<p>We propose new and efficient algorithms for semi-blind source extraction derived from the well-known FastICA for independent component analysis (ICA). The algorithms assume independence of the source of interest (SOI) and other signals in the mixture and simultaneously exploit references as sources of side information about the SOI. Depending on how strong this information is, the algorithms show improved global convergence and accuracy compared to their purely blind counterparts. In the paper, we also consider similar existing algorithms and compare them analytically and experimentally, revealing context and differences in terms of global convergence, accuracy, and computational complexity. The broad applicability of the methods is demonstrated by speech extraction and extraction of brain networks from functional magnetic resonance imaging data.</p>