Polynomial arithmetic

Abstract: Arb is a C library for arbitrary-precision interval arithmetic using the midpoint-radius representation, also known as ball arithmetic. It supports real and complex numbers, polynomials, power series, matrices, and evaluation of many special functions. The core number types are designed for versatility and speed in a range of scenarios, allowing performance that is competitive with non-interval arbitrary-precision types such as MPFR and MPC floating-point numbers. We discuss the low-level number representation, strategies for precision and error bounds, and the implementation of efficient polynomial arithmetic with interval coefficients.

Abstract: As integrated circuit technology plumbs ever greater depths in the scaling of feature sizes, maintaining the paradigm of deterministic Boolean computation is increasingly challenging. Indeed, mounting concerns over noise and uncertainty in signal values motivate a new approach: the design of stochastic logic, that is to say, digital circuitry that processes signals probabilistically, and so can cope with errors and uncertainty. In this paper, we present a general methodology for synthesizing stochastic logic for the computation of polynomial arithmetic functions, a category that is important for applications such as digital signal processing. The method is based on converting polynomials into a particular mathematical form --- Bernstein polynomials --- and then implementing the computation with stochastic logic. The resulting logic processes serial or parallel streams that are random at the bit level. In the aggregate, the computation becomes accurate, since the results depend only on the precision of the statistics. Experiments show that our method produces circuits that are highly tolerant of errors in the input stream, while the area-delay product of the circuit is comparable to that of deterministic implementations.

Abstract: Publisher Summary This chapter discusses the numerical condition related to polynomials Polynomials permeate much of classical numerical analysis, either in the role of approximators, or as gauge functions for a variety of numerical methods, or in the role of characteristic polynomials of one kind or another It seems appropriate, therefore, to study some of their basic properties as they relate to computation It is found that one particular aspect of polynomials, namely, the extent to which they, or quantities related to them, are sensitive to small perturbations Condition numbers, such as those proposed, cannot be expected to do more than convey general guidelines as to the susceptibility of the respective maps to small changes in their domains By their very definition, they reflect worst case situations and, therefore, are inherently conservative measures It is important to note that exponential growth of the condition is also observed for piecewise polynomial functions, if represented in terms of normalized B-splines The basis consisting of the Lagrange polynomials for Chebyshev nodes, therefore, is optimally conditioned among all Lagrangian bases, and indeed among all polynomial bases, in the sense of attaining the optimal growth rate