interested in direction of solution of linear system

Question

I have a system of linear equations $Ax = b$ where $x$ is unknown and $A, b$ are known. $A$ and $b$ are large, so there is a significant cost to compute the system In my particular applications, I only need the direction of $x$. Are there special techniques/tricks to compute only the direction of $x$?

Thanks in advance,

Answer 1

Not really, finding the direction of the solution and the solution itself is quite the same thing.

If you would know the direction $d$ of $x$, so that $x=\alpha d$ for some $\alpha$, then you could find the solution, e.g., by minimizing $$ f(\alpha)=\|b-\alpha Ad\|_2^2, $$ which gives $$ \alpha=\frac{b^Tv}{v^Tv}\quad\text{with}\quad v=Ad. $$