On the Number of Nonscalar Multiplications Necessary to Evaluate Polynomials
We present algorithms which use only $O(\sqrt n )$ nonscalar multiplications (i.e. multiplications involving “x” on both sides) to evaluate polynomials of degree n, and proofs that at least $\sqrt n $ are required. These results have practical application in the evaluation of matrix polynomials with scalar coefficients, since the “matrix $ \times $ matrix” multiplications are relatively expensive, and also in determining how many multiplications are needed for polynomials with rational coefficients, since multiplications by integers can in principle be replaced by several additions.
 , The evaluation of polynomials, Numer. Math., 6 (1964), 17–21 10.1007/BF01386049 MR0164440 0116.09002
 , Evaluation of polynomials and evaluation of rational functions, Bull. Amer. Math. Soc., 61 (1955), 163–
 , Methods of computing values of polynomials, Uspekhi Mat. Nauk, 21 (1966), 105–136, English transl., Russian Math. Surveys., 21(1966) 0173.17802
 , Fast evaluation of polynomials by rational preparation, Res. Rep., RC 3645, IBM, Yorktown Heights, N.Y., 1971
 , Modern Algebra, Frederick Ungar, New York, 1949 0033.10102
 , On the number of multiplications necessary to compute certain functions, Comm. Pure Appl. Math., 23 (1970), 165–179 MR0260150 0191.15804