A stochastic roundoff error analysis for the fast Fourier transform
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The accuracy of the output of the Fast Fourier Transform is studied by estimating the expectedvalue and the variance of the accompanying linear forms in terms of the expected value and variance ofThe relative roundoff errors for the elementary operations of addition and multiplication.Abstract:
We study the accuracy of the output of the Fast Fourier Transform by estimating the expected value and the variance of the accompanying linear forms in terms of the expected value and variance of the relative roundoff errors for the elementary operations of addition and multiplication. We compare the results with the corresponding ones for the direct algorithm for the Discrete Fourier Transform, and we give indications of the relative performances when different rounding schemes are used. We also present the results of numerical experiments run to test the theoretical bounds and discuss their significance.read more
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Computing Fourier Transforms and Convolutions on the 2-Sphere
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FFTs for the 2-Sphere-Improvements and Variations
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The Theory of Linear Models and Multivariate Analysis
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Accuracy of the Discrete Fourier Transform and the Fast Fourier Transform
TL;DR: Fast Fourier transform (FFT)-based computations can be far more accurate than the slow transforms suggest, but these results depend critically on the accuracy of the FFT software employed, which should generally be considered suspect.
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Discrete weighted transforms and large-integer arithmetic
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TL;DR: The concept of Discrete Weighted Transforms (DWTs) are introduced which substantially improve the speed of multiplication by obviating costly zero-padding of digits.
References
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An algorithm for the machine calculation of complex Fourier series
J.W. Cooley,John W. Tukey +1 more
TL;DR: Good generalized these methods and gave elegant algorithms for which one class of applications is the calculation of Fourier series, applicable to certain problems in which one must multiply an N-vector by an N X N matrix which can be factored into m sparse matrices.