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The process of frequency synthesis is the step-by-step approximation to the normalized output frequency. In this paper, three different approximation algorithms are briefly discussed, namely continued fractions, Cantor products, and Engel series. Their common disadvantage is their uniqueness; it is shown that this difficulty may be overcome by application of the new general approximation theorem, but the solution can be provided only by computer. Approximation errors are generally small, sometimes smaller than 1 × 10-12. The advantage of modified Cantor products or Engel series algorithms is the possibility of standardizing the hard ware of frequency synthesizers, particularly if the IC technology and the use of off-the-shelf circuits is emphasized.

The exact solutions are obtained for a one dimensional model of a scattering and absorbing medium. The results are given for both the reflected and transmitted radiance for any arbitrary surface albedo as well as for the interior radiance. These same quantities are calculated by the matrix operator method. The relative error of the solutions is obtained by comparison with the exact solutions as well as by an error analysis of the equations. The importance of an accurate starting value for the reflection and transmission operators is shown. A fourth order Runge-Kutta method can be used to solve the differential equations satisfied by these operators in order to obtain such accurate starting values.

Cumulative probabilities from the null distribution of the Wilcoxon signed rank statistic were approximated by the normal approximation and by Edgeworth expansions to terms of order 1/N and of order 1/N 2 for selected values of the statistic for values of N from 30 through 100. Use of the usual correction for continuity is advantageous at all probability levels in the Edgeworth expansions and in the normal approximation for probability levels greater than about 0.035. The accuracy of these approximations is summarized in terms of percent relative error incurred as a function of the exact probability being approximated.

Discussed are the main aspects of operation, physics, design, and application of toroidal cores as nondestructive read-out analog memory elements. The analog memory element and switch are the basic components of the analog memory device of the open-loop type (i.e., without negative feedback). The toroidal core has been shown, both theoretically and experimentally, to exhibit linear read-write characteristics in pulse remagnetization. Diode switches are the most preferable for writing analog information. Analysis of the output and read circuits required to determine the optimal value for the read circuit resistance is given. The complex studies carried out resulted in a simple analog memory device with ± 3 percent error and the write time of 0.3μs. Ways to improve these parameters are described.

Ising systems are approximated by a generalized mean-field approximation in which only some of the interactions are replaced by their expectation values. If all the interactions are positive then it is shown that a self-consistent solution exists for the magnetization in this model and that it gives an upper bound for the true magnetization. Using this mean-field approximation, a generalization of the random-phase approximation is obtained. This is shown to give an upper bound for two-spin correlation functions.

The nonlinear response of an atom to a near-resonant light pulse is studied using a novel approximation scheme. In first order, the approximate solution reduces to the well-known rate equations. The second-order approximation contains Grischkowsky's adiabatic-following approximation. In each order, the approximate solution of the Bloch equations is presented with a closed-form expression for the error that can be used to investigate its range of validity.

The chi-square distribution is one of the most widely used probability distributions; thus many functions have been advanced as approximations to the chi-square cumulative distribution function. One measure of the quality of an approximation is the maximum absolute error of the associated error function. This paper investigates the error functions for three normal approximations to the chi-square cumulative distribution function, those offered by Fisher, Wilson-Hilferty, and Kelley. In particular, a technique is developed for each approximation to find the maximum absolute error for fixed degrees of freedom. The maximum absolute error for each approximation and certain degrees of freedom is tabled.

A very simple, closed-form approximation to circular error probable (CEP) is determined analytically for the case when the underlying error distribution is elliptical. The accuracy of the approximation is quite good over a sizeable range of major-to-minor axis ratios. Accurate numerical CEP computations are shown, but for many applications only the approximation will be required. The inverse question of what percentage of elliptically distributed points lie within a circle of given radius is also considered.

The variation in standard deviation with transmittance for a large number of elements has been expressed in terms of four component error functions, each one characteristic of one (or more) possible sources of noise associated with the measurement of transmittance (or absorbance). The magnitude of the contribution of each of the component functions has been measured quantitatively, and it is shown that the major component in nearly every case is that related to noise associated with the dynamic nature of the flame. For many elements, the smallest relative error occurs at an absorbance between 0·35 and 0·61 units; the range from 0·2 to 0·7 absorbance units (50 × to 200 × the observed analytical sensitivity) is suggested as the most precise working range for a wide variety of elements.

