04/15/2008, 10:59 AM
Some remarks from my side:
1. This is a great undertaking
2. Dont mix different methods into one tetration function. Each method should have its own dedicated function name(or one function which you can provide with the method as parameter). This is essential if you want to compare different methods.
3. The computation of the regular iteration is quite well-developed in the literature. There is the direct numeric method and there is the method via powerseries coefficients. There is a numeric method to directly compute the regular Abel function, as well as the regular iteration (without explicitly using function inversion). You can also directly compute the powerseries coefficients of the exponential of the Abel function.
4. Though I trust Jay's intuition, I think there was no proof that the accellerated method indeed yields the same limit as the unaccelareted/your original method.
5. Determining error-bounds would be a great thing, I hope you can make it. However is this possible if we have no proof of the convergence of the coefficients yet? Is it on the line that *if* they converge, then they would have these errorbounds? Perhaps we need extra research about the error-bounds of the other methods. Generally I think properly dealing with error-bounds is a quite messy thing. You need some kind of function how the precision of the output depends on the precision of the input, so that you can compute the overall error of a computation. Often then it is necessary to drastically increase the precision of intermediate steps.
6. I am not quite sure whether to develop it in pseudo code really makes sense. I mean if this is several pages long, you dont know whether there are errors in it, and who wants to read that?
The dilemma here is that we are quite differently equipped with mathematical software. I would suggest to implement it in a real language (and hence have the ability of debugging) or/and describe all methods in a scientific enough way that it should be easy to implement. For the implementation the language of choice would be Sage, as nobody is restrained from its use.
1. This is a great undertaking
2. Dont mix different methods into one tetration function. Each method should have its own dedicated function name(or one function which you can provide with the method as parameter). This is essential if you want to compare different methods.
3. The computation of the regular iteration is quite well-developed in the literature. There is the direct numeric method and there is the method via powerseries coefficients. There is a numeric method to directly compute the regular Abel function, as well as the regular iteration (without explicitly using function inversion). You can also directly compute the powerseries coefficients of the exponential of the Abel function.
4. Though I trust Jay's intuition, I think there was no proof that the accellerated method indeed yields the same limit as the unaccelareted/your original method.
5. Determining error-bounds would be a great thing, I hope you can make it. However is this possible if we have no proof of the convergence of the coefficients yet? Is it on the line that *if* they converge, then they would have these errorbounds? Perhaps we need extra research about the error-bounds of the other methods. Generally I think properly dealing with error-bounds is a quite messy thing. You need some kind of function how the precision of the output depends on the precision of the input, so that you can compute the overall error of a computation. Often then it is necessary to drastically increase the precision of intermediate steps.
6. I am not quite sure whether to develop it in pseudo code really makes sense. I mean if this is several pages long, you dont know whether there are errors in it, and who wants to read that?
The dilemma here is that we are quite differently equipped with mathematical software. I would suggest to implement it in a real language (and hence have the ability of debugging) or/and describe all methods in a scientific enough way that it should be easy to implement. For the implementation the language of choice would be Sage, as nobody is restrained from its use.