Showing papers on "Function (mathematics) published in 1970"
TL;DR: In this article, a more detailed analysis of a class of minimization algorithms, which includes as a special case the DFP (Davidon-Fenton-Powell) method, has been presented.
Abstract: This paper presents a more detailed analysis of a class of minimization algorithms, which includes as a special case the DFP (Davidon-Fletcher-Powell) method, than has previously appeared. Only quadratic functions are considered but particular attention is paid to the magnitude of successive errors and their dependence upon the initial matrix. On the basis of this a possible explanation of some of the observed characteristics of the class is tentatively suggested. PROBABLY the best-known algorithm for determining the unconstrained minimum of a function of many variables, where explicit expressions are available for the first partial derivatives, is that of Davidon (1959) as modified by Fletcher & Powell (1963). This algorithm has many virtues. It is simple and does not require at any stage the solution of linear equations. It minimizes a quadratic function exactly in a finite number of steps and this property makes convergence of this algorithm rapid, when applied to more general functions, in the neighbourhood of the solution. It is, at least in theory, stable since the iteration matrix H,, which transforms the jth gradient into the /th step direction, may be shown to be positive definite. In practice the algorithm has been generally successful, but it has exhibited some puzzling behaviour. Broyden (1967) noted that H, does not always remain positive definite, and attributed this to rounding errors. Pearson (1968) found that for some problems the solution was obtained more efficiently if H, was reset to a positive definite matrix, often the unit matrix, at intervals during the computation. Bard (1968) noted that H, could become singular, attributed this to rounding error and suggested the use of suitably chosen scaling factors as a remedy. In this paper we analyse the more general algorithm given by Broyden (1967), of which the DFP algorithm is a special case, and determine how for quadratic functions the choice of an arbitrary parameter affects convergence. We investigate how the successive errors depend, again for quadratic functions, upon the initial choice of iteration matrix paying particular attention to the cases where this is either the unit matrix or a good approximation to the inverse Hessian. We finally give a tentative explanation of some of the observed experimental behaviour in the case where the function to be minimized is not quadratic.
2,854 citations
1 Feb 1970
TL;DR: By constructing long 'increasing' paths on appropriate convex polytopes, it is shown that the simplex algorithm for linear programs is not a 'good algorithm' in the sense of J. Edmonds.
Abstract: : By constructing long 'increasing' paths on appropriate convex polytopes, It is shown that the simplex algorithm for linear programs (at least with its most commonly used pivot rule) is not a 'good algorithm' in the sense of J. Edmonds. That is, the number of pivots or iterations that may be required is not majorized by any polynomial function of the two parameters that specify the size of the program. (Author)
1,085 citations
TL;DR: An optimum rejection rule is described and a general relation between the error and reject probabilities and some simple properties of the tradeoff in the optimum recognition system are presented.
Abstract: The performance of a pattern recognition system is characterized by its error and reject tradeoff. This paper describes an optimum rejection rule and presents a general relation between the error and reject probabilities and some simple properties of the tradeoff in the optimum recognition system. The error rate can be directly evaluated from the reject function. Some practical implications of the results are discussed. Examples in normal distributions and uniform distributions are given.
988 citations
TL;DR: In this article, the optimal control of linear time-invariant systems with respect to a quadratic performance criterion is discussed and an algorithm for computing FAST is presented.
Abstract: The optimal control of linear time-invariant systems with respect to a quadratic performance criterion is discussed. The problem is posed with the additional constraint that the control vector u(t) is a linear time-invariant function of the output vector y(t) (u(t) = -Fy(t)) rather than of the state vector x(t) . The performance criterion is then averaged, and algebraic necessary conditions for a minimizing F\ast are found. In addition, an algorithm for computing F\ast is presented.
949 citations
1 Jan 1970
TL;DR: A new algorithm is described for calculating the least value of a given differentiable function of several variables that may be preferable to current algorithms for solving many unconstrained minimization problems.
Abstract: A new algorithm is described for calculating the least value of a given differentiable function of several variables. The user must program the evaluation of the function and its first derivatives. Some convergence theorems are given that impose very mild conditions on the objective function. These theorems, together with some numerical results, indicate that the new method may be preferable to current algorithms for solving many unconstrained minimization problems.
503 citations
TL;DR: In this paper, the theory of penning ionization and related associative ionization (AI) was examined in a classical, semiclassical, and quantum mechanical framework, the correspondence between the several descriptions being explicitly deduced; formulas for total cross sections for PI and AI, angular distributions for PI, and the distribution of energies of the ionized electron were presented.