A new approximation formula for the critical current density is presented which is more exact than those proposed previously and allows the evaluation of the current density in the range of the initial magnetization curve. The relative error is calculated for one practical case and is plotted for the different branches of the magnetization curve.

The random phase approximation (RPA) or time-dependent Hartree–Fock approximation (TDHF) is reconsidered for the calculation of the dynamic polarizability for atoms. An integral equation which admits a simple numerical treatment is established. The asymptotic approximation for the electron propagator is tested for its applicability by means of comparisons with earlier results.

Quadratic approximation is a three-dimensional analogue of the two-dimensional Pade approximation. The advantages of employing quadratic approximation are demonstrated by several examples. With automatic computers having a relatively inexpensive square root instruction, quadratic approximation is an attractive alternative to Pade approximation if convergence of the Pade approximation is not rapid.

Optimum (in a minimax sense) linear phase FIR Hilbert transformers can be designed efficiently using a Remez optimization procedure. This paper presents useful design data on wideband Hilbert transformers with even and odd values of N, the impulse response duration (in samples) of the filter. Based on these data, the following observations can be made: (i) In the case of equal lower and upper transition regions, Hilbert transformers with odd values of N can be realized more efficiently than those with even values of N, assuming the same peak errors of approximation for both cases. This is because every other impulse response sample is exactly zero for odd values of N. (ii) The peak approximation error for Hilbert transformers with odd values of N is determined primarily by the minimum of the values of the lower and upper transition widths. (iii) The peak approximation error for Hilbert transformers with even values of N is determined primarily by the lower transition width of the filter. (iv) The smaller the bandwidth of the Hilbert transformer, the faster the decrease of peak error of approximation with decreasing bandwidth of the Hilbert transformer. (v) The larger the value of N, the faster the decrease of peak approximation error with decreasing bandwidth of the Hilbert transformer. These observations lead to the conclusion that the bandwidth of the Hilbert transformer should be made as small as possible, and that odd values of N should be used, whenever possible, for efficient direct form realizations. A set of tables of values of the impulse response coefficients is included for several different bandwidth Hilbert transformers and for both even and odd values of N.

With the use of the momentum approximation of classical scattering theory we derive an inversion formula for incomplete total scattering cross sections measured by ion-atom scattering experiments as a function of ion energy. The formula allows direct determination of repulsive interatomic potential functions without the use of trial functions for the potential and, in addition, an estimate of the absolute error as a function of interatomic distance. Analysis of the K+Ar cross sections of Amdur and coworkers shows that for the inversion to be only moderately accurate, the experimental data must cover an energy range of more than an order of magnitude.

An approximation scheme for a class of optimal control problems is presented. An order of convergence estimate is then developed for the error in the approximation of both the optimal control and the solution of the control equation.

Optimum (in a minimax relative error sense) linear phase FIR digital differentiators can be designed in an efficient manner using a Remez optimization procedure. This paper presents data on wideband differentiators designed with even and odd values of N, the filter impulse response duration in samples. Based on these data, several interesting observations can be made, including: (i) Differentiators with even values of N have peak relative errors which are approximately one to two orders of magnitude smaller than identical bandwidth differentiators with odd values of N, and with the same number of multiplications per sample in a direct convolution realization. (ii) The smaller the bandwidth of the differentiator, the faster the decrease of the peak relative error with increasing N. (iii) The larger the value of N, the faster the decrease of the peak relative error with decreasing bandwidth. These observations lead to the conclusions that the bandwidth of a differentiator should be made as small as possible, and that even values of N should be used whenever possible. Complete tables of values of the impulse response coefficients are included for several wideband differentiators for even and odd values of N.

Information-theoretic concepts are developed and employed to obtain conditions for a minimax error entropy stochastic approximation algorithm to estimate the state of a non-linear discrete time system baaed on noisy linear measurements of the state. Two recursive suboptimal error entropy estimation procedures are presented along with an upper bound formula for the resulting error entropy. A simple example is utilized to compare the optimal and suboptimal error entropy estimators and the minimum mean Square error linear estimator.

