The classical maximum eigenvalue detection (MED) algorithm has excellent performance in detecting correlated signals. However, with the increasing signal dimensionality, the MED algorithm faces serious problems in the calculation efficiency and implementation of test statistic and decision threshold, thus greatly limiting the further application of the algorithm in modern cognitive communication systems. To this end, a low-implementation complexity MED algorithm based on a numerical analysis theoretical framework is proposed. The new algorithm uses the Rayleigh quotient accelerated power method to iteratively compute the test statistic, which has a fast convergence rate in detecting high-dimensional signals compared with the classical power method; meanwhile, different from the classical look-up table method, a threshold calculation method based on the cubic spline interpolation method is proposed, which can quickly determine the decision threshold corresponding to any given target false-alarm probability. The proposed MED algorithm effectively improves the computational efficiency and reduces the complexity of algorithm implementation while maintaining the detection performance of the original algorithm, which is particularly attractive for spectrum sensing problems in high-dimensional conditions. Finally, the simulation results demonstrate the effectiveness of the proposed algorithm.