Abstract: The theory of Penning ionization (PI) and related associative ionization (AI) is developed and examined in a classical, semiclassical, and quantum mechanical framework, the correspondence between the several descriptions being explicitly deduced; formulas for total cross sections for PI and AI, angular distributions for PI, and the distribution of energies of the ionized electron are presented. The possibility of anomalous structure is seen to appear in the energy distribution of the ejected electrons if the difference between the A*–B and A–B+ potential curves has a local extremum as a function of internuclear distance. Classically, this appears as an infinity in the distribution of electron energies, but the quantum mechanical expressions are reduced to obtain a uniformly valid approximation; the transition region through the classical infinity is characterized by an Airy function.
409 citations
TL;DR: In this article, an infallible method of finding all required natural frequencies of undamped vibration of linearly elastic skeletal structures, when the members are analysed as continuous and uniform, using dynamic stiffnesses, and not as an approximately equivalent lumped-mass system, was described.
237 citations
TL;DR: In this paper, the Fourier transform of a function g(x) is quantized, and the function recovered by inverse transformation differs from g (x) by means of a biased limiter model.
Abstract: If the Fourier transform of a function g(x) is quantized, the function recovered by inverse transformation differs from g(x). By means of a biased limiter model, the effects of Fourier-domain phase...
181 citations
TL;DR: In this article, a unified approach for including the effects of exchange and correlation in the theory of simple metals is presented, and it is shown that the indirect potential acting on a conduction electron can be described approximately by a function of the scattering momentum G(q), and this result follows directly from general theoretical results.
Abstract: A unified approach for including the effects of exchange and correlation in the theory of simple metals is presented. It is first demonstrated that the indirect potential acting on a conduction electron can be described approximately by a function of the scattering momentum G(q), and that this result follows directly from general theoretical results. Exchange and correlation corrections to the Hartree expressions for model (or pseudo) potential form factors and energy-wavenumber characteristics are derived and these corrections are shown to have a simple dependence on G(q). The selfconsistent theory of Singwi et al. (1968) is used to obtain a pair of transform relations between G(q) and the pair correlation function g(r) which permit G(q) to be calculated directly from g(r). The pair correlation function for a tenuous electron gas is used to compute a G(q) which is found to be selfconsistent. A simple approximate form of G(q), which is also selfconsistent, is suggested.
170 citations
TL;DR: In this article, it was shown that almost all steady spatially periodic motions of a homogeneous conducting fluid will give dynamo action at almost all values of the conductivity, and the same result is obtained for motions periodic in space-time.
Abstract: It is established analytically that, in a precisely defined sense, almost all steady spatially periodic motions of a homogeneous conducting fluid will give dynamo action at almost all values of the conductivity. The same result is obtained for motions periodic in space-time. The asymptotic form of the growing field, for an arbitrary initial field of finite energy, is also presented. Dynamo action is first shown to require that for some real vector there is a magnetic field solution of the form B= H exp (pt+ij. x), where H is a complex function of position (or of position and time) with the same periodicity as the motion, and p has positive real part, indicating growth. This number p is an eigenvalue of a linear differential operator on the space of admissible functions H. The first term of a power series in j for the eigenvalues/) which vanish to zero order is studied. It is thus proved sufficient for dynamo action that the determinant of the symmetric part of a certain 3 x 3 tensor, a function of the motion and conductivity, is non-zero. Finally, it is shown that this determinant is an analytic function of the conductivity, and is non-zero in a small conductivity limit for nearly all motions. This proves the stated result.
169 citations
TL;DR: For instance, Lee as mentioned in this paper took his M.A. degree in psychology at Magdalene College, Cambridge, after war service with the R.F. and R.N.
Abstract: TERENCE LEE took his M.A. degree in psychology at Magdalene College, Cambridge, after war service with the R.A.F. and R.N. He was later appointed to a Rockefeller Research Fellowship and worked on urban neighborhood planning. He was awarded his Ph. D. in 1954. Since then he has been Research Fellow at the University of Exeter and Lecturer in Psychology, now Senior Lecturer, at the University of Dundee.
TL;DR: In this article, the Schrodinger equation for the potential function V =z4+Bz2 where B may be positive or negative is solved in reduced form and the first 17 eigenvalues are reported for 58 values of B in the range −50 ≤ B ≤ 100.
