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An alternative formulation of the microwave tomography problem is proposed, in order to reduce the computation time of the inversion procedure. The well-known multiplicative regularized contrast source inversion method is used as reference algorithm. Our formulation is based on a truncated singular value decomposition of the matrix involved in the most computationally intensive operations of the inversion algorithm. After theoretical and practical investigation of the behavior of the singular values, it is shown that the corresponding approximation does not yield any significant degradation of the quality of the solutions even for large complex permittivity contrasts, while the computation times are reduced by a factor ranging between 1.1 and 6 with respect to the original method.