Currently, MTSCAT MultivariateTaylorSeriesCategory(Coef: Ring,Var: OrderedSet)
exports
coefficient : (%,List(Var),List(NonNegativeInteger)) -> %
coefficient : (%,Var,NonNegativeInteger) -> %
coefficient : (%,IndexedExponents(Var)) -> Coef
and TaylorSeries(Coef: Ring) exports
coefficient : (%,NonNegativeInteger) -> Polynomial(Coef).
I believe that the intention of
coefficient : (%,Var,NonNegativeInteger) -> %
returning a TaylorSeries was to view the Taylor series as a univariate Taylor
series, and then return the coefficient of v^n. For example, after
(1) -> X := x::TS(FRAC INT); Y := y::TS(FRAC INT);
Type: TaylorSeries(Fraction(Integer))
(2) -> s := (1/(1-X-Y))::TS(FRAC INT)
(2)
2 2 3 2 2 3
1 + (y + x) + (y + 2x y + x ) + (y + 3x y + 3x y + x )
+
4 3 2 2 3 4
(y + 4x y + 6x y + 4x y + x ) + O(5)
Type: TaylorSeries(Fraction(Integer))
coefficient(s, y, 1) should probably give
1 + 2 x + 3 x^2 + ...
This was never implemented.
I think that this was a bad idea anyway, since it is confusing to have also the
coefficient function from AbelianMonoidRing(R: Ring,E: OrderedAbelianMonoid)
where
coefficient(s, monomial(2, y)$IndexedExponents(Var)) yields just 1. I.e., the
coefficient of y^2 x^0.
(not implemented either)
I think it would be better to have a coercion, like perhaps
coerce: % -> UTS(%, var, 0)
One could then say
s::UTS(TaylorSeries(Fraction(Integer)), y, 0) and get something similar to
(3) -> series(str)$UTS(UTS(EXPR INT, x, 0), y, 0)
(3)
1 3 5 1 2 1 4 5 1 1 3 5 2
x - - x + O(x ) + (1 - - x + -- x + O(x ))y + (- - x + -- x + O(x ))y
6 2 24 2 12
+
1 1 2 1 4 5 3 1 1 3 5 4 5
(- - + -- x - --- x + O(x ))y + (-- x - --- x + O(x ))y + O(y )
6 12 144 24 144
Type:
UnivariateTaylorSeries(UnivariateTaylorSeries(Expression(Integer),x,0),y,0)
Note however, that "coefficient : (%,Var,NonNegativeInteger) -> %" and
"coefficient: (%,List(Var),List(NonNegativeInteger))" would be (if implemented)
slightly more general, since, in the example (2) above, we could also ask for
coefficient(s, y, 2) which should probably give
1 + 3 x + 6 x^2 + ...
With my proposed semantics, this could be done via a mapping function (which
does not exist yet, I think...)
(27) -> map(c +-> coefficient(c, y, 2), [coefficient(s, n) for n in 0..])
2
(27) [0,0,1,3x,6x ,...]
Type: Stream(Polynomial(Fraction(Integer)))
If there are no comments, I'll try to implement the above behaviour.
Martin
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