A modification of the Blahut algorithm is proposed for decoding Reed-Solomon codes beyond half the minimum distance. An RS code is described as an (n, k) code, where the codeword consists of n symbols from a Galois field of q elements, k of which are information symbols, with r=(n-k) check symbols. We define the minimum distance, d=r+1, and the maximum number of error symbols that can be corrected, t. An effective method is offered for searching the unknown discrepancies needed for analytical continuation of the Berlekamp-Massey algorithm through two additional iterations. This reduces the search time by 2(q-1)n/((n+t+1)(n-t)) times compared to the Blahut algorithm. An architecture of a searcher for unknown discrepancies is given. The coding gain of the proposed algorithm is shown for some practical codes.