In this paper, we prove a general approximation theorem useful in obtaining order of convergence estimates for the approximation of the solutions of a class of variational inequalities. The theorem is then applied to obtain an "optimal" rate of convergence for the approximation of a second-order elliptic problem with convex set K = { υ ∈ H 0 1 ( Ω ) : υ ⩾ χ K = \{ \upsilon \in H_0^1(\Omega ):\upsilon \geqslant \chi a.e. in Ω \Omega }.

A set of piecewise-linear (PL) basis functions for signal or functional decomposition is introduced. These basis functions provide a PL approximation to the signal and an a priori determination of the required number for a finite term expansion to achieve a certain pointwise approximation error. Moreover, the expansion coefficients are linear combinations of samples of the function to be expanded and are virtually trivial to determine.

A mathematical model describing the unsteady stratified settling of sediment samples and nonuniform suspensions has been developed, taking into account the influence of the sediment concentration on the fall velocity. In principle, the resulting equations can be solved using the method of integration along characteristics. An estimating procedure for the settling-tube size required has been established. The procedure is based on the requirement that the relative error in the settling time of each particle, which is caused by possible concentration effects, viz. hindered settling and settling convection, be less than a prescribed value. Although a lack of unambiguous experimental data as regards settling convection prevents a positive statement, this phenomenon seems to be more severe than hindered settling. If settling convection occurs, it will apparently cause unacceptable errors in the analysis of the relatively large sample sizes (as compared to the settling-tube dimensions) recommended in the literature.

The separation of regional from local gravity anomalies by means of the application of two-dimensional linear filters is analyzed. It was found that optimization of the filter in the least squares sense leads to filters that produce strong localized concentrations of the error, which may erroneously be interpreted as anomalies. For this reason the maximum absolute value of the error is a more important criterion for the quality of the filter than the root mean square error. This maximum absolute error is minimized by the minimax filter. Intermediate filters are derived which give a transition zone which comes appreciably closer to that of the optimal filter at only a small price in terms of increase of the maximum absolute error.

The properties of best nonlinear approximations with respect to a generalized integral norm on an interval are studied. A necessary condition for an approximation to be locally best is obtained. The interpolatory properties of best approximations are related to the dimension of a Haar subspace in the tangent space. A sufficient condition for an approximation to be best only to itself is given for a class of norms including the Lp norms, 1 < p < ∞. A sufficient condition for the set of points at which the given approximated function and an approximant agree to be of positive measure is given. The results are applied to approximation by exponential families Vn: in the case of Lp approximation, 1 < p < ∞, degenerate approximations are best only to themselves and the error of a best approximation is either identically zero or has 2n sign changes.

This paper deals with characterization of best approximations to vector-valued functions. The approximations are themselves vector-valued functions with components depending nonlinearly on the approximation parameters. The constraint is imposed that certain of the parameters should be identical for all components. An application to exponen- tial approximation is discussed in some detail. 1. Introduction. The work reported in this paper was motivated by the following problem: Suppose a set of experimentally determined exponential decay curves is given. It is desired to approximate the curves by functions of the form a exp(fx), where A should be the same for the entire set of curves and a may vary from curve to curve. The problem is to determine how such approximation might best be made. This problem arises in a number of physical situations. In chemical kinetics, for example, monitoring of a chemical reaction which obeys a first-order rate law leads to just such exponential data, from which one wishes to extract a best A although the initial amount of material (a) varies from experiment to experiment. In a previous paper (1), this type of constrained vector-valued approximation was studied for the simpler situation where the approximating functions depend linearly on the parameters. In this paper, results for nonlinear approximation are presented. Section 2 contains a precise formulation of the problem and a characterization theorem applicable to the construction of best approximations from general classes of nonlinear families. In Section 3, the particular problem discussed in the preceding paragraph is taken up. A very simple alternation theorem is obtained, as well as an interesting theorem on uniqueness.

Improvements on the method of Lovell and Cole (1959) for the measurement of complex permittivity of liquids by means of the Boonton R-X Meter are presented. A new approach to the calibration of the cell and connecting line, considering ohmic losses, is described. Results obtained for the aliphatic alcohols compare closely with the values of complex permittivity measured by means of coaxial line techniques. The relative error in the real and imaginary components of permittivity are 1.2% and 2.5% respectively.

