Accumulation Of Round-off Errors Research Articles

Tetrahedral finite element matrix primitives

Program obtainable from: CPC Program Library, Queen’s element matrices for any such operator may be constructed University of Belfast, N. Ireland (see application form in this from three matrix primitives. The results are of use in threeissue) dimensional potential and flow problems, electromagnetics of plasnias, diffusion problems and elsewhere. Computer: Any PDP-1 1 for demonstration; any machine with ANSI Fortran compiler for subroutine package Method of solution The primitives are constructed for any specified order N of Operating system: Demonstration programs run under RT-1 1 polynomial approximation, by computing the standard tetrasystem. Subroutine package is system-independent hedron interpolation polynomials, then differentiating and integrating them as required. Differentiations are performed Programming language: Fortran IV analytically, while integrations use Newton—Cotes quadrature of sufficiently high order to g~.saranteezero discretization High speed storage required: approx. 25 Kwords error. The precision level attainable on the machine used is checked by the program at run-time. No. of bits in a word: 16 for demonstration programs. Subroutine package resets itself at run-time to the necessary preRestrictions on thecomplexity of theproblem cision level Matrices for interpolation ordersbeyond 7 or 8 may suffer from round-off error accumulation. Overlay structure: none Typical running time No. of magnetic tapesrequired: none On PDP-1 1/03 with KEy-h, about two hours for matrices of order 4. On IBM 370/165, about 8 mm. Otherperipherals used: none for subroutine package. Standard interactive terminal (VDU) for demonstration programs. Unusual featuresof theprogram The available machine precision is determined by the program No. of cards in combined program and test deck: 930 itself at run-time.

Fast progressive reconstruction of a transformed image (Corresp.)

Fast methods for performing progressive reconstruction of Fourier and Hadamard transformed images have been developed. Reconstruction of an <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N \times N</tex> point transformed image can be evaluated in order <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N^{2} \log_{2} N</tex> instructions. Accumulation of round-off errors due to iteration is reduced by the factor <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(\log_{2} N + 1) / N^{2}</tex> , compared with direct evaluation of the inverse transform.

An error analysis for 2-D block implemented digital filters

This correspondence considers error analysis of block-implemented 2-D digital filters. Expressions for error bound and mean-square error for roundoff error accumulation are derived using fixed-point arithmetic, and compared with the results obtained using ordinary 2-D difference equations.

A Logarithm Algorithm for a Five-Function Calculator

calculation yields 0.3010294. Computer simulations of this process indicate that our algorithm almost always underestimates the desired logarithm. The process appears to be accurate to about two decimal places fewer than the calculator will display. This error is due to the accumulation of round-off errors from the repeated divisions as well as the errors in our decimal approximations for the roots of ten. We can also construct a related algorithm to find antilogarithms. Actually, antiloglo(x) is lOx so we use the decimal digits of x as counters for our roots of ten in order to construct lOX by repeated multiplication. This method of calculating logarithms is not new; F. E. Hohn was aware of it in 1943 [2]. At that time he was correct in asserting that the method was not practical for pencil and paper computation; however, our pocket calculators have changed the situation. Now almost everyone has at least a simple four-function calculator, and many people are able to enjoy experimenting and playing with numbers. As noted at the beginning of the article, Hewlett-Packard calculators use a method similar to ours to calculate their logarithms. However, the H.P. algorithm is oriented toward machines while our method is oriented toward human beings. The differences are quite striking, as you will find by reading Egbert's article.

Development of mixing and isotropy in inviscid homogeneous turbulence

We have investigated a sequence of dynamical systems corresponding to spherical truncations of the incompressible three-dimensional Navier-Stokes equations in Fourier space. For lower-order truncated systems up to the spherical truncation of wavenumber radius 4, it is concluded that the inviscid Navier-Stokes system will develop mixing (and a fortiori ergodicity) on the constant energy-helicity surface, and also isotropy of the covariance spectral tensor. This conclusion is, however, drawn not directly from the mixing definition but from the observation that one cannot evolve the trajectory numerically much beyond several characteristic corre- lation times of the smallest eddy owing to the accumulation of round-off errors. The limited evolution time is a manifestation of trajectory instability (exponential orbit separation) which underlies not only mixing, but also the stronger dynamical charac- terization of positive Kolmogorov entropy (K-system).

