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Fast computation of binomial coefficients. (English) Zbl 1460.65019
Summary: One problem that arises in computation involving large numbers is precision. In certain situations, the result might be represented by the standard data type, but arithmetic precision might be compromised when dealing with large numbers in the course to the result. Binomial coefficients are an example that suffer from this torment. In the present paper, we review numerical methods to compute the binomial coefficients: Pascal’s recursive method; prime factorization to cancel out terms; gamma function approximation; and a simplified iterative method that avoids loss in precision. Acknowledging that the binomial coefficients might be obtained by a successive convolution of a simple discrete rectangular function, we propose a different approach based on the discrete Fourier transform and another based on its fast implementation. We analyze and compare performance and precision of all of them. The proposed methods overcome the previous ones when computing all coefficients for a given level \(n\), attaining better precision for large values of \(n\).
MSC:
65D20 Computation of special functions and constants, construction of tables
65T50 Numerical methods for discrete and fast Fourier transforms
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