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I have a sequence of unit vectors $\vec{v}_0,\vec{v}_1,\ldots,\vec{v}_k,\ldots$ with the following property: $\lim_{i\rightarrow\infty}\vec{v}_{i} = \vec{\alpha}$, i.e. the sequence converges to a finite unit vector.

As the sequence is generated by a poorly known process, I am interested in modelling $\vec{v}_k$ given previous generated vectors $\vec{v}_0,\vec{v}_1,\ldots,\vec{v}_{k-1}$.

What are the available mathematical tools which allows me to discover a vector function $\vec{f}$ such that $\vec{v}_k\approx \vec{f}(\vec{v}_{k-1},\vec{v}_{k-2},\ldots,\vec{v}_{k-n})$, for a given $n$, in the $L_p$-norm sense?

EDIT: I am looking along the lines of the Newton's Forward Difference Formula, which predicts interpolated values between tabulated points, except for two differences for my problem: 1) Newton' Forward Difference is applicable for a scalar sequence, and 2) I am doing extrapolation at one end of the sequence, not interpolation in between given values.

ADDITIONAL INFO: Below are plots of the individual components of an 8-tuple unit vector from a sequence of 200:

Plot of individual components of an 8-tuple unit vector

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    You can make it simpler by just considering one entry of the vector, then we can not answer that question instead of not answering your original question.2012-08-22
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    @none: I suspect each entry of $\vec{v}$ are interrelated, therefore I am looking for a solution whereby all elements are not considered separately.2012-08-22
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    If you know that the sequence will converge to $\vec\alpha$, wouldn't $\vec f = \vec\alpha$ be a good estimate?2012-08-24
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    Oh, I just see that your vectors are "unit vectors". Anyway, I still think $\vec f = \vec\alpha$ is a good estimate. But if you want something more, I guess you need to know more about the rate of convergence of $\vec v_i$.2012-08-24
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    I am kind of guessing what you are asking. I guess that you want a way to interpolate points on the $p$-norm unit sphere such that the result is still on the sphere. Am I right? I believe any interpolation/extrapolation methods can work as long as you normalize the result eventually. You do, however, need continuity of all maps involved to guarantee stability. That means a problem may arise when the result of interpolation/extrapolation is a zero vector, and you have to decide how to resolve it. (How would you estimate the mean of two points on the opposite poles of a sphere anyway?)2012-08-24
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    @Tunococ: To your first and second comments, computing the sequence to $\vec{\alpha}$ is computationally intensive. Therefore I would like to estimate $\vec{\alpha}$ given the currently computed sequence of $\vec{v_i}$ using some kind of interpolation scheme. To your third comment: Yes eventually the result from the interpolation scheme will be renormalized to $||\vec{v_i}||_{2} = 1$. You may safely assume that the interpolation will not be a zero vector.2012-08-24
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    What you want is a series acceleration. For the one dimensional case i know about Shanks transformations and Richardson extrapolations. These can be extended to several dimensions - i don't think that they are too effective in that case though. The problem is, that there are incredibly many ways to generalize them to three dimensions. Also, you have the special case of being restricted to a submanifold (the unit sphere) which can be used to reduce the number of dimensions, but then introduces the problem of singular points / parametrisations...2012-08-24
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    I see. I thought that you know $\vec\alpha$ beforehand, but now I know that you don't. I then think that a very good estimate of $\vec\alpha$ is the latest vector $\vec v_{k-1}$ that you have, especially when you say that the process is poorly known. Anyway, if anything is known, you should tell us. Different extrapolation methods perform differently depending on the underlying model.2012-08-25
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    @Tunococ: I have added plots of an 8-tuple unit vector sequence in the problem description. You may inspect the behaviour of how each component varies.2012-08-25

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