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You mean if you have an attracting fixed point then you simply get an arbitrary small x by applying

so that the precision becomes arbitrary big for the subsequent

and then you transform it back to the original value by applying

?

Hm, I dont know whether you loose the achieved precision by transforming it back. This also would only work for fractional iterations, while I was talking about general series.

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08/16/2007, 10:03 PM
(This post was last modified: 08/16/2007, 10:05 PM by jaydfox.)
bo198214 Wrote:You mean if you have an attracting fixed point then you simply get an arbitrary small x by applying so that the precision becomes arbitrary big for the subsequent and then you transform it back to the original value by applying ?

Exactly, except for f(z)=e^z-1, it's a repelling fixed point, so -n and n would be used instead of n and -n.

Quote:Hm, I dont know whether you loose the achieved precision by transforming it back. This also would only work for fractional iterations, while I was talking about general series.

You shouldn't lose much precision. The absolute value of the difference between

and

is about 2/(100^2). For

and

, it's about 2/(1000^2).

In other words, the precision you lose is on the order of 1/n^2. For n=1000, you only need an extra six decimal places of precision in your iterated step out function. Of course, each iteration will introduce additional errors, so really you need about 6+log_10(1000) = 9 to 10 extra digits of precision. Still, the iterating functions converge quite nicely, so the extra expense of using additional precision for the integer iterations is acceptable.

~ Jay Daniel Fox

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08/16/2007, 10:13 PM
(This post was last modified: 08/16/2007, 10:17 PM by bo198214.)
jaydfox Wrote:You shouldn't lose much precision.

Yes, thats right because you can compute

and

to arbitrary exactness.

So did you already determine the formula for the optimal truncation and its error function? (The answer however should go into the thread computing the iterated exp(x)-1).

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Not yet, I've been busy with my job most of today, just taking opportunities here and there to try to make a few posts. Also, I'm working on trying to figure out if there's a relationship between the graph of

and

. I think I'm getting close, conceptually, but I'm still not there. That's where most of my attention is focussed at the moment. If I can derive a formula for

, then I'll be quite satisfied that all the hard pieces are out of the way, and it'll just be a matter of formalizing everything, including providing proofs with all the gory details.

~ Jay Daniel Fox