Recently there have been renewed interests in single-hidden-layer neural networks (SHLNNs). This is due to its powerful modeling ability as well as the existence of some efficient learning algorithms. A prominent example of such algorithms is extreme learning machine (ELM), which assigns random values to the lower-layer weights. While ELM can be trained efficiently, it requires many more hidden units than is typically needed by the conventional neural networks to achieve matched classification accuracy. The use of a large number of hidden units translates to significantly increased test time, which is more valuable than training time in practice. In this paper, we propose a series of new efficient learning algorithms for SHLNNs. Our algorithms exploit both the structure of SHLNNs and the gradient information over all training epochs, and update the weights in the direction along which the overall square error is reduced the most. Experiments on the MNIST handwritten digit recognition task and the MAGIC gamma telescope dataset show that the algorithms proposed in this paper obtain significantly better classification accuracy than ELM when the same number of hidden units is used. For obtaining the same classification accuracy, our best algorithm requires only 1/16 of the model size and thus approximately 1/16 of test time compared with ELM. This huge advantage is gained at the expense of 5 times or less the training cost incurred by the ELM training.