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Thread: Tensor product constructs

  1. #1
    Jan 2010

    Tensor product constructs

    Hi all.
    I have already done linear interpolation in four variables and it works but the accuracy is not good, that's why I need to do non-linear interpolation for the same four variables.

    Now, I have since learnt that one of the "easiest" ways to do it is by use of tensor product constructs.
    Now, my question is, given, say, 4 univariate cubic polynomials (each of the 4 is interpolating its respective variable ), how does one multiply all four together ? (That is the definition of a tensor product construct in this context).
    Thus, given the cubic polynomials,
    var1 = aw^3 + bw^2 + bw + d
    var2 = ex^3 + fx^2 + gx + h
    var3 = iy^3 + jy^2 + ky + l
    var4 = mz^3 + nz^2 + oz + p

    F(w,x,y,z) = tensor product construct. How does one do that ?
    Now please, no references to wikipedia articles -- or any other articles for that matter. I have read and re-read lots of articles online and at this point, they wont help --- what I need now is for someone who knows how to do a tensor product construct of the 4 univariate cases above to produce a multivariate case in 4 dimensions.

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  2. #2
    Jan 2010

    I guess you don't know what a tensor product is.

    If you want tensor products you first need vector spaces.

    The idea is that you create a new space, called the tensor space. This space fullfilled all vector space axioms (abelian group for the inner composition, outer operation for the scalar multiplication and the structure like a monoid (but now real monoid) and a structure which looks like distributive laws but like the monoid no real distributive laws. People often says that a vector space has distributive laws but this is not true.) So a tensor is only an element of a tensor space. If you have polynomials, you have to interpretate them as vectors in the spaces of polynomials.

    If you take $\displaystyle n$ vectorspaces in the cartesian product (thats the reason why one say tensor to the $\displaystyle n$-th) and map them into a new space with the map $\displaystyle T$ (like tensor).

    1.) $\displaystyle T$ has to be multi linear, which means linear in each argument. Think about the determinant, which is an alternating multilinear form. Its the same thing

    2.) $\displaystyle T$ is universal. This means, that you have one unity map to an aribratary space such you can build $\displaystyle T$ as the composition of this.

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