# Huge Linear equation system

Do you know any repository of huge matrices of a linear system?, Or to tell a problem in which a lot of linear equations are needed?

I want to solve huge linear equations systems but have not had any luck when trying to find huge matrices.

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I'm not sure I understand your request. Could you maybe indicate in what context you want to do this? is the idea to test your own solver and benchmark it against others? if so why not generate a random matrix and independent vector and solve in the mean-square sense (say) Ax=b? Also, do you want to consider any kind of system or structured systems? –  tibL Apr 23 '12 at 16:56
Why not generate a huge matrix, generate a solution, construct the rhs, and then solve? –  Sasha Apr 23 '12 at 16:56
Ok, I understand, could you just tell a problem in which huge linear systems are used, like in economy or something... –  cMinor Apr 23 '12 at 16:59
Here is a nice one. Interpolation using Vandermone matrix. Generate a random polynomial of degree 10000000 or any bgillion degree you like. Warning though: numerically ill for very large degrees.. –  user2468 Apr 24 '12 at 2:04
@J.D. "numerically ill" - indeed, but the growth rate very much depends on your choice of interpolation points. If the interpolation points are the roots of unity, the Vandermonde matrix is, up to scaling, unitary (and thus, well-conditioned)... –  Ｊ. Ｍ. Apr 24 '12 at 5:03

You might try http://math.nist.gov/MatrixMarket/

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...and the links to other sites show other places to get nice examples! –  Ｊ. Ｍ. Apr 23 '12 at 17:11

Let $n\geq 1$ be an integer, and fix $n+1$ points in $\mathbb{R}^2$, say $(x_1,y_1)$, ... , $(x_n,y_n)$, $(x_{n+1},y_{n+1})$, such that $x_1<x_2<\ldots<x_n$.

Problem: find the coefficients of the unique polynomial $p(x)\in \mathbb{R}[x]$ of degree $n$ that interpolates all $n+1$ points, i.e., find coefficients $a_0,\ldots,a_n\in\mathbb{R}$ such that the polynomial $$p(x)=a_0+a_1x+\cdots+a_nx^n$$ satisfies $p(x_k)=y_k$ for all $k=1,\ldots,n+1$.

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Oops, I just noticed that J.D. pointed this one out in a comment. –  Álvaro Lozano-Robledo Apr 24 '12 at 4:13

Vast systems of linear equations are solved as part of some integer factorization algorithms, such as the Quadratic Sieve, and the Number Field Sieves (both special and general). Much literature can be found by searching on those terms.

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