The authors study shooting methods for boundary value problems in ordinary differential equations from the point of view of how the accuracy and robustness of the ''simple'' shooting procedure can be enhanced. The essential idea is to combine the fundamental matrices originating from both endpoints of the interval in such a way that the smaller singular values of these matrices and the corresponding orthogonal projections are eliminated. The reason for this strategy is that perturbations of a matrix have a much stronger effect on the smaller singular values than on the larger ones. There are some examples in which the authors compare the performance of this version with the well-known shooting method.