01.03.2011 00:34, Vinzent Steinberg пишет:
A. My fears that it will be hard to maintain behavior of O((x-x0)**n)
> for pure "asymptotic expansions" expression in general case (without
> knowledge of series structure). In this case O((x-x0)**n) must detect
> and catch (x-x0)**k terms in a whole expression for every operation. And
> detect and do not permit operations between expressions which contain
> O((x-x1)**n) and O((x-x2)**n) respectively (x1<> x2).
It would be a "for-the-moment" fix which is likely to be changed (and
should be marked as such). I think it is still better than no O() term
at all.
> B. I realize only maintaining of O behavior "asymptotic expansions" with
> knowing series structure ( 2) in [1] )
>
> Nevertheless, I observe consensus that A*or* B must be implemented
> obligatory.
> And what is more, both A*and* B implementations better.
Could you please explain what you mean with the last sentence?
Vinzent
Well, It will be easer to explain from the point of view of programmer.
It is clear that "A" is sympy Add class which consists of polynomials
terms "a_n (x-x0)**n" and O((x-x0)**n).
In "B" is imagined class TaylorExpansion that implement the "n-th order
Taylor expansion" so it track his precision (n), the list of [a_n],
variable x, and point x0.
[a_n] may by any sympy expression, not necessarily numbers.
It has self-depended method for representation it in sympy (print) to
print "a0 + a1*(x - x0) + ... + O(x - x0)"
It has method "asExpr()" to shape "A" variant for general propose.
It has method "truncate" to form Taylor polynomial. May be another class
too if it has not analog in polynomials module yet.
The reason of this class - that manipulations with its objects will be
easer.
F.e. the derivative of it will bring to the same object but the list of
[a_n] will be shifted throu its index by one and precision will be
decremented.
Summation equivalent to summation of lists (c_i = a_i + b_i), with
alignment of precision to minimal value of both.
Multiplication will be a little complex, but still without analyzing
(x-x0)**k parts of expression.
May be I became entangled now it with similar "exponential generating
function", but the last one is not derived from function (opposite to
Taylor), and it is not expansion (it is series).
--
Alexey U.
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