommutator: (%, %) -> %

from NonAssociativeSemiRng

argument: % -> %

from ComplexCategory %

asec: % -> %

from ArcTrigonometricFunctionCategory

asech: % -> %

from ArcHyperbolicFunctionCategory

asin: % -> %

from ArcTrigonometricFunctionCategory

asin: (%, JLSymbol) -> %

asin(x, repr) returns asin(x) using one of the repr representations: :default (since ‘default’ is a FriCAS keyword use jsym(String) to coerce it) :logarithm :direct

asinh: % -> %

from ArcHyperbolicFunctionCategory

associates?: (%, %) -> Boolean

from EntireRing

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

from NonAssociativeRng

atan: % -> %

from ArcTrigonometricFunctionCategory

atan: (%, JLSymbol) -> %

atan(x, repr) returns atan(x) using one of the repr representations: :default (since ‘default’ is a FriCAS keyword use jsym(String) to coerce it) :logarithm :direct or :arctangent

atanh: % -> %

from ArcHyperbolicFunctionCategory

basis: () -> Vector %

from FramedModule %

ceiling: % -> %

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

characteristic: () -> NonNegativeInteger

from NonAssociativeRing

characteristicPolynomial: % -> SparseUnivariatePolynomial %

from FiniteRankAlgebra(%, SparseUnivariatePolynomial %)

coerce: % -> %

from Algebra %

coerce: % -> JLObject

from JLObjectType

coerce: % -> OutputForm

from CoercibleTo OutputForm

coerce: Complex Integer -> %

coerce(z) coerces z. Convenience function.

coerce: Fraction Integer -> %

coerce(q) coerces q. Convenience function.

coerce: Integer -> %

from NonAssociativeRing

coerce: NMAlgebraicNumber -> %

coerce(qbar) coerces qbar. Convenience function.

coerce: NMFraction NMInteger -> %

coerce(q) coerces q. Convenience function.

coerce: PositiveInteger -> %

coerce(pi) coerces pi. Convenience function.

commutator: (%, %) -> %

from NonAssociativeRng

complex: (%, %) -> %

from ComplexCategory %

complexNormalForm: % -> %

complexNormalForm(x) returns x rewritten using standard transformations. See the NM.jl documentation fo more informations.

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

from PolynomialFactorizationExplicit

conjugate: % -> %

from ComplexCategory %

conjugate: (%, JLSymbol) -> %

conjugate(x, repr) returns conjugate(x) using one of the repr representations: :default (since ‘default’ is a FriCAS keyword use jsym(String) to coerce it) :deep (recursively) :shallow (a new extension, ā, is used if there is no express simplification)

convert: % -> SparseUnivariatePolynomial %

from ConvertibleTo SparseUnivariatePolynomial %

convert: % -> String

from ConvertibleTo String

convert: % -> Vector %

from FramedModule %

convert: SparseUnivariatePolynomial % -> %

from MonogenicAlgebra(%, SparseUnivariatePolynomial %)

convert: Vector % -> %

from FramedModule %

coordinates: % -> Vector %

from FramedModule %

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

from FiniteRankAlgebra(%, SparseUnivariatePolynomial %)

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

from FiniteRankAlgebra(%, SparseUnivariatePolynomial %)

coordinates: Vector % -> Matrix %

from FramedModule %

cos: % -> %

from TrigonometricFunctionCategory

cos: (%, JLSymbol) -> %

cos(x, repr) returns cos(x) using one of the repr representations: :default (since ‘default’ is a FriCAS keyword use jsym(String) to coerce it) :exponential :tangent :direct or :sine_cosine

cosh: % -> %

from HyperbolicFunctionCategory

cot: % -> %

from TrigonometricFunctionCategory

coth: % -> %

from HyperbolicFunctionCategory

csc: % -> %

from TrigonometricFunctionCategory

csch: % -> %

from HyperbolicFunctionCategory

csign: % -> %

csign(x) is an extension of the real sign function, x/sqrt(x^2) unless x is 0 (0 in this case).

