Successive Corrections Research Articles

Wigner distribution functions and the representation of canonical transformations in quantum mechanics

Shows how for classical canonical transformations the authors can pass, with the help of Wigner distribution functions, from their representation U in the configurational Hilbert space to a kernel K in phase space. The latter is a much more transparent way of looking at representations of canonical transformations, as the classical limit is reached when h(cross) to 0 and the successive quantum corrections are related with the power of h(cross)2n, n=1,2, et seq.

Corrections to the Smoluchowski equation in the presence of hydrodynamic interactions

The systematic procedure for deriving the Smoluchowski equation and successive corrections to it from the Fokker-Planck equation is modified and extended in such a way that it now also covers the case of several interacting Brownian particles with hydrodynamic interactions. This is done by means of a suitable adaptation of the Chapman-Enskog method. The expression found for the first correction term to the Smoluchowski equation is worked out in full detail for the special case of two identical, spherically symmetric Brownian particles.

Quantum chemistry by random walk: Method of successive corrections

The random-walk method of solving the Schrödinger equation is reformulated to allow the direct calculation of the difference δ between a true wavefunction ψ and a trial wave-function ψ 0. For a trial wavefunction from any source the difference δ may be calculated and used to correct the trial wavefunction. Successive calculations offer the possibility of further corrections and wavefunctions of unlimited accuracy. The calculation of δ is illustrated for the cases of the particle-in-a-box and the hydrogen atom. Energies are determined directly from the random-walk calculations and indirectly from computation of the expectation values for the corrected wavefunctions.

Fast on-line implementation of two-dimensional median filtering

A fast two-dimensional median filtering technique is proposed for on-line image-processing applications, using a minicomputer. The histogram of the picture elements enclosed in the filter window (L × L) is computed and the median element is sequentially evaluated by successive corrections of the previous output. A comparison with an efficient sorting method is carried out in terms of computing time and memory requirements.

Exact realisation of 2-dimensional digital filters by separable filters

A scheme for the exact realisation of 2-dimensional digital filters by using separable filters is proposed. The desired impulse response is obtained by a process of successive corrections. The performance criterion in terms of computing speed is compared with that of another existing technique.

Objective Analysis of Aircraft Data in Tropical Cyclones

A method for objective analysis of aircraft observations in tropical cyclones has been developed. Quasi-horizontal fields of motion, temperatures, mixing ratios, and D-values are analyzed using a modified version of the method of successive corrections. The weighting functions are specified so that the high degree of circular symmetry characteristic of tropical cyclones is incorporated in the analyses. The analyses are performed on a 25 by 25 Cartesian grid of 5 n mi spacing which is centered on the storm. A special feature is the analysis of vertical motions as determined from aircraft flight characteristics. Three Atlantic storms are analyzed in detail: Hurricanes Inez (1966), Debbie (1969), and Ginger (1971). The analyses show the significant larger-scale features and major asymmetries of these storms. Both Inez and Debbie, which were well organized hurricanes, display characteristic vertical motion patterns in which a ring of strong ascent is found immediately surrounding the eye, with marked descent just outside of the annulus of strong ascent. Maximum ascent and descent rates were each indicated to be a few meters per second in these storms. Ginger was a marginal hurricane with poorly organized eye structure and relatively weak and disorganized patterns of vertical motion.

Numerical calculation of nonunique solutions of a two-dimensional sinh-Poisson equation

We give a method for computing nontrivial solutions to the nonlinear partial differential equation ∇ 2 Ψ + λ 2sinh Ψ = 0, with Ψ = 0 on a square boundary. The method consists of a Newton-Raphson iteration, in which successive corrections to Ψ must satisfy a linearized partial differential equation. We give a direct solution algorithm for the linearized equation, which is suitable for small meshes. Using this method, we establish the nonuniqueness of solutions by finding six solutions for the same value of λ. Calculation of these solutions required from 4 to about 40 iterations each, depending upon the accuracy of the initial approximation. These solutions are the lowest-mode members of three classes of solutions possessing (1) rectangular, (2) quasicylindrical, and (3) diagonal symmetry.