The effects of pole-zero pairs (doublets) on the frequency response and settling time of operational amplifiers are explored using analytical techniques and computer simulation. It is shown that doublets which produce only minor changes in circuit frequency response can produce major changes in settling time. The importance of doublet spacing and frequency are examined. It is shown that settling time always improves as doublet spacing is reduced whereas the effect of doublet frequency is different for 0.1 and 0.01 percent error bands. Finally it is shown that simple analytical formulas can be used to estimate the influence of frequency doublets on amplifier settling time.

An attractive alternative to least‐squares data modeling techniques is the use of absolute value error criteria. Unlike the least‐squares techniques the inclusion of some infinite blunders along with the data will hardly affect the solution to an otherwise well‐posed problem. An example of this great stability is seen when an average is, determined by using the median rather than the arithmetic mean. Algorithms for absolute error minimization are often approximately as costly as least‐squares algorithms; however, unlike least‐squares, they naturally lend themselves to inequality or bounding constraints on models.

A new Remes-type algorithm is developed for the actual practical computation of the best approximation to a continuous function f from an n-dimensional space of generalized polynomials p, where the error of approximation is measured by the uniform norm of errors related to the relative error ${{(f(x) - p(x))} / {\max (| {f(x)} |,| {p(x)} |)}}$. In this algorithm, the system of nonlinear equations whose (approximate) solution at each step has previously been necessary in problems involving such nonlinear errors is replaced by a weighted linear system, the weights being easily calculated from information obtained in the previous step of the algorithm.

Our main theorem states that under a certain existence hypothesis a varisolvent family does not permit a best approximation with a constant error. We deal with real valued continuous functions on a compact real interval using the Chebyshev (uniform) norm. Our result is applied to simultaneous approximation to show that a constant error cannot arise there. Further topics such as restricted range approximation, a betweenness property, and approximation on a proper compact subset of an interval are also studied. The existence hypothesis of the main theorem appears to be satisfied by most known varisolvent families. Examples are given.

In this note, we state general conditions which imply that the numerical range of a function mapping a setSinto a normed linear spaceYis the closed convex hull of the spatial numerical range of the function. This conclusion is of interest since, for example, it is equivalent to an extension to non-compact spaces of Kolmogoroff's characterization of functions of best approximation.

Pressure data from 20° cone-cylinder models tested in a blowdown wind tunnel for the Mach number range 0.2 to 5.0 are compared to an interference-free standard to determine wall interference effects as a function of test section blockage. Four models representing a range of blockage from approximately 1% to 6% were compared to curve fits of the interference-free standard at each Mach number, and errors were determined at each pressure tap location. The average of the absolute values of the percent error over the length of the model based on an interference-free standard was determined and used as the criterion for evaluating model blockage interference effects. The results are presented in the form of the percent error as a function of model blockage and Mach number.

The relationship among the machine precision, computation error and condition number in solving linear algebraic systems is derived for floating elimination method by using backward error analysis. It is shown that the negative of the logarithm of the relative error is proportional to the machine precision for fixed system and to the negative of the logarithm of the condition of the system for fixed t. It is also shown that for fixed relative error, the machine precision required to achieve this is proportional to the logarithm of the condition of the system. Numerical experiments have been carried out and the observed data do coincide with what was expected. (Author)

Two simple heuristic algorithms for piecewise-linear approximation of functions of one variable are described. Both use a limit on the absolute value of error and strive to minimize the number of approximating segnents subject to the error limit. The first algorithm is faster and gives satisfactory results for sufficiently smooth functions. The second algorithm is not as fast but gives better approximations for less well-behaved functions. The two algorithms are ilustrated by several examples.

Using graphs for representing computational processes, relative error propagation is described. It is shown how this relates to the condition of a problem and to the property of a process to be benign, i.e., to have only harmless effects of rounding errors. In particular, composition of processes is studied under these aspects. Several examples illustrate the theory.

Best possible error estimates are proved for spline semi-discrete approximations to dissipative initial value problems. Error bounds are also established for suitable difference quotients.

New mixing rules which are general and easy to use have been developed to predict the properties of mixtures from the BWR equation. The rules contain a binary interaction parameter, readily obtainable from the experimental values of the second virial cross-coefficients. The mixing rules were used to predict the densities of binary mixtures of carbon dioxide with methane, ethane and propane at elevated pressures. For any of the three mixtures, the standard error of the predictions was approximately equal to or less than the sum of the standard errors of fit for the pure components. The BWR parameters for the pure components, required in the above predictions, were determined by minimizing the sum of relative error squares in specific volume.

