The art of computer programming, volume 2 (3rd ed.): seminumerical algorithms
The art of computer programming, volume 2 (3rd ed.): seminumerical algorithms
On Euclid's Algorithm and the Computation of Polynomial Greatest Common Divisors
Journal of the ACM (JACM)
The Calculation of Multivariate Polynomial Resultants
Journal of the ACM (JACM)
Computation of powers of multivariate polynomialsover the integers
Journal of Computer and System Sciences
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Suppose we are given a polynomial P(x"1,...,x"r) in r=1 variables, let m bound the degree of P in all variables x"i, 1@?i@?r, and we wish to raise P to the nth power, n1. In a recent paper which compared the iterative versus the binary method it was shown that their respective computing times were O(m^2^rn^r^+^1) versus O(mn)^2^r) when using single precision arithmetic. In this paper a new algorithm is given whose computing time is shown to be O((mn)^r^+^1). Also if we allow for polynomials with multiprecision integer coefficients, the new algorithm presented here will be faster by a factor of m^r^-^1n^r over the binary method and faster by a factor of m^r^-^1 over the iterative method. Extensive empirical studies of all three methods show that this new algorithm will be superior for polynomials of even relatively small degree, thus guaranteeing a practical as well as a useful result.