ロランによる船位要素ならびに空間波補正値の求め方

This report is a description of a method of solution for obtaining ship's sailing parameters (latitude, longitude, speed, course) and sky wave correction from Loran data. It consist of applying successive differential corrections, obtained by a least squares treatment, to a set of initial approximations for the ship's sailing parameters and sky wave correction. Chapter 4 presents the results when the method of solution was applied to Loran measurements of T.U.M.M. Training Ship Shioji Maru on 23rd of July 1974. The parameters, which were determined, agree very well with those published by cross bearings and radar. The computed height of reflection of wave of frequency 100kHz, have been obtained with this experiment, was about 66.5km and the angle of incidence was about 81.5 degree.

A scheme for multi-level objective analysis of contour heights

The results of experiments performed with a scheme for objective analysis of the contour heights of five levels of the atmosphere are presented here. Whereas objective analysis so far produced in India related to only 500-mb level in this study a scheme has been prepared for the analysis of 850, 700, 500, 300 and 200mb levels simultaneously. The method known as 'successive corrections by weighted averages has been employed. To maintain vertical consistency of analysis, thickness between two layers and the vector wind shear at the reporting stations have bean taken into account. Use of satellite pictures and AIREP messages bas been discussed.

The Adjustment of Surface Wind and Pressure by Sasaki's Variational Matching Technique

Abstract The sea level pressure and surface wind fields operationally produced at Fleet Numerical Weather Central are adjusted by using numerical variational analysis. The analysis region is a global band extending from 40S to 6ON and the successive corrections method is used to generate the initial or input fields. These fields are then adjusted within the framework of the variational method by requiring that they satisfy certain governing dynamical equations. A detailed study of this method is made in the Atlantic Ocean region on 4 January 1971. There is convincing evidence that small-scale wind information is incorporated into the pressure field and that the adjusted wind field has been modified to account for ageostrophic motion.

Sensitivity Calibration of a Dual-Beam Vertically Pointing FM-CW Radar

Abstract A method of calibrating a fixed vertically pointing radar is presented. The technique involves the firing of B-B shot of known radar cross section through the beam while making successive trajectory corrections until the absolute maximum signal is attained. The results agree closely with an independent calibration of antenna gain. The approach is particularly suited to an FM-CW radar with high range resolution because the pellets reach heights well in excess of the minimum range and errors in range are negligible. Corrections are presented for the reduction in maximum two-way gain resulting from intersecting beams whose full gain is attained only at the point of intersection. It is also shown that Probert-Jones’ k2 factor is significantly smaller for this system, and possibly for others, than the generally accepted value of unity. The method can be readily extended to any sufficiently sensitive pulsed radar by using small elevation angles and direct measurements of range rather than those obtaine...

SPECTRAL MODIFICATION BY OBJECTIVE ANALYSIS1

The power transfer function for the objective analysis of scalar fields by linear interpolation is determined with a simple statistical model of the data distribution. The method of successive corrections is used in a computational verification of the result. It is shown that the spectral filtering by interpolation is due to distance-dependent weighting and discrete sampling. True spectra can be inferred from interpolated waves that are long with respect to average sampling interval.

OPTIMUM INFLUENCE RADII FOR INTERPOLATION WITH THE METHOD OF SUCCESSIVE CORRECTIONS1

Abstract A method of selecting optimum influence radii for the objective analysis of a scalar field using the method of successive corrections is presented for an arbitrary weight function. The Cressman weight function is used in a computational verification of the result. A well-defined first pass optimum radius is found that increases with station separation, observational error, and wavelength of the true field for the average taken as the guess field.

Variations on variable-metric methods. (With discussion)

In unconstrained minimization of a function f f , the method of Davidon-FletcherPowell (a "variable-metric" method) enables the inverse of the Hessian H H of f f to be approximated stepwise, using only values of the gradient of f f . It is shown here that, by solving a certain variational problem, formulas for the successive corrections to H H can be derived which closely resemble Davidon’s. A symmetric correction matrix is sought which minimizes a weighted Euclidean norm, and also satisfies the "DFP condition." Numerical tests are described, comparing the performance (on four "standard" test functions) of two variationally-derived formulas with Davidon’s. A proof by Y. Bard, modelled on Fletcher and Powell’s, showing that the new formulas give the exact H H after N N steps, is included in an appendix.

