Fast Fourier Transforms

Gentleman, W. Morven; Sande, Gordon

doi:10.1145/1464291.1464352

Cited by 326 publications

(95 citation statements)

References 5 publications

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“…The pure crystal density of states go(A) is evaluated using the Monte Carlo method to the desired accuracy. Use is then made of a numerical fast Fourier transform procedure (Gentleman and Sande 1966) to evaluate the Fourier coefficients an of go(A). This enables the real part R(A) to be readily calculated since (Mahanty 1966)…”

Section: Impure Crystal Density Of Statesmentioning

confidence: 99%

Extended Theory of Magnon Sideband Shapes. I. Impure Three-dimensional Ferromagnets

Richardson¹

1976

Aust. J. Phys.

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A calculation of magnon sideband lineshapes is given which uses a simple phenomenological Hamiltonian allowing of an exact solution. The perturbation Hamiltonian has an improved form compared' with that used by Richardson (1974). We consider the effect of a small concentration of substitutional spin impurities on the sideband. An example of an f.c.c. ferromagnetic crystal is studied in detail, and the conditions for the appearance of local modes due to the impurity are discussed. The possibility of resonance modes occurring within the absmption band is considered but, for the f.c.c. crystal studied here, this appears to be unlikely to within the accuracy of the numerical calculations undertaken.

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Section: Impure Crystal Density Of Statesmentioning

confidence: 99%

Extended Theory of Magnon Sideband Shapes. I. Impure Three-dimensional Ferromagnets

Richardson¹

1976

Aust. J. Phys.

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“…where Gp(x) is a well-known approximation (Gentleman and Sande [8]) to g"(x), namely m/2 (2.2) G"ix) = 2 J2" («'r;1' cos 2irrx + b\mvu sin 2-wrx).…”

Section: The Discretisation Error F(x) -/(X)mentioning

confidence: 99%

Computational Techniques Based on the Lanczos Representation

Lyness¹

1974

Mathematics of Computation

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Abstract. In his book Discourse on Fourier Series, Lanczos deals in some detail with representations of f(x) of the type f(x) = hv-,(x) + g/x) where Ap_i(^:) is a polynomial of degree p -1 and gp(x) has the property that its full range Fourier coefficients converge at the rate r-*.In Part I, some properties of Ap(x) and of the series {hAx)\,oe are described. These properties are used here to provide criteria for the convergence or divergence of the Euler-Maclaurin series, in the case when f(x) is an analytic function. The similarities and differences between this series and the Lidstone and other two-point series are briefly mentioned.In Part II, the Lanczos representation is employed to derive an approximate representation This representation is a powerful one. The drawback is that it requires derivatives. Most of Part II is devoted to the effect of using only approximate derivatives. It is shown that when these are successively less accurate with increasing order (the sort of behaviour encountered using finite difference formula), then the representation is still powerful and reliable. In a computational context the only penalty for using inaccurate derivatives is that a larger value of m may-or may not-be required to attain a specific accuracy.

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“…The required multiplication count would then bê (Ei^i ^0-Third, the above operation counts do not take into account the fact that many of the complex exponentials are ±1 or ±V -1. And fourth, (2.2) holds for the Sande-Tukey algorithm as well as the Cooley-Tukey algorithm but with different diagonal matrices, Z>¡ (see [1]). …”

Section: Introductionmentioning

confidence: 99%

“…By comparing upper bounds, Gentleman and Sande [1] show that accumulated floating-point roundoff error is significantly less when one uses the FFT than when one computes (1.1) directly. In [2], Welch derives approximate upper and lower bounds on the RMS error in a fixed-point power-of-two algorithm.…”

Section: Introductionmentioning

confidence: 99%

“…are diagonal matrices of complex exponential elements (called rotation factors by Singleton [13], twiddle factors by Gentleman and Sande [1]), and B,(l = 1, 2, • • • , M) are block-diagonal matrices whose blocks on the diagonal are identical square submatrices, each the matrix of a complex Fourier transform of dimension N¡. In this case, the required number of operations is reduced to N(M -1 + 22f-i Ni) complex multiplications and N(22î'-i (N¡ -1)) complex additions.…”

Section: Introductionmentioning

confidence: 99%

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Roundoff Error Analysis of the Fast Fourier Transform

Ramos¹

1971

Mathematics of Computation

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Abstract. This paper presents an analysis of roundoff errors occurring in the floatingpoint computation of the fast Fourier transform. Upper bounds are derived for the ratios of the root-mean-square (RMS) and maximum roundoff errors in the output data to the RMS value of the output data for both single and multidimensional transformations. These bounds are compared experimentally with actual roundoff errors.

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Fast Fourier Transforms

Cited by 326 publications

References 5 publications

Extended Theory of Magnon Sideband Shapes. I. Impure Three-dimensional Ferromagnets

Extended Theory of Magnon Sideband Shapes. I. Impure Three-dimensional Ferromagnets

Computational Techniques Based on the Lanczos Representation

Roundoff Error Analysis of the Fast Fourier Transform

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