GRAPPA (Generalized Autocalibrating Partially Parallel Acquisition) is a widely used parallel MRI reconstruction technique. The processing of data from multichannel receiver coils may increase the storage and computational requirements of GRAPPA reconstruction. Random projection on GRAPPA (RP-GRAPPA) uses random projection (RP) method to overcome the computational overheads of solving large linear equations in the calibration phase of GRAPPA, saving reconstruction time. However, RP-GRAPPA compromises the reconstruction accuracy in case of large reductions in the dimensions of calibration equations. In this paper, we present the implementation of GRAPPA reconstruction method using potential iterative solvers to estimate the reconstruction coefficients from the randomly projected calibration equations. Experimental results show that the proposed methods withstand the reconstruction accuracy (visually and quantitatively) against large reductions in the dimension of linear equations, when compared with RP-GRAPPA reconstruction. Particularly, the proposed method using conjugate gradient for least squares (CGLS) demonstrates more savings in the computational time of GRAPPA, without significant loss in the reconstruction accuracy, when compared with RP-GRAPPA. It is also demonstrated that the proposed method using CGLS complements the channel compression method for reducing the computational complexities associated with higher channel count, thereby resulting in additional memory savings and speedup.