Solve a particular linear system efficiently in julia
<p>I use extensively the julia's linear equation solver <code>res = X\b</code>. I have to use it millions of times in my program because of parameter variation. This was working ok because I was using small dimensions (up to <code>30</code>). Now that I want to analyse bigger systems, up to <code>1000</code>, the linear solver is no longer efficient.</p>

<p>I think there can be a work around. However I must say that sometimes my X matrix is dense, and sometimes is sparse, so I need something that works fine for both cases. </p>

<p>The <code>b</code> vector is a vector with all zeroes, except for one entry which is always <code>1</code> (actually it is always the last entry). Moreover, I don't need  all the <code>res</code> vector, just the first entry of it.</p>