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)).

0: %: from AbelianMonoid

1: %: from MagmaWithUnit

*: (%, %) -> %: from Magma
*: (%, Fraction Integer) -> %: from RightModule Fraction Integer
*: (%, Fraction R) -> %: from RightModule Fraction R
*: (%, R) -> %: from RightModule R
*: (Fraction Integer, %) -> %: from LeftModule Fraction Integer
*: (Fraction R, %) -> %: from LeftModule Fraction R
*: (Integer, %) -> %: from AbelianGroup
*: (NonNegativeInteger, %) -> %: from AbelianMonoid
*: (PositiveInteger, %) -> %: from AbelianSemiGroup
*: (R, %) -> %: from LeftModule R

+: (%, %) -> %: from AbelianSemiGroup

-: % -> %: from AbelianGroup
-: (%, %) -> %: from AbelianGroup

/: (%, %) -> %: from Field

=: (%, %) -> Boolean: from BasicType

^: (%, Integer) -> %: from DivisionRing
^: (%, NonNegativeInteger) -> %: from MagmaWithUnit
^: (%, PositiveInteger) -> %: from Magma

~=: (%, %) -> Boolean: from BasicType

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

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

approximants: % -> Stream Fraction R: approximants(x) returns the stream of approximants of the continued fraction spadvar{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: % -> OutputForm: from CoercibleTo OutputForm
coerce: Fraction Integer -> %: from Algebra Fraction Integer
coerce: Fraction R -> %: from Algebra Fraction R
coerce: Integer -> %: from NonAssociativeRing
coerce: R -> %: from Algebra R

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

complete: % -> %: complete(x) causes all entries in spadvar{x} to be computed. Normally entries are only computed as needed. If spadvar{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 spadvar{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 spadvar{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 spadvar{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 spadvar{n} entries in the continued fraction spadvar{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 spadvar{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 spadvar{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 spadvar{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 spadvar{x}. That is, if x = continuedFraction(b0, [a1, a2, a3, ...], [b1, b2, b3, ...]), then partialQuotients(x) = [b0, b1, b2, b3, ...].

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

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 spadvar{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 spadvar{x}. That is, if x = continuedFraction(b0, [a1, a2, a3, ...], [b1, b2, b3, ...]), then wholePart(x) = b0.