I am looking for a representation of approximations forGiven a finite but very long integer sequence. The actual sequence has on the orderfunction of $2^{150}$ elementsreal numbers f(x), so it would be impossible to calculate or store the entire sequence. The plan is to create a program thatI can come up withcreate approximations of the sequence and through evolution slowly improve the approximation as it is runningto arbitrary precision using Taylor polynomials.
What I have:
- I would start with small and random approximations.
- I have a good test for comparing two approximations to see which one is better. Better approximations will "mate" with each other and generate new approximations.
- In addition, some random changes will be introduced as "mutations" for some of the children.
ThingsIs there something equivalent in the discrete case when I have considered:
Polynomials, but they are not going to give me integers, and also have the downside that they will be computationally expensive to when they reach high orders, which they quickly would.
I looked into a Mersenne Twister, but this will probably never yield good approximations.
I have also considered a net of NAND gates. I would start with a small random net and let the evolution reuse portions of the net in subsequent generations. This is the best I can come up with on my own.
What type of structure or formula can approximate a finite integer sequence of integers that I want to approximate to arbitrary precision, while at the same time be fast to evaluate? The approximations does not need to be finite, but they must start at zero and be defined and computable for every integer input up to $2^{150}$ or so.