Numerical evaluation of integrals containing a spherical Bessel function by product integration

A method is developed for numerical evaluation of integrals with k-integration range from 0 to ∞ that contain a spherical Bessel function jl(kr) explicitly. The required quadrature weights are easily calculated and the rate of convergence is rapid—only a relatively small number of quadrature points is needed—for an accurate evaluation even when r is large. The quadrature rule is obtained by the method of product integration. With the abscissas chosen to be those of Clenshaw–Curtis and the Chebyshev polynomials as the interpolating polynomials, quadrature weights are obtained that depend on the spherical Bessel function. An inhomogenous recurrence relation is derived from which the weights can be calculated without accumulation of roundoff error. The procedure is summarized as an easily implementable algorithm. Questions of convergence are discussed and the rate of convergence demonstrated for several test integrals. Alternative procedures are given for generating the integration weights and an error analysis of the method is presented.

Error analysis of recursive digital filters implemented with logarithmic number systems

This paper presents an analysis of roundoff error accumulation in the direct realization of a logarithmic-arithmetic digital filter and formulas for computing the ratio of the rms value of the roundoff error and that of the output signal of the digital filter. It is shown that for the case of stochastic input, the theoretical error to signal ratio is less using logarithmic number systems as compared to using floating-point number systems. The theoretical results of the roundoff error analysis were verified by simulation experiments. The experimental results demonstrate that a logarithmic number system gives filtering performance superior to that of a floating-point system of equivalent word length and range, and that the limitation in implementing digital filters with logarithmic number systems is a function of the range of the numbers which may be represented.

Roundoff accumulation at weak horizontal positions

This paper examines computer roundoff error accumulation in the least-squares adjustment of horizontal geodetic networks containing positions of very poor geometry. Roundoff error accumulates at the weak unknowns but does not disturb the solution of well-determined positions. Numerical results are given.

Partial double precision with residue retention: a new approach to solving differential equations on microprocessors

This paper proposes the use of partial double and triple precision with residue retention as a new arithmetic structure for solving differential equations on microprocessors. It is shown that a residue register, which is the distinguishing feature of the digital differential analyser, improves solution accuracy considerably by suppressing the accumulation of roundoff error, which is generally a problem on short-wordlength machines. Both theory and simulation reveal that by employing partial triple precision with residue retention, better than double-precision accuracy may be achieved with only a single-precision multiplication, whereas, without residue retention, single-precision accuracy only is possible.

Partial double precision with residue retention: a new approach to solving differential equations on microprocessors

This paper proposes the use of partial double and triple precision with residue retention as a new arithmetic structure for solving differential equations on microprocessors. It is shown that a residue register, which is the distinguishing feature of the digital differential analyser, improves solution accuracy considerably by suppressing the accumulation of roundoff error, which is generally a problem on short-wordlength machines. Both theory and simulation reveal that by employing partial triple precision with residue retention, better than double-precision accuracy may be achieved with only a single-precision multiplication, whereas, without residue retention, single-precision accuracy only is possible.

Accumulation of roundoff errors in floating point FFT

A statistical model for roundoff error is used to predict the output noise to signal ratio of the two common FFT algorithms, the decimation in time and the decimation in frequency algorithms. A unified approach is used to obtain the error in both algorithms. Results for radix 2 and for arbitrary radix are presented. Multidimensional FFT is also discussed.

Recursive realization of finite impulse filters using finite field arithmetic

Recursive filter design techniques are described and developed for finite impulse filters using finite field arithmetic. The finite fields considered have the form GF(q^{2}) , the Galois field of q^{2} elements, and are analogous to the field of complex numbers when q is a prime such that (-1) is not a quadratic residue. These filters can be designed to yield either a desired finite impulse or finite frequency response function. This filtering technique has other possible applications, including the encoding or decoding of information and signal design. Infinite signal trains can be decomposed naturally into orthogonal sequences which may be useful in the encoding and decoding process and may provide another approach to convolutional coding. Since the recursive filters developed here do not have the accumulation of round-off or truncation error that one might expect in recursive computations, such filters are noise-free transducers in the sense of Shannon.

Heuristic optimization of the cascade realization of fixed-point digital filters

In the cascade realization of fixed-point digital filters under dynamic range constraints, the output noise due to accumulation of roundoff errors is highly dependent upon the order of the sections. For recursive filters it also depends on the pole-zero pairing that forms the individual second-order sections. The output noise may vary over several orders of magnitude for different cascade realizations of high-order filters. Therefore an optimization procedure to find a good ordering and pairing is very desirable. We propose a heuristic optimization procedure for finding a "near optimal" solution. The procedure is completely automatic and does not require any knowledgeable judgment. The number of function evaluations required for a filter of N-cascaded sections is proportional to N <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> . By using this procedure, "near optimal" solutions have been found for a 22nd-order recursive filter in 23 s, and for a 55th-order nonrecursive filter in 37.5 s, on an IBM 360-91 computer.

