Search

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.

This paper presents a design of PID controller for furnace temperature control system with disturbance. Currently, PID controller has been used to operate in electric furnace temperature control system because its structure is simpler compared to others. However, the issue of tuning and designing PID controller adaptively and efficiently is still open. This paper presents an improved PID controller efficiency from tuning by Nelder Mead method. The parameters of PID controller shall be obtained from the Nelder Mead optimization procedure. Errors between desired magnitude response and actual magnitude response are calculated by using the Integral of Absolute Error (IAE). The proposed Nelder Mead based PID design method is simpler, more efficient and effective than the existing traditional methods included Ziegler Nichols, Cohen-Coon and Direct Synthesis. Simulation result shows that the performance of PID controller using this proposed method is better than traditional methods and resistant to disturbance.

N a recent Note, A. D. Fine1 has shown that the multiaxial stress-strain diagram can be approximated by the uniaxial stress-strain diagram. He has also noted that for a linear strain hardening material (monotonically increasing loading) the diagrams are identical if Poisson's ratio JJL is 0.5, and that the approximate error is of the order of 10% for M = 0. This writer would like to make the following comments. Although Mr. Fine's paper is a valuable tool for engineers and analysts, the results and conclusions are hardly surprising or new. Several years ago Vol'mir2 showed that the uniaxial stress-strain diagram can be used to approximate the inelastic behavior of biaxially loaded structural members. He has proved that for an incompressibl e material (/z = 0.5) the diagrams coincide. His proof is based on an assumption made by Kachanov,3 namely that the stress-strain expressions can be separated into an elastic and an inelastic part. The final conclusion reached is that the secant moduli are identical in the two diagrams. Vol'mir's conclusions are therefore valid for any arbitrary stress-strain relation and are not restricted to the linear strain hardening case. The relative error estimate (about 10% when JJL = 0) in Ref. 1 is of academic interest only as far as structural materials are concerned. It is a matter of experimental evidence that most common materials are characterized by a Poisson ratio of about 0.5 in the plastic range. The results and conclusions of Refs. 2 and 3 have been used quite extensively in the Russian literature over the past decade since they are of great practical importance. It is felt by this writer that the results of the aforementioned papers will be of importance in elastic-plast ic analyses, since they lend themselves readily to computer applications.

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 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.

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 }.

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.

An improved Newton iteration procedure for computing pth roots from best Chebyshev or Moursund initial approximations is developed. It differs from the usual New- ton method by a multiplicative factor at each step. This multiplier halves the relative error by translating the usual one-sided error curve into a two-sided one, and then adjusting to make a Moursund-like fit. The generalized logarithmic error is used in determining this Introduction. The concept of generalized logarithmic error was introduced by Phillips (6) so that best Chebysbev fits to the pth root could be readily transformed into best Moursund fits (1) for more accurate Newton iteration. And Taylor (8) independently showed the existence of these transformations for finding rational Moursund fits to x', 0 < a < 1. Such procedures correspond to exploiting the usual logarithmic error for the square root case (2), (3), (4). In another paper (5), Ninomiya has shown how Newton iterates of Moursund fits for the square root can be improved from one-sided Chebyshev-like fits to Moursund two-sided fits for the next iteration; each improved fit has a maximum relative error that is only about half that of the one-sided fit. By using the generalized logarithmic error, we will derive similar improved two-sided fits for the pth root, with p being any positive integer; these fits in turn will form the basis for an improved Newton iteration method to compute integral roots.

In this paper, a generalized rational Chebyshev approximation problem is considered. The problem is this: To minimize the maximum absolute value of the "criterion function" of the error. By imposing a rather natural restriction on the criterion function, the problem is solved completely; the existence, the uniqueness and the characterization of the best approximation are clarified and interesting relationships between the best approximations corresponding to different criterion functions are found. The theory is applied to the starting rational approximation for Newton iteration for x 1 / n {x^{1/n}} .

are now well understood. In particular the equivalence of (i) with a linear programming problem permits its solution under very general conditions (see, for example, Kelley (1959), Stiefel (1960), Rice (1964), Osborne and Watson (1967, 1968)). In this paper, we consider the solution of the corresponding nonlinear minimax approximation problems, by solving a sequence of linear discrete T-problems. Certain properties of the solution of the linear problem are required, in particular, properties relevant to its solution as a linear programming problem, and these we now summarise. If we write r=f-Aa, (1.1)

The second Remes algorithm as originally established for polynomials, may converge or not when the approximating functions are rational. However, the few results known in this domain show how efficient the algorithm can be to obtain approximations with a small error, much more than in the polynomial case, in which the best approximation can be very nearly approached directly by a series development. The aim of this paper is to investigate the limitations of the applicability of certain extensions of the algorithm to the case where the approximations are rational as well as to present some numerical results.