Abstract: The one-dimensional Schrodinger equation in reduced form is solved for the potential function V=z4+Bz2 where B may be positive or negative. The first 17 eigenvalues are reported for 58 values of B in the range −50 ≤ B ≤ 100. The interval of B between the tabulated values is sufficiently small so that the eigenvalues for any B in this range can be found by interpolation. At the limits of the range of B the potential function approaches that of a harmonic oscillator with only small anharmonicity. The effect of a small Cz6 term on this potential is studied and it is concluded that a previously reported approximation formula is quite applicable but only for positive values of B. The success of the quartic–harmonic potential function for the analysis of the ring-puckering vibration is shown; it is also demonstrated that the same potential serves as a useful approximation for many other systems, especially those of the double minimum type.
TL;DR: In this article, the authors present Church's thesis, which is the reducibility axiom for constructive mathematics, focusing on the development of abstract constructions by use of the notion of graspable domain and a connection between Scott's model and the theory of lawless sequences.
Abstract: Publisher Summary This chapter presents Church's thesis, which is the reducibility axiom for constructive mathematics. The chapter focusses on the development of the theory of abstract constructions by use of the notion of graspable domain and a connection between Scott's model and the theory of lawless sequences. The chapter considers the abstract notion of arbitrary subset and the power set operation (collecting all subsets of a given set); correspondingly, the notion of constructible set, obtained by iterating the process of collecting the subsets that are definable by means of formulae in the language of set theory, using names for elements of the given set. The chapter also considers the (abstract) notion of constructive function (with integral arguments and values) as understood in intuitionistic mathematics; correspondingly, the notion of recursive function, defined by means of recursion equations. The notions and laws of mechanics (e.g. the equations for incompressible fluids) were derived from general qualitative experience, not from delicate measurements, which can be stated only in terms of the theoretical notions.
TL;DR: In this paper, the life function of a system expresses the life length of the system in terms of the life lengths of its components, and the application of life functions to reliability problems is discussed.
Abstract: The life function of a system expresses the life length of the system in terms of the life lengths of its components. This paper illustrates the application of life functions to reliability problems. The principal results are two characterizations of the life functions of coherent systems, which are used to obtain a number of properties for such systems.
Journal Article•
TL;DR: In this paper, the authors established the existence of a non-random limit for a wide class of stationary ergodic potentials under the assumption that the potential is Markovian, and the argument is based on the well-known theorems of Sturm.
Abstract: Let be the number of eigenvalues not exceeding for the selfadjoint boundary problem with random potential , and let Our problem is to clarify the conditions under which this function will exist and to indicate methods for calculating it.In the present article we establish the existence of a nonrandom limit for a wide class of stationary ergodic potentials. This limit is calculated under the assumption that the potential is Markovian, and the argument is based on the well-known theorems of Sturm.At the end of the article we consider an example in which is a Markov process with two states. In this case the calculations can all be carried out completely in a practical way, with the result that we obtain a formula expressing by means of integrals of elementary functions.Bibliography: 9 items.
TL;DR: In this article, the response function of a single-channel digital filter can be specified in terms of scalar-valued weighting coefficients, while the corresponding response function for a multichannel filter is more conveniently described by matrix-valued weighted coefficients.
Abstract: The transition from single‐channel to multichannel data processing systems requires substantial modifications of the simpler single‐channel model. While the response function of a single‐channel digital filter can be specified in terms of scalar‐valued weighting coefficients, the corresponding response function of a multichannel filter is more conveniently described by matrix‐valued weighting coefficients. Correlation coefficients, which are scalars in the single‐channel case, now become matrices. Multichannel sampled data are manipulated with greater ease by recourse to multichannel z‐transform theory. Exact inverse filters are calculable by a matrix inversion technique which is the counterpart to the computation of exact single‐channel inverse operators by polynomial division. The delay properties of the original filter govern the stability of its inverse. This inverse is expressible in the form of a two‐stage cascaded system, whose first stage is a single‐channel recursive filter. Optimum multichannel ...
TL;DR: In this article, a 2-energy-level model is proposed for the Fe2+ ions, which removes most, though not all, discrepancies between predicted and experimental results, and shows a transition from a temperature independent region to a temperature activated region.
Abstract: Measurements are reported of the resistivity and the thermoelectric power as a function of temperature for nickel-iron ferrites. The thermoelectric power shows a transition from a temperature independent to a temperature activated region. The results are broadly in agreement with the Jonker hopping model but some disagreement with the accepted simple theory is found. A 2-energy-level model is proposed for the Fe2+ ions; this removes most, though not all, discrepancies between predicted and experimental results.
