<p>Both packages do the same. LSMR is based on Fong &amp; Saunders algorithm from 2010 (see <a href="http://arxiv.org/abs/1006.0758" rel="noreferrer">paper</a>), and has been introduced in scipy very recently (ie, version 0.10 and earlier won't have it). According to the paper, LSMR should converge faster than LSQR, which uses the Paige &amp; Saunders algorithm that has been around for almost 30 years. </p>
differences between scipy.sparse.linalg.lsmr and scipy.sparse.linalg.lsqr
<p>Does anybody know when is better to choose which? They seem the same to me...</p>

<p><a href="http://docs.scipy.org/doc/scipy/reference/generated/scipy.sparse.linalg.lsmr.html#scipy.sparse.linalg.lsmr" rel="noreferrer">lsmr</a>
<a href="http://docs.scipy.org/doc/scipy/reference/generated/scipy.sparse.linalg.lsqr.html#scipy.sparse.linalg.lsqr" rel="noreferrer">lsqr</a></p>