On Wednesday, June 10, 2015 at 5:53:16 PM UTC+2, Shivam Vats wrote:
>
> Mario, the algorithm looks quite similar to the Newton
> method you have used for functional inverses for 
> expanding tan and tanh. We could also expand `sin/cos`.
> Which one do think should be faster?
>
> Also, can the convergence of the method be proven?
> I could not find any reference to it (except the algorithm
> in Computer Arithmetic by Paul Zimmerman).
>

Series reversion is a purely formal operation, so it always converges.

For a faster algorithm to compute the reversion of a generic power series, 
see my paper http://fredrikj.net/math/reversion.html (a "generic" power 
series being given by a list of coefficients -- there are better methods 
for computing inverse series of functions composed from elementary 
functions like sin(sin(x))).

Fredrik

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