Approximate Complex Polynomial Evaluation in Near Constant Work Per Point

Abstract

Given the n complex coefficients of a degree n-1 complex polynomial, we wish to evaluate the polynomial at a large number $m \ge n$ of points on the complex plane. This problem is required by many algebraic computations and so is considered in most basic algorithm texts (e.g., [A. V. Aho, J. E. Hopcroft, and J. D. Ullman, The Design and Analysis of Computer Algorithms, Addison-Wesley, 1974]). We assume an arithmetic model of computation, where on each step we can execute an arithmetic operation, which is computed exactly. All previous exact algorithms [C. M. Fiduccia, Proceeding} 4th Annual ACM Symposium on Theory of Computing, 1972, pp. 88--93; H. T. Kung, Fast Evaluation and Interpolation, Carnegie-Mellon, 1973; A. B. Borodin and I. Munro, The Computational Complexity of Algebraic and Numerical Problems, American Elsevier, 1975; V. Pan, A. Sadikou, E. Landowne, and O. Tiga, Comput. Math. Appl., 25 (1993), pp. 25--30] cost at least work $\Omega(\log^2 n)$ per point, and previously, there were no known approximation algorithms for complex polynomial evaluation within the unit circle with work bounds better than the fastest known exact algorithms. There are known approximation algorithms [V. Rokhlin, J. Complexity, 4 (1988), pp. 12--32; V. Y. Pan, J. H. Reif, and S. R. Tate, in Proceedings 32nd Annual IEEE Symposium on Foundations of Computer Science, 1992, pp. 703--713] for polynomial evaluation at real points, but these do not extend to evaluation at general points on the complex plane. We provide approximation algorithms for complex polynomial evaluation that cost, in many cases, near constant amortized work per point. Let $k = \log(|P|/\epsilon)$, where |P| is the sum of the moduli of the coefficients of the input polynomial P(z). Let {\it ${\tilde{P}}(z_j)$ be an $\epsilon$-approx of $P(z)$} if $\epsilon$ upper bounds the modulus of the error of the approximation ${\tilde{P}}(z_j)$ at each evaluation point zj, that is, $|P(z_j)-{\tilde{P}}(z_j)| \le \epsilon;$ note that $\epsilon$ is an absolute error bound rather than a relative error bound. In many applications (particularly in signal processing) the evaluation points zj are fixed and require only polylogarithmic $k = \log(|P|/\epsilon) = O(\log^{O(1)} n)$; for these cases we get a surprising reduction in work by use of approximation algorithms, as compared to the fastest known exact algorithms. We $\epsilon$-approx complex degree n-1 polynomial evaluation at $m \ge n\log n/\log^2 k $ fixed points on or within the unit disk in the complex plane in amortized work O(log2 k) per point, which is O(log2 log n) for polylogarithmic k. If the m points are not fixed, then we have increased amortized work O(log2 k + log m) per point, which is O(log m) for polylogarithmic k and $m \ge n\log n/\log k,$ and is still substantially below the previous bound of $\Omega(\log^2 m)$ for known exact algorithms. We further reduce our amortized bounds for special sets of evaluation points widely used in signal processing applications. The chirp transform is equivalent to evaluating a complex degree n-1 polynomial at the chirp points, which are $\zeta^j, j = 0,\dots,m-1$, for some fixed complex number $\zeta.$ We $\epsilon$-approx complex degree $n-1$ polynomial evaluation at these $m$ chirp points, where $m \ge n \log n/\log^2 k$ and $|\zeta| \le 1$ % or (ii) $m \ge n$ and $|\zeta| \le $ a function that limits to $1$ %for %$k = o(n)$ and large $n$) in amortized work O(log k) per point, whereas the previous best bounds for exact evaluation (via the chirp transform) were $\Omega(\log m)$ per point [A. V. Aho, K. Steiglitz, and J. D. Ullman, SIAM J. Comput., 4 (1975), pp. 533--539]. Using instead a reduction to approximate real polynomial evaluation (by interpolation at the Chebyshev points), in total work O(n log k), we $\epsilon$-approx the evaluation of a degree n polynomial at the first n powers of the n'th root of unity, where $n' \ge \Omega(n^2/k), $ and $\epsilon$-approx the n-point DFT for certain inputs with descending coefficient magnitude. All of our results require polylogarithmic (that is, logO(1)n)depth with the same work bounds.We also provide a lower bound for a wide class of schemes for approximate evaluation of a degree n-1 polynomial on the unit circle; namely, we prove that if a scheme uses an approximation polynomial of degree k-1, then it can be convergent only over a small fraction O(k/n) of the unit circle. We believe this is the first lower bound of this sort proved, and the proof uses an interesting reduction to the approximation of a matrix product by a matrix of reduced rank.