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Some methods are described for the generation of fractional and negative powers of any formal series, such as Poisson series or Chebyshev series. It is shown that, with the use of the three elementary operations of addition, subtraction, and multiplication, all rational (positive and negative) powers of a series con be constructed. There are basically two approaches: the binomial theorem and the iteration methods. Both methods are described here, and the relationship between them is pointed out. Some well-known classical formulas are obtained as particular cases, and it is shown how the convergence properties of these formulas can be improved with very little additional computations. Finally, at the end of the article, some numerical experiments are described with Chebyshev series and with Fourier series. [ABSTRACT FROM AUTHOR]

An algorithm for Filon quadrature is described. Considerable attention has been devoted to an analysis of the round-off and truncation errors. The algorithm includes an automatic error control feature. [ABSTRACT FROM AUTHOR]

This article discusses the ALGOL convolution procedure based on the fast Fourier transform. The operation of convolution is used in computing autocorrelation and cross-correlation functions, in digital filtering of time series, and many other applications. Procedure Convolution computes the convolution of two real vectors of dimension. The special features of this procedure are the usual reordering of the fast Fourier transform results is avoided, and the return from frequency to time is made with a transform of dimension n/2 instead of n. The two vectors A and B are first transformed with a single complex Fourier transform of dimension n. Procedure Convolution uses-procedure REALTRAN but repeated here with revisions to improve accuracy on computers using truncated floating-point arithmetic. Procedures FFT4 and REVFFT4 are also used and perform the same computation as procedures FFT2 and REVFFT2 given for use on a system with virtual memory. The transform procedures given are organized without regard to the problem of memory overlay.

The article presents the procedures for computing the finite Fourier transform of a complex data vector which is based on the Cooley-Tukey algorithm. An overview of the complextransform and realtransform's procedures are offered. Either the complex Fourier transform or its inverse are the procedure computed by complextransform, while realtransform computes either the Fourier coefficients of a sequence of real data points or evaluates a fourier series. The topic also includes orthogonal transformation, permutations, and spectral analysis.

The article presents information about a method for computing Fourier integrals by using algorithm 418. The most commonly used formula for calculating Fourier integrals is Filon's formula. The formula is based on the approximation of the function by a quadratic equation in each double interval. In order to obtain a better approximation, the cube spline fit is used. The obtained formulas do not need the explicit calculation of the spline fit. However, in addition to the function values, at all intermediate points, the values of the first and second derivatives at the boundary points are required. However, these values are often obtained from symmetry conditions. If the derivatives at the end points are unknown, they may be calculated from a cubic spline fit, for example by using some exterior points or by using two extra interior conditions for the spline fit. It can also be noted that in certain periodic cases the terms containing the derivatives will cancel and their vales will be superfluous.

The article focuses on an algorithm related to quadrature. FSER1 evaluates the integrals using the Filon quadrature algorithm. The user may request an evaluation of C only, S only, or both C and S. FSER1 contains an automatic error-control feature which selects an integration step size on the basis of an error parameter supplied by the user. The Filon quadrature formulas, truncation error, rounding error, and automatic error control are described in a companion paper by the authors. The calling parameters for this subroutine are defined as follows. F is the name of a FUNCTION subprogram F(X), supplied by the user, which evaluates F(X) appearing in the integrand, EPS is the name for "e" appearing in inequalities. It is used in the error control portion of the algorithm. The error in the computed values of C and S is related to "e" by the inequality. The user must assign a value to EPS before calling FSER1. MAX specifies the maximum number of halvings of the step size that are allowed.

The article reports on the International Business Machine Corporation which use a technique of Fourier series analysis putting in the computer. The company said that the technique achieved a logarithmic reduction in the number of calculation made in such an analysis. The analysis which widely used to study complex wave patterns generated by earthquakes, sources of light, atoms and other sources.