Fixed-Size Least Squares Support Vector Machines (FS-LSSVM) is a powerful tool for solving large scale classification and regression problems. FS-LSSVM solves an over-determined system of M linear equations by using Nyström approximations on a set of prototype vectors (PVs) in the primal. This introduces sparsity in the model along with ability to scale for large datasets. But there exists no formal method for selection of the right value of M . In this paper, we investigate the sparsity-error trade-off by introducing a second level of sparsity after performing one iteration of FS-LSSVM. This helps to overcome the problem of selecting a right number of initial PVs as the final model is highly sparse and dependent on only a few appropriately selected prototype vectors (SV) is a subset of the PVs. The first proposed method performs an iterative approximation of L0-norm which acts as a regularizer. The second method belongs to the category of threshold methods, where we set a window and select the SV set from correctly classified PVs closer and farther from the decision boundaries in the case of classification. For regression, we obtain the SV set by selecting the PVs with least minimum squared error (mse). Experiments on real world datasets from the UCI repository illustrate that highly sparse models are obtained without significant trade-off in error estimations scalable to large scale datasets.