In this paper, the relations between the individual errors of a gear and the angular position error of the gear are studied.The angular position error of a gear can be expressed with a Fourier expansion formula, and the equation which expresses the relation between the pitch error and the angular position error is obtained. From the study following results are obtained.1. In evaluating the angular position error from the individual errors, the essential errors are accumulative pitch error and the tooth profile error. The former is to be used to evaluate the lower harmonics components of the Fourier expansion of the angular position error and the latter is to be used to evaluate the higher harmonics components of it.2. It is not advisable to obtain the maximum value of the accumulative pitch error from the measured values of the individual pitch error. As the measurement error in the individual pitch error measurements are also accumulated and maximum value has largely changed. The value is rather to be obtained from the measured value of z/2 pitch error measurement or from the value of run-out error measurement.

The experiments with numerical forecasting using the barotropic model have been continued and a number of 24, 48 and 72 hour forecasts are presented. The initial data for these forecasts covered an area of about 9,000 by 12,000 km. The results indicate that the boundary influences have been reduced to be unimportant in the centre of the area in the 24-hour forecasts but they may still cause errors in the 72-hour forecasts. In view of the approximations made very good 72-hour forecasts have been obtained in some cases. The most successful one gave a correlation of 0.87 and a relative error of the forecast height changes of 0.58. Most of these extended forecasts should be of definite value in forecasting the weather. Large errors are, however, still obtained in some cases. The neglection of baroclinic process is one of the reasons for these errors but it is quite obvious that the largest errors are of a different nature. Experiments are at present conducted to find the sources of these errors. One line of this research is a general re-analysis of the assumptions made in deriving the barotropic model which is presented in the latter part of this paper. A number of improvements are suggested for further tests. DOI: 10.1111/j.2153-3490.1955.tb01139.x

Let $M(x)$ denote the expected value at level $x$ of the response to a certain experiment. $M(x)$ is assumed to be a monotone function of $x$ but is unknown to the experimenter, and it is desired to find the solution $x = \theta$ of the equation $M(x) = \alpha$, where $\alpha$ is a given constant. We give a method for making successive experiments at levels $x_1,x_2,\cdots$ in such a way that $x_n$ will tend to $\theta$ in probability.

This paper describes a new type of digital reactor period meter utilizing integrated circuits. The period is calculated from the number of counts N 2 during t 2 at t=τ, where t 2 is equal to t 1 which is the interval required for N 1=104 at t=0. Its output is the digital inverse period signal. The statistical error is delimited by fixing the number of counts instead of the counting interval. The absolute error is about 0.015 sec−1 in the case of a delay time of τ = 1 sec. For τ≧2 sec, it becomes smaller than 0.008 sec−1. The delay time τ is set to a value chosen among 1, 2, 4, 8, 16 and 32 sec, depending upon the count rate. The period meter contains about 250 integrated circuits, of which approximately 80 circuits are used for display and auxiliary functions not directly related to the primary purpose of period calculation.

Problems concerning approximation of real-valued continuous functions of a real variable by polynomials of degree smaller than n with various linear restrictions have been studied by several authors. This paper is an attempt to provide a unified approach to these problems. In particular, the notion of restricted derivatives approximation is seen to fit into the theory and includes as special cases the notions of monotone approximation and restricted range approximation. Also bounded coefficients approximation, ε \varepsilon -interpolator approximation, and polynomial approximation with interpolation fit into our scheme.

This paper is concerned with computing accurate approximations to the eigenvalues and eigenfunctions of regular Sturm–Liouville differential equations. The method consists of replacing the coefficient functions of the given problem by piecewise polynomial functions and then solving the resulting simplified problem. Error estimates in terms of the approximate solutions are established and numerical results are displayed. Since the asymptotic properties for Sturm–Liouville systems are preserved by the approximation, the relative error in the higher eigenvalues is much more uniform than is the case for finite difference or Rayleigh–Ritz methods.

This paper deals with existence and characterization of best approximations to vector-valued functions. The approximations are themselves vector-valued functions with components taken from a linear space, but the constraint is imposed that certain of the approximation parameters should be identical for all components. 1. Introduction. The following situation often occurs in the analysis of ex- perimental data. A set of experimental curves, all of which should roughly fit the same theoretical formula, has been obtained, and one desires to determine certain parameters in the theoretical formula. The experimental conditions have been such that, although some of the parameters should be the same for all of the given curves, others vary with the particular experiment. One would like, therefore, to be able to find the set of theoretical curves which simultaneously best approximates the set of given curves, under the constraint that some of the theoretical parameters should take on the same value for all members of the approximating set. As a simple example, one might want to fit a set of experimental curves by a set of straight lines, all with the same slope but with varying axis-intercepts. In the next section, we give a precise formulation of this problem for approximation from linear families. Following this, we provide existence and characterization theo- rems and discuss some examples.

