I am now refactoring PartialFraction. One thing I noted is that
PartialFraction is underspecified. First, simplest partial
fraction decomposition uses prime-power denominators. However,
partial fraction decomposition makes sense also with more general
denominators, just one need to ensure that denominators are
coprime. Since partialFraction accepts Factored with not
necessarily prime factors, it is possible to create such fractions.
But then there are troubles with arithmetic because when adding
to fractions denominators can have nontrivila common factors,
so simple comparison of denominators is not enough.
In fact, there are possible troubles even with prime-power
denominators: we can multiply prime by a unit and we again
get a prime. So correct arithmetic would need to check if
denominators are equivalent, which is not done in current
code. Alternatively, we could assume canonicalUnitNormal,
but ATM argument to PartialFraction is just an EuclideanDomain.
We could also have different spec and require that all factors
of denominators are pairwise coprime and chosen in canonical
way. This is more powerful, but may be tricky for naive
users.
There is also question of nonunique representation. Namely
currently we have:
(1) -> p1 := 1 + partialFraction(-3, 7)
3
(1) 1 - -
7
Type: PartialFraction(Integer)
(2) -> p2 := partialFraction(4, 7)
4
(2) -
7
Type: PartialFraction(Integer)
(3) -> (p1 = p2)@Boolean
(3) false
Type: Boolean
(4) -> (p1::FRAC(INT) = p2::FRAC(INT))@Boolean
(4) true
Type: Boolean
So current '=' in PartialFraction is wrong (that is easy to fix).
But there is question if we should try to make representation
more unique. In general there are multiple possible remainders,
so different ways to write mathematically the same partial
fraction. But in case of OrderedRing we could try to find
positive remainder. In case of Integer this would lead to
unique representation (assuming given denominators).
Actally, I noticed the problem testing different implementation
of 'partialFraction'. I got:
(10) -> partialFraction(1,factorial 10)
97 58 13 6
(10) - -- + -- + -- - -
8 4 2 7
2 3 5
Type: PartialFraction(Integer)
which is mathematically correct, but differs from previous
result which has different signs.
--
Waldek Hebisch
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