Two-Dimensional Digital Filtering and its Error Analysis

The concept of a two-dimensional recursive digital filter is introduced and block diagram representations are given. Expressions for error bounds and mean-square, error for roundoff error accumulation are derived. Other sources of error are described. Simulation of the implementation of digital filters is discussed, and corresponding to this simulation, calculation of error is performed. Numerical examples are given. The derived analytic results are shown to be in good agreement with the simulation results.

On the Importance of Precision for Short-Range Forecasting and Climate Simulation

Abstract This paper presents results of experiments designed to test the importance of arithmetic precision in short-term forecasts and long-term climate simulations with the NCAR global circulation model. It is expected that the next-generation computers will have a sizable speed gain when using lower-precision arithmetic. To determine how precision affects the model results, we compare several short- and long-term experiments using 48-bit mantissa arithmetic (normal for the CDC 6600 and 7600 computers) with corresponding experiments using 24- and 21-bit mantissa arithmetic. The errors due to the lower precision are much smaller than typical observational errors. In addition, it appears that in the short-term experiments the rapid error growth of the model dominates the round-off error accumulation resulting from the lower-precision arithmetic. Therefore, the lower precision used by the next-generation computers should not have a detrimental effect on short-term forecast accuracy. The long-term climate s...

Complex gamma function with error control

An algorithm to compute the gamma function and the loggamma function of a complex variable is presented. The standard algorithm is modified in several respects to insure the continuity of the function value and to reduce accumulation of round-off errors. In addition to computation of function values, this algorithm includes an object-time estimation of round-off errors. Experimental data with regard to the effectiveness of this error control are presented. A Fortran program for the algorithm appears in the algorithms section of this issue.

On the structure of error estimates for finite-difference methods

In this paper we study in an abstract setting the structure of estimates for the global (accumulated) error in semilinear finite-difference methods. We derive error estimates, which are the most refined ones (in a sense specified precisely in this paper) that are possible for the difference methods considered. Applications and (numerical) examples are presented in the following fields: 1. Numerical solution of ordinary as well as partial differential equations with prescribed initial or boundary values. 2. Accumulation of local round-off error as well as of local discretization error. 3. The problem of fixing which methods out of a given class of finite-difference methods are "most stable". 4. The construction of finite-difference methods which are convergent but not consistent with respect to a given differential equation.

Error analysis of digital filter employing floating-point arithmetic

The error accumulation in a digital filter employing floatingpoint arithmetic is studied. Sources of errors are considered to originate from 1) roundoff in the arithmetic operations, 2) quantization of both the input and the initial states, and 3) effects of coefficient inaccuracies. Existing methods of analysis for the case of deterministic input (case A) and the case of stochastic input (case B) are reviewed and presented in the context of the canonical realization of a digital filter. A unified treatment of all these sources of errors is indicated, and results complementary to the existing results on the direct realization are thus obtained. A comparison is made of the roundoff error accumulation of the canonical realization with that of the direct realization, with the general conclusion that both realizations of any filter exhibit the same roundoff error distortion for any arbitrary wide-sense-stationary stochastic input.

Accumulation of Round-Off Error in Fast Fourier Transforms

The fast Fourier transform (FFT) is an algorithm to compute the discrete Fourier coefficients with a substantial time saving over conventional methods. The finite word length used in the computer causes an error in computing the Fourier coefficients. This paper derives explicit expressions for the mean square error in the FFT when floating-point arithmetics are used. Upper and lower bounds for the total relative mean square error are given. The theoretical results are in good agreement with the actual error observed by taking the FFT of data sequences.

Effect of Limited Wordlength on Satellite Prediction Accuracy

Round-off error accumulation in numerical integration of the equations of motion in an inverse square field has been studied for approximately circular orbits. The second order equations of motion in Cowell formulation were assumed to be integrated with a one-step method of arbitrary order after reduction to a first order system. Analytical expressions for the first two statistical moments of the accumulated error in one position component have been derived via the adjoint method for particular classes of fixed and floating point arithmetics. The error mean increases linearily or quadratically with time dependent on the mechanization of the ADD-instruction regardless of number system (sign-magnitude or two’s complement) and method of scaling (fixed or floating point). The error variance increases with third power of time.