On Wednesday, June 17, 2015 at 9:04:54 PM UTC+2, Shivam Vats wrote:
>
>
>
> Is your question related to series somehow? What kind of compositions
>> do you have in mind?
>
> Yes, Ondrej. Since, we have already implemented many elementary
> functions in ring_series, it will be nice if we have faster methods
> to invert them and their compositions (eg sin(sin(x)))
>
> The general idea is that computing logarithmic and inverse trigonometric
>> functions of formal power series is just algebraic operations on power
>> series followed by formal (term by term) integration, e.g. log(f(x)) = int
>> f'(x) / f(x) dx. From there, Newton iteration allows you to compute
>> exponential and forward trigonometric functions.
>
> Right! We have attempted something similar in `ring_series`.
> Formula based expansion as `_atan_series` and this method
> as `rs_atan` are implemented here
> <https://github.com/sympy/sympy/blob/master/sympy/polys/ring_series.py#L627>.
> `_atan_series` seems to
> be much faster than `rs_atan`. As of now, we expand `tan`
> from `atan` using Newton iterations and sin/cos from `tan`
> using half angle formula. What do you suggest?
>
Yes, for atan this is the right approach.
For sin/cos, exp, tan, etc. Newton iteration is optimal for long power
series if you have asymptotically fast polynomial multiplication. When your
multiplication is O(n^2), the following O(mn) algorithm for exponentials
(where m is the length of the input and n is the length of the output) is
better:
def exp_series(A, n):
B = [exp(A[0])]
for k in range(1, n):
B.append(sum(j*A[j]*B[k-j]/k for j in range(1,min(len(A),k+1))))
return B
It's easy to turn this into an algorithm for sin/cos (and hence also tan),
and there is an analog for raising a series to a constant power (if I
remember correctly, it's in Knuth vol 2).
Fredrik
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