For generalized-minimum-distance decoding, a successful decoding can require up to $d/2$ applications of an errors-and-erasures decoding algorithm to find all possible candidate codewords ($d$ is the minimum distance of the code). For Reed-Solomon codes, each application of an errors-and-erasures decoding algorithm such as the Sugiyama algorithm or the Berelkamp-Massey algorithm requires up to $O(d\sp2)$ multiplications and/or additions. These two algorithms are modified so that the original algorithm need be applied only once. The additional applications of the errors-and-erasure decoding algorithm are replaced by an update algorithm. The update algorithm reuses the intermediate results from the original application of the errors-and-erasures algorithm to find the next candidate. It is shown that the total number of multiplications and/or additions needed to find the error locator polynomial and the error evaluator polynomial is reduced to $O(d\sp2)$.