A Stochastic Roundoff Error Analysis for the Fast Fourier Transform

Abstract

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.

Discrete Weighted Transforms and Large-Integer Arithmetic

“…The problem of bounding convolution errors is extremely difficult. Though some interesting general theorems are known on the subject of FFT errors [3], the general results do not always give a fair picture of the errors obtained in practice. One problem is the fact of more than one source of error; for example, one suffers from both roundoff error and errors in the representations of sin and cos.…”

A Stochastic Roundoff Error Analysis for the Convolution

“…In the course of the rounding error analysis of the circular convolution we utilize the results of a similar type of analysis for the Radix-2 fast Fourier transform [2]. The bounds for the expected value and variance of the contribution to the linear approximation of the absolute roundoff error coming from additions and multiplications for the Radix-2 fast Fourier transform are functions of the expected values and variances of the relative rounding errors for addition and multiplication, pa , a2, pm , o^ , respectively.…”

“…In computation, The bounds for the entries of the covariance matrix of the global error for the particular Radix-2 FFT considered here can be found by utilizing the results of [2]. In particular, since coyFka, = o2\Y^r--m+iZ2Wk^x.-.W/ + WlZ2_xW/ + zA , where Z2 is the n x n diagonal matrix such that…”

“…Owing to the dimensions of the vectors involved and to the need for repeated computations, the direct evaluation of the convolution product is usually prohibitively expensive [8]. While some studies of the rounding error for the fast Fourier transform can be found in the literature (see [2,6,7,9,12,14]), the issue of the numerical stability of circular convolution is only briefly addressed in [6].…”