Subspace methods such as MUSIC and Minimum Norm estimators are popular for their high resolution property in sinusoidal and directions of arrival (DOA) estimation, but they are also known to be of high computational demand. In this paper, new fast algorithms for DOA and sinusoidal frequency estimation which do not require the exact eigendecomposition of the covariance matrix are presented. These algorithms approximate the required subspace using rational and power-like methods applied to the sample covariance matrix. A substantial computational saving would be gained compared with those associated with the eigendecomposition-based methods. Simulations results have shown that these approximated estimators have comparable performance at low signal-to-noise ratio (SNR) to their standard counterparts and are robust against overestimating the number of impinging signals.