ContinuedFraction(R)

R : EuclideanDomain

ContinuedFraction implements general continued fractions.This version is not restricted to simple, finite fractions and uses the Stream as a representation.The arithmetic functions assume that the approximants alternate below/above the convergence point. This is enforced by ensuring the partial numerators and partial denominators are greater than 0 in the Euclidean domain view of R (i.e. sizeLess?(0, x)).

* : (%, %) -> %: from Magma

* : (%, R) -> %: from RightModule(R)

* : (%, Fraction(R)) -> %: from RightModule(Fraction(R))

* : (%, Fraction(Integer)) -> %: from RightModule(Fraction(Integer))

* : (R, %) -> %: from LeftModule(R)

* : (Fraction(R), %) -> %: from LeftModule(Fraction(R))

* : (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 BasicType

^ : (%, Integer) -> %: from DivisionRing

^ : (%, NonNegativeInteger) -> %: from MagmaWithUnit

^ : (%, PositiveInteger) -> %: from Magma

annihilate? : (%, %) -> Boolean: from Rng

antiCommutator : (%, %) -> %: from NonAssociativeSemiRng

approximants : % -> Stream(Fraction(R)): approximants(x) returns the stream of approximants of the continued fraction x. If the continued fraction is finite, then the stream will be infinite and periodic with period 1.

associates? : (%, %) -> Boolean: from EntireRing

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

characteristic : () -> NonNegativeInteger: from NonAssociativeRing

coerce : % -> %: from Algebra(%)

coerce : R -> %: from Algebra(R)

coerce : Fraction(R) -> %: from Algebra(Fraction(R))

coerce : Fraction(Integer) -> %: from Algebra(Fraction(Integer))

coerce : Integer -> %: from NonAssociativeRing

coerce : % -> OutputForm: from CoercibleTo(OutputForm)

commutator : (%, %) -> %: from NonAssociativeRng

complete : % -> %: complete(x) causes all entries in x to be computed. Normally entries are only computed as needed. If x is an infinite continued fraction, a user-initiated interrupt is necessary to stop the computation.

continuedFraction : (R, Stream(R), Stream(R)) -> %: continuedFraction(b0, a, b) constructs a continued fraction in the following way: if a = [a1, a2, ...] and b = [b1, b2, ...] then the result is the continued fraction b0 + a1/(b1 + a2/(b2 + ...)).

continuedFraction : Fraction(R) -> %: continuedFraction(r) converts the fraction r with components of type R to a continued fraction over R.

convergents : % -> Stream(Fraction(R)): convergents(x) returns the stream of the convergents of the continued fraction x. If the continued fraction is finite, then the stream will be finite.

denominators : % -> Stream(R): denominators(x) returns the stream of denominators of the approximants of the continued fraction x. If the continued fraction is finite, then the stream will be finite.

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

euclideanSize : % -> NonNegativeInteger: from EuclideanDomain

expressIdealMember : (List(%), %) -> Union(List(%), "failed"): from PrincipalIdealDomain

exquo : (%, %) -> Union(%, "failed"): from EntireRing

extend : (%, Integer) -> %: extend(x, n) causes the first n entries in the continued fraction x to be computed. Normally entries are only computed as needed.

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

multiEuclidean : (List(%), %) -> Union(List(%), "failed"): from EuclideanDomain

numerators : % -> Stream(R): numerators(x) returns the stream of numerators of the approximants of the continued fraction x. If the continued fraction is finite, then the stream will be finite.

one? : % -> Boolean: from MagmaWithUnit

opposite? : (%, %) -> Boolean: from AbelianMonoid

partialDenominators : % -> Stream(R): partialDenominators(x) extracts the denominators in x. That is, if x = continuedFraction(b0, [a1, a2, a3, ...], [b1, b2, b3, ...]), then partialDenominators(x) = [b1, b2, b3, ...].

partialNumerators : % -> Stream(R): partialNumerators(x) extracts the numerators in x. That is, if x = continuedFraction(b0, [a1, a2, a3, ...], [b1, b2, b3, ...]), then partialNumerators(x) = [a1, a2, a3, ...].

partialQuotients : % -> Stream(R): partialQuotients(x) extracts the partial quotients in x. That is, if x = continuedFraction(b0, [a1, a2, a3, ...], [b1, b2, b3, ...]), then partialQuotients(x) = [b0, b1, b2, b3, ...].

plenaryPower : (%, PositiveInteger) -> %: from NonAssociativeAlgebra(%)

prime? : % -> Boolean: from UniqueFactorizationDomain

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

quo : (%, %) -> %: from EuclideanDomain

recip : % -> Union(%, "failed"): from MagmaWithUnit

reducedContinuedFraction : (R, Stream(R)) -> %: reducedContinuedFraction(b0, b) constructs a continued fraction in the following way: if b = [b1, b2, ...] then the result is the continued fraction b0 + 1/(b1 + 1/(b2 + ...)). That is, the result is the same as continuedFraction(b0, [1, 1, 1, ...], [b1, b2, b3, ...]).

reducedForm : % -> %: reducedForm(x) puts the continued fraction x in reduced form, i.e. the function returns an equivalent continued fraction of the form continuedFraction(b0, [1, 1, 1, ...], [b1, b2, b3, ...]).

rem : (%, %) -> %: from EuclideanDomain

rightPower : (%, NonNegativeInteger) -> %: from MagmaWithUnit

rightPower : (%, PositiveInteger) -> %: from Magma

rightRecip : % -> Union(%, "failed"): from MagmaWithUnit

sample : () -> %: from AbelianMonoid

sizeLess? : (%, %) -> Boolean: from EuclideanDomain

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

wholePart : % -> R: wholePart(x) extracts the whole part of x. That is, if x = continuedFraction(b0, [a1, a2, a3, ...], [b1, b2, b3, ...]), then wholePart(x) = b0.