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I want to find the matrix-vector product:

$b=C \times a$

where $b=[b_1, ..., b_n]$ is the unknown vector, $a=[a_1, ..., a_n]$ is the known vector, and C is the coefficients matrix.

The complexity to calculate b is $O(n^2)$

Assuming that the C matrix is a Cauchy matrix $n \times n$ matrix as follows:

\begin{align} C[i,j] = \frac{1}{x_i+y_i} \end{align}

Is it possible to calculate b with less complexity?

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migrated from Nov 29 '13 at 18:47

This question came from our site for professional mathematicians.

this is the problem of multiplying a vector by a matrix, not solving linear equations... – Dima Pasechnik Nov 29 '13 at 17:28
@DimaPasechnik If you replace "solving set of linear equations" by "computing the matrix-vector product", the question is completely OK. It is about whether you can compute a product of this matrix (depending on $2n$ parameters) with smaller complexity than $O(n^2)$ for general MV product. – Algebraic Pavel Nov 29 '13 at 19:57
I've edited the question. Hopefully someone reviews the changes. – Dima Pasechnik Dec 4 '13 at 13:38

Here's a brief summary (see Wikipedia and linked arXiv article for more)

  1. $b=Ca$ can be approximated in $O(n\log n)$ time using the Fast multipole method
  2. LU factorization of $C$ using the GKO algorithm, this takes $O(n^2)$ time. See this paper for how to implement this well (i.e., $a=C^{-1}b$)
  3. If you are not worried about approximations or numerical instability, then you can handle this in $O(n\log^2 n)$ time (i.e., compute $a \approx C^{-1}b$)
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