Nonlinear optimization: complexity issues
Nonlinear optimization: complexity issues
The art of computer programming, volume 2 (3rd ed.): seminumerical algorithms
The art of computer programming, volume 2 (3rd ed.): seminumerical algorithms
On the Multivariate Horner Scheme
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Program transformation for numerical precision
Proceedings of the 2009 ACM SIGPLAN workshop on Partial evaluation and program manipulation
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Formal Methods in System Design
Parallel computation of the minimal elements of a poset
Proceedings of the 4th International Workshop on Parallel and Symbolic Computation
Efficient evaluation of large polynomials
ICMS'10 Proceedings of the Third international congress conference on Mathematical software
A computer algebra user interface manifesto
ACM Communications in Computer Algebra
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For univariate polynomials f(x1), Horner's scheme provides the fastest way to compute a value. For multivariate polynomials, several different version of Horner's scheme are possible; it is not clear which of them is optimal. In this paper, we propose a greedy algorithm, which it is hoped will lead to good computation times.The univariate Horner scheme has another advantage: if the value x1 is known with uncertainty, and we are interested in the resulting uncertainty in f(x1), then Horner scheme leads to a better estimate for this uncertainty that many other ways of computing f(x1). The second greedy algorithm that we propose tries to find the multivariate Horner scheme that leads to the best estimate for the uncertainty in f(x1,...,xn).