Variations on Variable-Metric Methods

In unconstrained minimization of a functions, the method of Davidon-Fletcher- Powell (a variable-metric method) enables the inverse of the Hessian H of f to be ap- proximated stepwise, using only values of the gradient of f. It is shown here that, by solv- ing a certain variational problem, formulas for the successive corrections to H can be de- rived which closely resemble Davidon's. A symmetric correction matrix is sought which minimizes a weighted Euclidean norm, and also satisfies the condition. Numerical tests are described, comparing the performance (on four standard test functions) of two variationally-derived formulas with Davidon's. A proof by Y. Bard, modelled on Fletcher and Powell's, showing that the new formulas give the exact H after N steps, is included in an appendix. 1. The DFP Method. The class of gradient methods for finding the uncon- strained minimum of a function f(x)* in which the direction Sk of the next iterative step from Xk to Xk+l is computed from a formula such as:

Stability of Spaceframes

Nonlinear equations governing the behavior of spaceframes whose members undergo finite deflections and moderate chord rotations are presented. The beam-column effect including the effect of flexural shortening is taken into account, while twist due to torsion is assumed to be small. The method of solution is based upon the Newton method, and the corresponding tangent stiffness matrix is given. The solution for a particular loading is obtained by applying successive corrections to the approximate solution by means of the tangent stiffness matrix. A tangent stiffness matrix which is not positive, definitely implies that the potential energy of the corresponding equilibrium configuration is not at a minimum, and thus the configuration is unstable. Therefore, the lowest value of the load for which the determinant of the tangent stiffness matrix vanishes gives the critical buckling load for the structure. The formulation may be used for predicting the buckling loads and the load-deformation history for arches or shell-like spaceframes. Buckling of a shallow arch is investigated as an illustration. The first two branching points on the load-deflection curve, indicating the asymmetric modes of buckling, are determined.

Natural orbits of the first kind in the restricted problem

Lindstedt's method is applied to expand analytically the natural families of periodic orbits of the first kind in the restricted problem of three bodies. The mass ratio is kept as a literal variable; the series proceed in the powers of Hill's ratio of periods. They cover all four cases at once, namely, the direct or retrograde orbits for either inferior planets or for satellites. When the mass ratio is given a numerical value, the expansions contain fewer terms and thus can be carried up to degree 17 within a 32K core of an IBM 7094. For the system Sun-Jupiter, the initial conditions provided by the series have been corrected to yield the beginnings of all four natural families, and the characteristic exponents have been computed. Comparison between the values computed out of the series and their improvement by numerical integrations and successive variational corrections show that, within an accuracy of one part in ten thousand, the series represent moderately well even the orbit of J VIII; the retrograde planetary orbits are fairly well covered up to 90% of the distance from Sun-Jupiter, whereas the direct planetary orbits are covered only up to 60% of that distance.

A Technique for Fitting Nonlinear Models to Biological Data

A technique for determining best parameter values for nonlinear mathematical models by an iterative least—squares method is presented. The process is given in detail to enable nonmathematicians to implement it successfully. The technique is based upon a Taylor series expansion and involves solving for successive corrections to trial parameter values until the best possible parameter values are found. A table is given showing for combinations of scatter in the data and error in estimating initial or trial parameter values the number of iterations required for convergence to best parameter values.

Higher corrections to the WKB approximation

A systematic method of successive approximations is described, of which the first step is the WKB approximation. The higher corrections consist of successive recombinations of the two WKB waves, rather than a perturbation expansion in their interaction. At the same time the local wave number (or refractive index) undergoes successive corrections. Using this method it is shown that the quantummechanical reflection by a potential barrier is a small quantity of order ħ ∞, provided that the barrier is smooth, and so low that there is no classical turning point.

Approximate Solution of Hill's Equation

A modified WKB approximation, amenable to successive corrections, to the solution of the linear differential equation of second order having a periodic coefficient in normal form is presented. Considered as an application of the related equation method, it uses the free-particle wave equation rather than, as in an alternative approach, Mathieu's equation. Particular attention is given the instances, where two simple turning points appear in the period and where there are no turning points. With respect to the one-dimensional crystal, it is shown how the energy band structure can be gleaned directly from the given periodic potential.