TL;DR: The lobe function and cartesian (spherical harmonic) gaussian are compared with reference to calculations for second-row atoms in this article, showing that the two types of functions are in excellent agreement.
Abstract: The lobe function and cartesian (spherical harmonic) gaussian are compared with reference to calculations for second-row atoms. Single and grouped gaussian basis sets which have been reported for cartesian functions are taken over directly to construct corresponding lobe function bases with identical sets of exponents and with lobe separations chosen by a scaling procedure. Total and orbital energies and SCF coefficients resulting from calculations on the second-row atoms using the two types of functions for both primitive and grouped gaussian basis sets are seen to be in excellent agreement, thereby emphasizing the essential equivalence of lobe functions and cartesian gaussians, at the very least with respect to calculation of energy surfaces.
TL;DR: In this paper, two related algorithms are given to find a monotone function of one or more variables that best approximates in the least squares sense given function values that are not already monotones.
Abstract: SUMMARY Two related algorithms are given to find a monotone function of one or more variables that best approximates in the least squares sense given function values that are not already monotone. For one independent variable, such algorithms are well known; see, for example, Bartholomew (1959). This case is briefly repeated in ? 2 for completeness and as an introduction to the more difficult case of two or more independent variables. Of the two algorithms given in ?? 3 and 5, the second one should generally be shorter. Section 4 contains an auxiliary algorithm that is needed in connexion with both ?? 3 and 5. The paper closes with an example in ? 6 and some remarks in ? 7.
TL;DR: The Fletcher-Powell algorithm for minimizing a function of several variables and its use is discussed and illustrated and it is shown that the algorithm can be used also when the variables satisfy certain equality constraints.
Abstract: The Fletcher-Powell algorithm for minimizing a function of several variables is described. A package of FORTRAN IV subroutines that follows this algorithm with some modifications is given and its use is discussed and illustrated. It is shown that the algorithm can be used also when the variables satisfy certain equality constraints.
TL;DR: An algorithm for obtaining all simple disjunctive decompositions of a switching function using the operations AND, EXCLUSIVE OR, and complementation, which differs from existing methods in that it attempts to test fewer bound sets at the expense of additional analysis.
Abstract: An algorithm for obtaining all simple disjunctive decompositions of a switching function is described. It operates on a function given as an expression using the operations AND, EXCLUSIVE OR, and complementation. It uses necessary conditions for the existence of a decomposition to eliminate sets of bound sets from consideration. Thus this technique differs from existing methods in that it attempts to test fewer bound sets at the expense of additional analysis. The algorithm can also be applied to functions given in a canonical form. It is shown that for a collection of functions of n variables chosen at random, the time required grows as n3. Previous methods, on the other hand, have an exponential growth rate.
1 Jan 1970
TL;DR: In this article, the problem of determining a best lp approximation to discrete data is recast as a nonlinear program, and the resulting program involves linear constraints and a non-linear objective function.
Abstract: The problem of determining a best lp approximation to discrete data is recast as a nonlinear program. For a linear approximating function the resulting program involves linear constraints and a nonlinear objective function. This objective function is concave for 0 < p < 1 and convex for 1 < p < ∞. For p = 1 or p = ∞ the determination of best approximations can be accomplished by linear programming.
Computational aspects of these formulations are discussed, and some numerical and theoretical results are presented.
TL;DR: In this paper, the authors considered the relaxation problem for the linear hard-sphere gas using a Rayleigh-Ritz expansion of the energy kernel for transitions in a test-particle/heatbath system at variable mass ratio γ.
Abstract: The relaxation problem for the linear hard‐sphere gas is considered using a Rayleigh–Ritz expansion of the energy kernel for transitions in a test‐particle/heat‐bath system at variable mass ratio γ. The basis sets used are the “exact” eigenfunctions for the problem in the Rayleigh limit γ → 0 and the required expansion matrices can be obtained to order at least 20 × 20 by algebraic procedures. Good, converged eigenvalues λk are obtained in the discrete region 0 λ* are also determined and their use to “represent” the continuum in the initial‐value problem is discussed. A number of applications are considered. We obtain the relaxation of distribution functions P(x,t) and mean energies 〈x(t)〉 and investigate the energy‐autocorrelation function S(t) for equilibrium fluctuations, as a function of mass ratio. The latter proves to be very nearly Gaussian f...