On the Euclidean scheme for polynomials having interlaced real zeros

If p(z) and q(z) are polynomials of degree n and n 1, respectively, having reai interlaced zeros, all the coefficients of the polynomials generated by the Euclidean scheme applied to p(z) and q(x) can be computed by using 0(log3 n) parallel arithmetic steps and n* processors. The number of arithmetic operations is within a polylogarithmic factor from the straightforward lower bound. This result can be applied to the solution of the inverse eigenvalue problem for Jacobi matrices and can be extended to any pair of polynomials provided that the Euclidean scheme is carried out in n steps. The algorithm is obtained by reducing the Euclidean scheme computation to the evaluation of the Cholesky factorization of a positive definite Hankel matrix. The same reduction can be used to compute the GCD of two general polynomials by evaluating a vector of minimum “degree” in the kernel of a Hankel matrix. This new characterization of the GCD yields an algorithm for its computation in O(log* n) parallel steps using n2 processors.