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This work aims at reducing the memory demand of the data structures that are usually pre-computed and stored for fast computation of the i-vectors, a compact representation of spoken utterances that is used by most state-of-the-art speaker recognition systems. We propose two new approaches allowing accurate i-vector extraction but requiring less memory, showing their relations with the standard computation method introduced for eigenvoices, and with the recently proposed fast eigen-decomposition technique. The first approach computes an i-vector in a Variational Bayes (VB) framework by iterating the estimation of one sub-block of i-vector elements at a time, keeping fixed all the others, and can obtain i-vectors as accurate as the ones obtained by the standard technique but requiring only 25% of its memory. The second technique is based on the Conjugate Gradient solution of a linear system, which is accurate and uses even less memory, but is slower than the VB approach. We analyze and compare the time and memory resources required by all these solutions, which are suited to different applications, and we show that it is possible to get accurate results greatly reducing memory demand compared with the standard solution at almost the same speed.