Coriolis coupling constants have been calculated from band contours for the molecules 12 CF 4 , 13 CF 4 , 28 SiF 4 , 29 SiF 4 , 30 SiF 4 , GeF 4 and SO 3 according to the most recent literature values. With the help of a modified gradient method these ζ-values serve to establish the force fields. For some isotopic substitutions the force field is altered and these changes as well as the approximated absolute error are discussed.

This paper presents a method of network synthesis for a prescribed unit-impulse response f(t) , taken to be zero outside the interval (0, t_{0}) . The imposed response f(t) is approximated by another one h(t) , whose Laplace transform H(s) is always physically realizable as the transfer function of a passive network with lumped parameters. A finite number of equidistant values of the specified function f(t) is used for the approximation. The pulse transfer function that has been used in elaborating the method no longer appears in the final design relations. The approximation criterion is the mean-square error in the interval (0, t_{0}) , and outside this interval h(t) is smaller than some imposed value. The transfer function H(s) results are expanded into partial fractions. The necessary computations are simple. As a particular case, the method permits the determination of the response h(t) in such a way, that it takes certain given values at regular intervals of time, thus making the method also useful for sampled-data systems. Illustrative examples and experimental results are given.

THE PROBLEM of replacing a complicated function with a simple one — having approximately the same properties often arises both in mathematics and physics. Let us suppose that we have an algorithm for computing the value of the complicated function at any chosen point and we do not have a good estimate about the number and place of points where the value of the complicated function is to be computed in order to get the simple one which may replace it. In this case a random point selection appears advantageous and we give an estimate for the error of the approximation in the cases where this kind of point selection is applied.

The need for a general approximation method for the design of cooler-condensers is discussed and the recent papers on the condensation of vapour in the presence of a non-condensable gas are reviewed. The various approximation methods that can be used for design are presented including the Colburn [5] method, in which enthalpy is assumed as the driving force. A new approximation method is given for obtaining the mean value of the heat flux, (Uδt), which is based on the assumption that a plot of the heat flux point values throughout the cooler-condenser, against the heat transferred per unit time, q, is a parabola. The areas calculated by the use of the various approximation methods are compared with the areas obtained for cooler-condensers designed by the rigorous Colburn and Hougen [7] method, for six examples that have been fully worked out in the literature. It is concluded that the method involving the mean value of (UΔt) is considerably simpler to use than the Colburn and Hougen method and is also generally more accurate than the Colburn approximation method for the cases studied. The error in the use of the Colburn method varied from −0·7% to +95%. The error in the use of the mean heat flux method varied from −0·7% to +2·0%, except for one case which gave an error of −42%. The methods involving various logarithmic means of the terminal values are shown to lead to serious errors.

Completely ab initio calculations of the perturbation parameters of the a 3Π and A 1Π states of CO have been carried out and compared with the recent experimental results of Field et al. Use has been made of a new program for calculating off‐diagonal matrix elements of the B L · S and spin‐orbit coupling operators. By including configuration interaction functions built from Hartree‐Fock orbitals optimized for each state, good agreement has been obtained between calculated and experimental constants in most cases. Our results show that the perturbations have a strong dependence on R and that they are different for different pairs of states. The single configuration approximation gives a relative error up to 40% and the constancy of perturbation values found by Field et al. cannot be attributed to dependence on a single, constant parameter.

In distributed lag models we often parameterize the lag distribution's form so that only small finite numbers of parameters are required even when it is likely that the model so written involves some specification error. The effects of such error depend on the autocorrelation properties of the independent variable; quasi-difference transforms of the data will have effects, possibly undesirable, on the nature of error due to approximation. Certain hypotheses, e.g., those concerning the sum of coefficients or the mean lag of the distribution, may be untestable in time series regressions in the presence of approximation error of this type.

A PREVIOUS PAPER1 has proposed the ?probabilistic error? as measure of performance and basis of design for control systems. The probabilistic error or ?end-sigma error? was defined as the average value of the suitably penalized error occurring at such times when the output of the system is actually utilized. This measure of performance was mathematically formulated1 and, for squared-error penalizing and invariant linear systems, was evaluated in a general way in the s or the t domain. An integral equation was also obtained1 which defines the optimum time invariant linear system in the sense that it has the smallest probabilistic error.