For purposes of actual evaluation, mathematical functions f are commonly replaced by approximation polynomials p. Examples include floating-point implementations of elementary functions, quadrature or more theoretical proof work involving transcendental functions. Replacing f by p induces a

A bound on the error introduced by truncating a quantum addition is given. This bound shows that only a few controlled rotation gates will be necessary to get a reliable computation.

Let F and A be two linear function spaces defined on some domain \Omega, and let |*| be a vector semi-norm for the space A + F. We consider here the question of how well A approximates F in the sense of the metric |*|. Global error measures are insuficiently informative when the space A

of the exact error of an FMA (by giving more general conditions and providing a formal proof). Then, we present a new algorithm that computes an approximation to the error of an FMA, and provide error bounds and a formal proof for that algorithm.

the approximation performance of a special FIR prefilter. The results show that the convergent rate of the prefiltered projection is the same as that of the orthogonal projection. In addition, for bandlimited signals, the quantitative estimates of the upper bounds of the three types of errors are obtained

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variety of target functions, yet not need an excessive number of examples to ensure that a learning algorithm can choose a near-optimal approximation to the target function. We show that these two objectives are incompatible, in the sense that as the approximation error of the hypothesis class decreases

these tradeoffs. One of the objectives of this paper is to suggest that there is the potential for developing a more formal approach, including utilizing current research in Computer Science on Approximate Processing and one of its central concepts, Incremental Refinement. Toward this end, we first summarize a

. A new sequential data assimilation method is discussed. It is based on forecasting the error statistics using Monte Carlo methods, a better alternative than solving the traditional and computationally extremely demanding approximate error covariance equation used in the extended Kalman filter

simplification algorithm which can rapidly produce high quality approximations of polygonal models. The algorithm uses iterative contractions of vertex pairs to simplify models and maintains surface error approximations using quadric matrices. By contracting arbitrary vertex pairs (not just edges), our algorithm