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

from DifferentialExtension %

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

from DifferentialExtension %

definingPolynomial: () -> SparseUnivariatePolynomial %

from MonogenicAlgebra(%, SparseUnivariatePolynomial %)

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

from MonogenicAlgebra(%, SparseUnivariatePolynomial %)

differentiate: % -> % if % has DifferentialRing

from DifferentialRing

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

from DifferentialExtension %

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

from DifferentialExtension %

differentiate: (%, NonNegativeInteger) -> % if % has DifferentialRing

from DifferentialRing

discriminant: () -> %

from FramedAlgebra(%, SparseUnivariatePolynomial %)

discriminant: Vector % -> %

from FiniteRankAlgebra(%, SparseUnivariatePolynomial %)

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

from EuclideanDomain

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

from Finite

erf: % -> %

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

erfc: % -> %

erfc(x) is the complementary error function evaluated at x.

erfi: % -> %

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

euclideanSize: % -> NonNegativeInteger

from EuclideanDomain

eulerGamma: () -> %

eulerGamma() returns the Euler's constant gamma (γ).

exp1: () -> %

exp() returns ℯ (exp(1)).

exp: % -> %

from ElementaryFunctionCategory

exp: () -> %

exp() returns ℯ (exp(1)).

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

from PrincipalIdealDomain

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

from ComplexCategory %

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

from EuclideanDomain

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

from EuclideanDomain

factor: % -> Factored %

from UniqueFactorizationDomain

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

from PolynomialFactorizationExplicit

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

from PolynomialFactorizationExplicit

floor: % -> %

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

Gamma: % -> %

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

gcd: (%, %) -> %

from GcdDomain

gcd: List % -> %

from GcdDomain

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

from GcdDomain

generator: () -> %

from MonogenicAlgebra(%, SparseUnivariatePolynomial %)

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

from Hashable

imag: % -> %

from ComplexCategory %

imaginary?: % -> Boolean

imaginary?(x) checks whether or not x is imaginary.

imaginary: () -> %

from ComplexCategory %

index: PositiveInteger -> % if % has Finite

from Finite

infinity?: % -> Boolean

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

infinity: % -> %

infinity(x) returns signed infinity depending on x sign.

infinity: () -> %

infinity() returns unsigned infinity.

integer?: % -> Boolean

integer?(z) checks whether or not z 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

jlId: % -> JLInt64

from JLObjectType

jlNMRing: () -> String

from NMRing

jlObject: () -> String

from NMRing

jlOptions: % -> JLObjDict

jlOptions(x) returns the options used during compoutations.

jlRef: % -> SExpression

from JLObjectType

jlref: String -> %

from JLObjectType

jlType: % -> String

from JLObjectType

jnecf: (Fraction Integer, Fraction Integer) -> %

jnecf(real,cplx) coerces real and cplx to a NM exact complex number where real is the real part and cplx the complex part.

jnecf: (NMFraction NMInteger, NMFraction NMInteger) -> %

jnecf(real,cplx) coerces real and cplx to a NM exact where real is the real part and cplx the complex part.

jnecf: Fraction Integer -> %

jnecf(fi) coerces fi to a NM real exact number.

jnecf: NMAlgebraicNumber -> %

jnecf(qbar) coerces qbar to a NM exact real or complex number.

jnecf: NMFraction NMInteger -> %

jnecf(nr) coerces nr to a NM exact real number.

