# ContinuedFraction RΒΆ

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

from RightModule Fraction R

*: (%, R) -> %

from RightModule R

*: (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: (%, %) -> %
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: (%, %, %) -> %
characteristic: () -> NonNegativeInteger
coerce: % -> %

from Algebra %

coerce: % -> OutputForm
coerce: Fraction Integer -> %
coerce: Fraction R -> %

from Algebra Fraction R

coerce: Integer -> %
coerce: R -> %

from Algebra R

commutator: (%, %) -> %
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)
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 %
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) -> %
prime?: % -> Boolean
principalIdeal: List % -> Record(coef: List %, generator: %)
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 %
squareFreePart: % -> %
subtractIfCan: (%, %) -> Union(%, failed)
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`.

zero?: % -> Boolean

from AbelianMonoid

AbelianGroup

AbelianMonoid

AbelianSemiGroup

BasicType

BiModule(%, %)

BiModule(R, R)

CancellationAbelianMonoid

canonicalsClosed

canonicalUnitNormal

CommutativeRing

CommutativeStar

DivisionRing

EntireRing

EuclideanDomain

Field

GcdDomain

IntegralDomain

LeftOreRing

Magma

MagmaWithUnit

Monoid

NonAssociativeRing

NonAssociativeRng

NonAssociativeSemiRing

NonAssociativeSemiRng

noZeroDivisors

PrincipalIdealDomain

Ring

Rng

SemiGroup

SemiRing

SemiRng

SetCategory

TwoSidedRecip

UniqueFactorizationDomain

unitsKnown