A limited disproof to the conjecture of evaluating the matrix polynomial I+A+A/sup 2/+/spl middot//spl middot//spl middot/+A/sup N/spl minus/1/

The problem of evaluating matrix polynomial I+A+A/sup 2/+/spl middot//spl middot//spl middot/+A/sup N/spl minus/1/, has been considered. The proposed algorithms require at most 3/spl middot//spl lsqb/log/sub 2/ N/spl rsqb/ and 2/spl middot//spl lsqb/log/sub 2/ N/spl rsqb//spl minus/1 matrix multiplications, respectively. If the binary representation of N is (i/sub t/i/sub t/spl minus/1//spl middot//spl middot//spl middot/i/sub 1/i/sub 0/)/sub 2/, then the number of the matrix multiplication for the evaluation of this polynomial is at least 2/spl middot//spl lsqb/log/sub 2/ N/spl rsqb//spl minus/2+i/sub t/spl minus/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.<
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