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From: Ebrahim Mousavi on 4 Aug 2010 19:47
Hi everybody,

I'm using \ to solve an under-determined system of equations Ax=y (A is m x n, n = c*m, c ~ 1.5-3). The whole system is rather small (m ~ 10-100). As is common, I'm very interested in the most sparse solution. Is there any guarantee for the existence of the solution or an upper bound for the nnz(x), other than rank of A or m?

Matlab's doc says that \ works this way (ignoring permutations):
[Q, R] = qr(A);
x = R\(Q'*y) (*)

but then, (*) is another under-determined system by itself with R being upper-triangular, so is there a trivial way to solve (*)?

Thanks,
Meysam
From: Greg Heath on 5 Aug 2010 09:15
On Aug 4, 7:47 pm, "Ebrahim Mousavi" <si_moos...(a)yahoo.com> wrote:
> Hi everybody,
>
> I'm using \ to solve an under-determined system of equations Ax=y (A is m x n, n = c*m, c ~ 1.5-3). The whole system is rather small (m ~ 10-100). As is common, I'm very interested in the most sparse solution. Is there any guarantee for the existence of the solution or an upper bound for the nnz(x), other than rank of A or m?

What is nnz(x)???

> Matlab's doc says that \ works this way (ignoring permutations):
> [Q, R] = qr(A);
> x = R\(Q'*y)    (*)
>
> but then, (*) is another under-determined system by itself with R >being upper-triangular, so is there a trivial way to solve (*)?

Backslash will automatically generate one of the
many sparse "basic" solutions. Which one depends
on the pivots during column pivoting.

Search the CSSM archives in Google groups using

greg heath "basic solution"

The following threads may be relevant:

1. Algorithm for "BASIC" solutions?
2. difference between pseduoinversion and backslash operator in ...
3. Finding Linearly dependent rows/columns
4. The effect of linear dependence on linear regression (Part1)

Hope this helps.

Greg






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