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For the single-group multicast scenario, where K users are served with the same data by a base station equipped with N antennas, we present two beamforming algorithms which outperform state-of-the-art multicast filters and feature a drastically reduced complexity at the same time. For the power minimization problem, where QoS constraints need to be satisfied, we introduce a successive beamforming filter computation approach aiming at satisfying at least one additional SNR constraint per orthogonal filter update. As long as the number of users K is smaller than the number N of transmit antennas, this procedure delivers excellent results. Our second approach is an iterative update algorithm for the max-min problem subject to a limitation of the transmit power. Given a low-complexity initialization beamformer, we search within the local vicinity of this filter vector for a filter-update preserving the transmit power and achieving a larger minimum SNR. To this end, we improve the weakest user's SNR during each iteration and keep on applying this procedure as long as the updates increase the smallest SNR. Otherwise, we adapt the step-size and continue investigating the local vicinity. It turns out that this novel approach is superior to existing state-of-the-art multicast beamformers for an arbitrary number of users.