PartialFraction R¶

The domain PartialFraction implements partial fractions over a euclidean domain R. This requirement on the argument domain allows us to normalize the fractions. Of particular interest are the 2 forms for these fractions. The compact\ '' form has only one fractional term per prime in the denominator, while the p-adic'' form expands each numerator p-adically via the prime p in the denominator. For computational efficiency, the compact form is used, though the p-adic form may be gotten by calling the function padicFraction. For a general euclidean domain, it is not known how to factor the denominator. Thus the function partialFraction takes an element of Factored(R) as its second argument.

0: %

from AbelianMonoid

1: %

from MagmaWithUnit

*: (%, %) -> %

from LeftModule %

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

from RightModule R

*: (Fraction Integer, %) -> %
*: (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: (%, %) -> %
associates?: (%, %) -> Boolean

from EntireRing

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

from Algebra %

coerce: % -> Fraction R

coerce(p) sums up the components of the partial fraction and returns a single fraction.

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

coerce(f) takes a fraction with numerator and denominator in factored form and creates a partial fraction. It is necessary for the parts to be factored because it is not known in general how to factor elements of R and this is needed to decompose into partial fractions.

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

from Algebra R

commutator: (%, %) -> %
compactFraction: % -> %

compactFraction(p) normalizes the partial fraction p to the compact representation. In this form, the partial fraction has only one fractional term per prime in the denominator.

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

from EuclideanDomain

euclideanSize: % -> NonNegativeInteger

from EuclideanDomain

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

from EntireRing

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

from EuclideanDomain

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

from EuclideanDomain

factor: % -> Factored %
fractionalTerms: % -> List Record(num: R, den: Factored R)

fractionalTerms(p) extracts the fractional part of p to a list of Record(num : R, den : Factored R). This returns [] if there is no fractional part.

gcd: (%, %) -> %

from GcdDomain

gcd: List % -> %

from GcdDomain

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

from GcdDomain

hash: % -> SingleInteger

from SetCategory

hashUpdate!: (HashState, %) -> HashState

from SetCategory

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

numberOfFractionalTerms: % -> Integer

numberOfFractionalTerms(p) computes the number of fractional terms in p. This returns 0 if there is no fractional part.

one?: % -> Boolean

from MagmaWithUnit

opposite?: (%, %) -> Boolean

from AbelianMonoid

padicallyExpand: (R, R) -> SparseUnivariatePolynomial R

padicallyExpand(p, x) is a utility function that expands the second argument x p-adically'' in the first.

padicFraction(q) expands the fraction p-adically in the primes p in the denominator of q. For example, padicFraction(3/(2^2)) = 1/2 + 1/(2^2). Use compactFraction to return to compact form.

partialFraction: (R, Factored R) -> %

partialFraction(numer, denom) is the main function for constructing partial fractions. The second argument is the denominator and should be factored.

partialFraction: Fraction R -> % if R has UniqueFactorizationDomain

partialFraction(f) is a user friendly interface for partial fractions when f is a fraction of UniqueFactorizationDomain.

prime?: % -> Boolean
principalIdeal: List % -> Record(coef: List %, generator: %)
quo: (%, %) -> %

from EuclideanDomain

recip: % -> Union(%, failed)

from MagmaWithUnit

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(p) extracts the whole part of the partial fraction p.

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