It is known that solving systems of linear equations is reducible to SVD in a straightforward way; if you want to solve $\mathbf{Ax}=\mathbf{b}$, then you can perform SVD on $\mathbf{A}$ and minimize $||\mathbf{UDVx}-\mathbf{b}||$.

However, is there a reverse reduction that is also very efficient? That is, if you can solve linear equations, you can solve SVD?

EDIT: Because of Denis's comment/answer below, it looks like there isn't a reduction *in general*. But I'm interested in these problems over $\mathbb{C}$; so, the new question is: If we can solve linear equations over $\mathbb{C}$ exactly or approximately, can we perform an "approximate" SVD (for some suitable notion of "approximate")?

The answer still seems to be in the negative, but I defer to people who actually know something about this.