This paper studies on a fast approach for the eigenproblems of correlation matrices used in direction-of-arrival (DOA) estimation algorithms, especially for the case that the number of arriving waves is a few. The eigenvalues and the corresponding eigenvectors can be obtained in a very short time by the algebraic solvent of up to quartic polynomials. We also confirm that the present approach does not make the accuracy worse when it is implemented by finite word-length processors like digital signal processor (DSP) or field programmable gate array (FPGA).