This paper presents a method of improving the Curtis-Godson approximation for computing transmission along a non-homogeneous path. The present approximation, as well as those by Curtis-Godson and by Goody, are applied to model atmospheres with typical distributions of three important absorbers, CO2, H2O and O3. Error estimates for each are discussed.

This paper presents a method for determining the transfer function coefficients of a system from its frequency response. The method is based on Bush's transfer function decomposition technique and Chen-Shen's response conversion formulas. It involves no prejudgment and therefore has no cumulative error.

Unitarity in all three channels for the πN system is used to show that (a) one can, given the first LπN partial waves and the πN total cross sections, extrapolate to find the NN → ππ waves, the whole amplitude, and the πN waves, in the unphysical regions and this within absolute error bands, and (b) from either of those quantities, we construct Pade approximants which are then shown to converge to the cuts (4μ2 ⩽ t ≲ 2M2) again here within an explicit error band. It is shown that the method is stable under perturbations, giving also the explicit form in which such perturbations propagate. The analysis is carried first for amplitudes with positivity, and then extended to any amplitude. This paper is the first of a series, and only a general outline is presented.

Anbar (1971) and, independently, Abdelhamid (1971) have shown that if the density $g$ of the errors of estimates of function values is known, a transformation of observations leads to stochastic approximation methods which under mild conditions produce asymptotically efficient estimators (the first author considers the RM case, the second the RM and KW cases). This paper treats the RM case and shows that the same asymptotic result can be achieved without the knowledge of the density $g$.

There are fixed sample size methods for estimating the unknown size N of a finite population by tagging the elements of a first sample and then counting the number of tagged elements of a second sample [1, p. 43]. Less well known are sequential methods [2], which have the advantage that the total sample size automatically adjusts itself to the unknown N to assure a desired accuracy of the estimate. All these methods only provide estimates for which the relative error is likely to be small. Suppose, however, that we want PN (estimate = N)⪴α =.99, say, no matter what the value N = 1, 2, • • •. How can this be done?

The process of transmitting speech signals over digital channels requires sampling and quantizing the signals into a discrete number of levels occurring at discrete times. Speech signals quantized into 128 levels and sampled 8000 times a second are considered good quality for communications purposes and require a transmission rate of 56 000 binary pulses per second. It is the purpose of this paper to describe a coding technique that allows comparable quality speech to be sent over digital channels at pulse rates of from 20 000 to 40 000 binary pulses per second. The coding scheme consists of filtering the speech into two frequency bands and digitalizing the bands separately. The high‐frequency band can be quantized into fewer amplitude levels and the low‐frequency band requires fewer samples. The speech is remade at the receiving terminal by adding the outputs of the two channels. Because the average spectrum of speech signals falls off at high frequencies, the use of few levels in the high‐frequency band introduces only a small relative error in the total signal. A typical system might use a 1000‐cps dividing frequency and 64 and 4 levels and 2000 and 8000 samples, respectively, to produce a pulse rate of 28 000 binary pulses per second.

Various writers have dealt with the subject of optimal starting approxi- mations for square-root calculation by Newton's method. Three optimality criteria that have been used can be shown to lead to closely related approximations. This fact makes it surprisingly easy to choose a starting approximation of some pre- scribed form so that the maximum relative error after any number of Newton iterations is as small as possible. U 1. Introduction. The choice of polynomial and rational starting approximations for square-root calculation by Newton's method has been the subject of various investigations (e.g., (1)-(5)). These approach the problem from several different points of view. In this paper we will show how approximations obtained from these different viewpoints are related and how some of them can be derived from others. The problem of evaluating V/x for any x > 0 is easily reduced to the problem of evaluating V/x for x in some closed interval (a, b) such that 0 < a < b. Here (a, b) depends on the radix of the floating-point number system of the computer to be used; typical possibilities are (1/16, 1) and (-, 2). The following procedure is used to compute an approximate value for VIx. Using a polynomial or rational approxima- tion f(x) to V/x, valid in (a, b), compute a starting value yo = f(x) and then obtain Y1, Y22 .2 Yn by means of the relation Yk+1 = 12 (Yk + Xl8k), k = O. 1, * ,n - 1 . Then Yn V/x. It is customary not to test for convergence, since the number of iterations required in practice is quite small. Instead, the number of iterations n is

It has been shown by Butler et al. that a good approximation to the Bethe-Goldstone wavefunction can be constructed from eigenfunctions of the free two-nucleon system. The approximation is therefore closely related to the T-matrix. In this paper, it is used to derive an approximate G-matrix in terms of the T-matrix. As an illustration of this approach, the resulting approximate G-matrix is compared with the reference spectrum approximation of Bethe, Brandow, and Petschek for the simple case of a pure hard core potential.

The following corrections should be made in the authors' paper [2]: Replace p. 12, line 4 by ‖f−Nn‖p, = 0(1/nα),The author's theorems are contained as special cases in [1], Theorem 1.