Calculation of the characteristic polynomial of a matrix

We consider efficient algorithms of calculation of the characteristic polynomials of matrices over commutative rings. We give estimates of complexity treated as the number of ring operations, and for the ring of integers the estimates are presented in terms of the number of multiplication operations over the machine words. We suggest a new algorithm to calculate the characteristic polynomial which has the best estimate of complexity in the ring operations. We give recommendations concerning applications of the algorithm of calculation of the characteristic polynomials depending on the size of the matrix, in particular, the algorithm suggested in this paper is recommended to be applied to integer-element matrices of size greater than 60.