Polynomial and matrix computations (vol. 1): fundamental algorithms
Polynomial and matrix computations (vol. 1): fundamental algorithms
Parallel complexity of tridiagonal symmetric Eigenvalue problem
SODA '91 Proceedings of the second annual ACM-SIAM symposium on Discrete algorithms
Supereffective slow-down of parallel computations
SPAA '92 Proceedings of the fourth annual ACM symposium on Parallel algorithms and architectures
Analysis of Asynchronous Polynomial Root Finding Methods on a Distributed Memory Multicomputer
IEEE Transactions on Parallel and Distributed Systems
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Let lg stand for log2, lg(0)n = n, lg(h)n = lg lg(h-1)n, h = 1, …, lg*n, lg*n = min{h,lg(h)n ≤ 1}. Given natural N, h, 1 ≤ h ≤ lg*N, and polynomial p(x), p(O) ≠ O, we compute r(x) = p(x)-1 mod xN for the cost OA(t, P), t = h lg N, P = (N/h)lg(h)N, under the PRAM arithmetic model, that is, we need O(t) steps and O(P) processors (with t and P as above), provided DFT(m) costs OA(lg m, m). For h = lg* N, the cost bounds turn into OA(lg N lg*N, N/lg*N). The results improve [G] and apply to various related computations [BP].Basis observation. For i = 0, 1, …,vi(x) = r(x) mod xk2i, deg(vi(x)) ≤ (i + 1)k2i - 2i+1 + 1 if vo(x) = r(x) mod xk; pi(x) p(x) mod xk2i, vi+1(x) = (2 - vi(x)pi(x))vi(x). Let 4 ≤ &phgr; ≤ n, lg(h)n, j = lg lg &phgr; be integers. Algorithm. Input: n, &phgr;, the coefficients of p(x) and r(x) mod xk, k = n/&phgr;. Output: r(x) mod x&ggr;, &ggr; = n/lg2&phgr;. 1. Set vo(x) = r(x) mod xk; compute vo(x), pi(x) = p(x) mod xk2i, i = 1,…, j - 1, j = lg &phgr; - 2lg lg &phgr;, at the &ggr;-th roots of 1 (k DFTs(&ggr;)). 2. Apply (1) pointwise, for i = 1, …, j - 1, to evaluate vj(x) at the &ggr;-th roots of 1. 3. Compute the coefficients of vj(x) (DFT(&ggr;)). Basis observation implies correctness of algorithm 1, whose cost is OA(logn,n).Apply the algorithm for &phgr; = lg(i), i = 0,1, … , J, J ≤ 2h to compute the first ⌈n/lg(h)n⌉ coefficients of r(x). If lg(h)n j = lg(h+1)n, to compute r(x) mod xn for the additional cost OA(lg n,n(lgh+1)n)2) ≤ OA(h lg n, n/h lg(h)n).Finally apply h steps of Sieveking-Kung's algorithm, starting with r(x) mod xn, and arrive at r(x) mod xN, for N = n2h. These steps cost OA(h lg N, N/h). Acknowledgement. NSF CCR-8805782. Reference [BP] Bini, D., and V. Pan, On polynomial and matrix computations, to appear. [G] Georgiev, R. E., Inversion of triangular Toeplitz matrices by using the fast Fourier transform, J. New Gener. Comput. Syst., vol. 2, 3, pp. 247-256, 1989.