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Register a new userSuper-high-efficiency approximate calculation of series sum and discrete Fourier transform
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Authors
Xin-Zhong Yan
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We present a super-high-efficiency approximate computing scheme for series sum and discrete Fourier transform. The summation of a series sum or a discrete Fourier transform is approximated by summing over part of the terms multiplied by corresponding weights. The calculation is valid for the function under the transform being piecewise smooth in the continuum variable. The scheme reduces significantly the requirement for computer memory storage and enhances the numerical computation efficiency without losing accuracy. By comparing with the known results of examples, we show the accuracy and the efficiency of the scheme. The efficiency can be higher than $10^6$ for the examples.
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In this work, we introduce a definition of the Discrete Fourier Transform (DFT) on Euclidean lattices in $R^n$, that generalizes the $n$-th fold DFT of the integer lattice $Z^n$ to arbitrary lattices. This definition is not applicable for every lattice, but can be defined on lattices known as Systematic Normal Form (SysNF) introduced in cite{ES16}. Systematic Normal Form lattices are sets of integer vectors that satisfy a single homogeneous modular equation, which itself satisfies a certain number-theoretic property. Such lattices form a dense set in the space of $n$-dimensional lattices, and can be used to approximate efficiently any lattice. This implies that for every lattice $L$ a DFT can be computed efficiently on a lattice near $L$. Our proof of the statement above uses arguments from quantum computing, and as an application of our definition we show a quantum algorithm for sampling from discrete distributions on lattices, that extends our ability to sample efficiently from the discrete Gaussian distribution cite{GPV08} to any distribution that is sufficiently smooth. We conjecture that studying the eigenvectors of the newly-defined lattice DFT may provide new insights into the structure of lattices, especially regarding hard computational problems, like the shortest vector problem.
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