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

lift: % -> SparseUnivariatePolynomial %

from MonogenicAlgebra(%, SparseUnivariatePolynomial %)

log: % -> %

from ElementaryFunctionCategory

lookup: % -> PositiveInteger if % has Finite

from Finite

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

from FullyEvalableOver %

max: (%, %) -> %

from OrderedSet

min: (%, %) -> %

from OrderedSet

minimalPolynomial: % -> SparseUnivariatePolynomial %

from FiniteRankAlgebra(%, SparseUnivariatePolynomial %)

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

from EuclideanDomain

mutable?: % -> Boolean

from JLObjectType

negativeInfinity: () -> %

negativeInfinity() returns negtive infinity.

norm: % -> %

from ComplexCategory %

nothing?: % -> Boolean

from JLObjectType

nthRoot: (%, Integer) -> %

from RadicalCategory

number?: % -> Boolean

number?(x) checks whether or not x is a number, i.e. not an infinity or an undefined value.

one?: % -> Boolean

from MagmaWithUnit

opposite?: (%, %) -> Boolean

from AbelianMonoid

pi: () -> %

pi() returns π.

plenaryPower: (%, PositiveInteger) -> %

from NonAssociativeAlgebra Fraction Integer

positiveInfinity: () -> %

positiveInfinity() returns positive infinity.

pow: (%, Integer, JLSymbol) -> %

pow(x, i, repr) x raised to power i using one of the repr representations: :default (since ‘default’ is a FriCAS keyword use jsym(String) to coerce it) :arithmetic See the NM documentation for more information.

prime?: % -> Boolean

from UniqueFactorizationDomain

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

from PrincipalIdealDomain

quo: (%, %) -> %

from EuclideanDomain

random: () -> % if % has Finite

from Finite

random: (Integer, Integer) -> %

random(depth, bits) returns a random number with size of coefficients up to bits. if depth is nonzero, apply a random arithmetic operation/function to operands produced using recursive calls with depth - 1.

random: (Integer, Integer, JLSymbol) -> %

random(depth, bits, type) returns a random number with size of coefficients up to bits. if depth is nonzero, apply a random arithmetic operation/function to operands produced using recursive calls with depth - 1. depth is not used for rationals. type can be one of: :rational (returns a rational) :null (returns value with default settings) :special (returns a special value of a number - can throw an error if it isn't a number).

rank: () -> PositiveInteger

from FramedModule %

rational?: % -> Boolean

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

real?: % -> Boolean

real?(z) checks whether or not z is a real number.

real: % -> %

from ComplexCategory %

recip: % -> Union(%, failed)

from MagmaWithUnit

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

from MonogenicAlgebra(%, SparseUnivariatePolynomial %)

reduce: SparseUnivariatePolynomial % -> %

from MonogenicAlgebra(%, SparseUnivariatePolynomial %)

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

from LinearlyExplicitOver %

reducedSystem: Matrix % -> Matrix %

from LinearlyExplicitOver %

regularRepresentation: % -> Matrix %

from FramedAlgebra(%, SparseUnivariatePolynomial %)

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

from FiniteRankAlgebra(%, SparseUnivariatePolynomial %)

rem: (%, %) -> %

from EuclideanDomain

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

from FiniteRankAlgebra(%, SparseUnivariatePolynomial %)

represents: Vector % -> %

from FramedModule %

retract: % -> %

from RetractableTo %

retract: % -> NMComplexBall

convert(x) converts x.

retractIfCan: % -> Union(%, failed)

from RetractableTo %

retractIfCan: % -> Union(NMAlgebraicNumber, failed)

retractIfCan(z) retracts if possible z to a NMAlgebraicNumber.

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

retractIfCan(z) retracts if possible z to a NMFraction(NMInteger).

retractIfCan: % -> Union(NMInteger, failed)

retractIfCan(z) retracts if possible z to a Integer.

retractIfCan: % -> Union(NMRealBall, failed)

convert(x) converts x.

rightPower: (%, NonNegativeInteger) -> %

from MagmaWithUnit

rightPower: (%, PositiveInteger) -> %

from Magma

rightRecip: % -> Union(%, failed)

from MagmaWithUnit

sample: %

from AbelianMonoid

sec: % -> %

from TrigonometricFunctionCategory

sech: % -> %

from HyperbolicFunctionCategory

sign: % -> %

sign(x) returns sign of x.

signedInfinity?: % -> Boolean

signedInfinity?(x) checks whether or not x is a signed infinity.

