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Use Plancherel's theorem to evaluate this integral. The basic idea is that for two functions $f,g$, $$I=\int_{-\infty}^{\infty} f(x) g^*(x)dx = \int_{-\infty}^{\infty} F(k) G^*(k) dk$$ where $F,G$...

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Expanding upon my comment above, if you only need a few digits of accuracy you can probably use the method of stationary phase. We can follow the procedure on Wikipedia. We can write the transform as ...

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Depending on how large $n$ can get and how many evaluation points $x$ you wish to use, this summation problem is well-suited to the use of fast multipole methods (FMMs); for instance, see the black-...

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Provided that your geometry conforms to the cylindrical coordinate system, the separation of variables solution should look something like  \Phi(\mathbf{r}) = \sum_{n,m} a_{nm} \left\lbrace \begin{...

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Simple way: Compute the Euclidean distance between each item and the target feature specification. Sort by distance. Take the first 50 elements. Fast way for arbitrary target features: Since your ...

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You can think of each vector as a point in your linear space. As such, we can use a simple quadtree/octree-like algorithm to map your points into boxes, with "nearby" vectors assigned to the same or ...

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The comments above give some very good reasons for using octrees (i.e., recursively halving the computational cube in each dimension as opposed to a more general orthogonal bisection). Symmetry and ...

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Since convolution can be written as a matrix-vector product $Ax=b$ of a circulant or Toeplitz matrix $A$ acting on a vector $x$, you can invert or pseudoinvert via SVD $A$ to obtain $x=A^{-1}b$. That ...

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You're using the orthonormalized version of the Legendre polynomials, while Hesthaven is not. The polynomials in your matrix are normalized by a factor of $\sqrt{\frac{2n+1}{2}}$, i.e. $\sqrt{\frac{2(... View answer 1 answers 0 votes 65 views Accepted answer 0 votes First, in an FMM implementation, you are always working in some coordinate frame relative to the center of some box, hence the$P-Q\$ vector defining the primed coordinate system above. Make sure this ...