A limited disproof to the conjecture of evaluating the matrix polynomial I+A+A2+···+AN-1

The problem of evaluating matrix polynomial I+A+A²+···+A^N-1, has been considered. The proposed algorithms require at most 3·[log₂ N] and 2·[log₂ N]-1 matrix multiplications, respectively. If the binary representation of N is (i_ti_t-1···i₁i ₀)₂, then the number of the matrix multiplication for the evaluation of this polynomial is at least 2·[log₂ N]-2+i_t-1. In the present communication the authors prove that for many values of N there exists an algorithm requiring a fewer number of matrix multiplications, thus disproving Lei-Nakamura´s conjecture