sin: % -> %

from TrigonometricFunctionCategory

sin: (%, JLSymbol) -> %

sin(x, repr) return sin(x) using one of the repr representations: :default (since ‘default’ is a FriCAS keyword use jsym(String) to coerce it) :exponential :tangent :direct

sinh: % -> %

from HyperbolicFunctionCategory

size: () -> NonNegativeInteger if % has Finite

from Finite

sizeLess?: (%, %) -> Boolean

from EuclideanDomain

smaller?: (%, %) -> Boolean

from Comparable

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

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

tan: % -> %

from TrigonometricFunctionCategory

tan: (%, JLSymbol) -> %

tan(x, repr) returns tan(x) using one of the repr representations: :default (since ‘default’ is a FriCAS keyword use jsym(String) to coerce it) :exponential :direct or :tangent :sine_cosine

tanh: % -> %

from HyperbolicFunctionCategory

trace: % -> %

from FiniteRankAlgebra(%, SparseUnivariatePolynomial %)

traceMatrix: () -> Matrix %

from FramedAlgebra(%, SparseUnivariatePolynomial %)

traceMatrix: Vector % -> Matrix %

from FiniteRankAlgebra(%, SparseUnivariatePolynomial %)

undefined?: % -> Boolean

undefined?(x) checks whether or not x is undefined.

undefined: () -> %

undefined() returns the undefined special value.

unit?: % -> Boolean

from EntireRing

unitCanonical: % -> %

from EntireRing

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

from EntireRing

unknown?: % -> Boolean

unknown?(x) checks whether or not x is unknown.

unknown: () -> %

unknown() returns the unknown special value.

unsignedInfinity?: % -> Boolean

unsignedInfinity?(x) checks whether or not x is an unsigned infinity.

zero?: % -> Boolean

from AbelianMonoid

AbelianGroup

AbelianMonoid

AbelianSemiGroup

Algebra %

Algebra Fraction Integer

ArcHyperbolicFunctionCategory

ArcTrigonometricFunctionCategory

BasicType

BiModule(%, %)

BiModule(Fraction Integer, Fraction Integer)

CancellationAbelianMonoid

canonicalsClosed

canonicalUnitNormal

CoercibleFrom %

CoercibleTo OutputForm

CommutativeRing

CommutativeStar

Comparable

ComplexCategory %

ConvertibleTo SparseUnivariatePolynomial %

ConvertibleTo String

DifferentialExtension %

DivisionRing

ElementaryFunctionCategory

EntireRing

EuclideanDomain

Field

FiniteRankAlgebra(%, SparseUnivariatePolynomial %)

FramedAlgebra(%, SparseUnivariatePolynomial %)

FramedModule %

FullyEvalableOver %

FullyLinearlyExplicitOver %

FullyPatternMatchable %

FullyRetractableTo %

GcdDomain

HyperbolicFunctionCategory

IntegralDomain

JLObjectRing

JLObjectType

JLRing

JLType

LeftModule %

LeftModule Fraction Integer

LeftOreRing

LinearlyExplicitOver %

Magma

MagmaWithUnit

Module %

Module Fraction Integer

MonogenicAlgebra(%, SparseUnivariatePolynomial %)

Monoid

NMCommutativeRing

NMRing

NMType

NonAssociativeAlgebra %

NonAssociativeAlgebra Fraction Integer

NonAssociativeRing

NonAssociativeRng

NonAssociativeSemiRing

NonAssociativeSemiRng

noZeroDivisors

OrderedSet

PartialOrder

Patternable %

PolynomialFactorizationExplicit

PrincipalIdealDomain

RadicalCategory

RetractableTo %

RightModule %

RightModule Fraction Integer

Ring

Rng

SemiGroup

SemiRing

SemiRng

SetCategory

TranscendentalFunctionCategory

TrigonometricFunctionCategory

TwoSidedRecip

UniqueFactorizationDomain

unitsKnown