The authors construct the reduced forms of circulant matrices and quasi-skew circulant matrices. Then they show that the problem of computing the circulant square roots of a circulant matrix
can be reduced to that of computing the square roots of two half size matrices
. Two efficient algorithms are presented to compute their square roots. Those methods are faster than the traditional algorithm which is based on the Schur decomposition. They further consider circulant
-matrices with positive diagonal entries and develop two algorithms for computing their principal square roots. Those two algorithms are based on
iteration and the modified Schulz iterative method, respectively. Some numerical experiments are presented.