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Showing papers on "Linear approximation published in 1986"


Journal ArticleDOI
TL;DR: This work presents an algorithm that finds a piecewise linear curve with the minimal number of segments required to approximate a curve within a uniform error with fixed initial and final points.
Abstract: Two-dimensional digital curves are often uniformly approximated by polygons or piecewise linear curves. Several algorithms have been proposed in the literature to find such curves. We present an algorithm that finds a piecewise linear curve with the minimal number of segments required to approximate a curve within a uniform error with fixed initial and final points. We compare our optimal algorithm to several suboptimal algorithms with respect to the number of linear segments required in the approximation and the execution time of the algorithm.

293 citations


Journal ArticleDOI
TL;DR: In this paper, the authors proposed a method to analyze systems in a time scale which is varied depending on the state such as dt/d\tau = s(x) (where t and τ are the actual time scale and that of new one, respectively, and s is the function which they call time scaling function).
Abstract: In this note, we propose a method to analyze systems in a time scale which is varied depending on the state such as dt/d\tau = s(x) (where t and τ are the actual time scale and that of new one, respectively, and s(x) is the function which we call time scaling function). Analysis of the system in the new time scale τ enables us to investigate the intrinsic structure of the system. A linearization problem in the new time scale is formulated as wide-sense feedback equivalence and is solved. It is also shown that the time scaling function which makes the system linear is derived as the solution of differential equations.

144 citations


Journal ArticleDOI
TL;DR: The preconditionded conjugate gradient method for solving a linear algebraic system of equations is recast to a form that permits sequential element-by-element calculations suitable for computations with finite element methods.
Abstract: The preconditionded conjugate gradient method for solving a linear algebraic system of equations is recast to a form that permits sequential element-by-element calculations suitable for computations with finite element methods. This strategy has been implemented for solving the linear systems arising in a finite element approximation for the standard test example of Laplace's equation. The element-by-element strategy has also been applied to the sequence of linear systems obtained using a successive approximation scheme for a representative class of nonlinear problems. Little storage is needed for these schemes, and test computations have been made on microprocessor, minicomputer, main-frame computers and special processors. The approach also appears appealing for calculations on parallel processors since individual element computations can be done in parallel.

107 citations


Journal ArticleDOI
TL;DR: In this article, the stability analysis of known Bianchi type cosmological models is performed for vacuum, comoving perfect fluids and non-comoving (tilted) perfect fluids.

103 citations


Journal ArticleDOI
TL;DR: The stochastic gradient algorithm using a simplified arithmetic using a power-of-two quantizer is used for the input of the multiplier to reduce the multiplication to at most a simple shift.
Abstract: The stochastic gradient algorithm using a simplified arithmetic is analyzed in this paper. A power-of-two quantizer is used for the input of the multiplier to reduce the multiplication to at most a simple shift. In spite of its simple implementation, the performance is shown to be comparable to the classical LMS algorithm. A linearized approximation to the quantizer is first derived, followed by the analysis of an exact nonlinear model. The derivation is based on the Gaussian assumption, and the effects of removing the Gaussian assumption are later considered. The roundoff error due to the finite-bit computation is calculated. Computer simulation results are provided to support the analysis.

60 citations




Journal ArticleDOI
TL;DR: An algorithm based on a linear function of the standard deviation of the random nutrient within each feedstuff, for which a penalty parameter is iterated in a search for a desired probability, yielded results very similar to the exact solutions found by nonlinear programming methods.

36 citations



Book ChapterDOI
01 Jan 1986

32 citations


Journal ArticleDOI
TL;DR: In this paper, the determination of l'ordre exact d'approximation de certains espaces de fonctions-splines regulieres a deux variables
Abstract: Determination de l'ordre exact d'approximation de certains espaces de fonctions-splines regulieres a deux variables

Journal ArticleDOI
TL;DR: In this article, the authors present a Gebrauch bestimmt ausschließlich für den persönlichen, nicht kommerziellen Gebrauchs, which is a rechtschutzbestimmter gebrauch, and gilt vorbehaltlich der folgenden Einschränkungen.
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Journal ArticleDOI
TL;DR: In this article, a method for solving free vibrations of conical shells with variable thickness is presented, where three components of displacements are assumed to have quadratic expressions of thickness, and the characteristic equations are solved by a new method.
Abstract: A method is presented for solving free vibrations of conical shells with variable thickness. Three components of displacements are assumed to have quadratic expressions of thickness. The characteristic equations are solved by a new method. Frequencies and distributions of displacements and stresses are presented and the new method is compared with Mirsky method and a linear approximation one.

Journal ArticleDOI
01 Oct 1986
TL;DR: In this article, the authors used area coordinates to obtain algebraic expressions for the computation of two-dimensional integrals with oscillatory integrands over triangular subregions that are simpler than those published recently.
Abstract: By using area coordinates one obtains algebraic expressions for the computation of two-dimensional integrals with oscillatory integrands over triangular subregions that are simpler than those published recently [1]. All the advantages of previous methods [1], [2] are retained. Moreover, the new equation is evaluated only once for each triangular subregion.

Journal ArticleDOI
TL;DR: The theory of absolute and convective instabilities is discussed in this paper, and the authors argue that the basis of the theory is questionable, since it describes the linear development of instabilities by their behaviour in the time asymptotic limit.
Abstract: The theory of absolute and convective instabilities is discussed and we argue that the basis of the theory is questionable, since it describes the linear development of instabilities by their behaviour in the time asymptotic limit. In order to make sensible predictions on the linear development of instabilities, the problem should be studied on the finite time scale implied by the linear approximation.

Journal ArticleDOI
TL;DR: In this paper, a variational derivation technique for the Green's functions and the self-energy is presented, which is formally closed and should be an appropriate starting point for any kind of iteration or approximation.
Abstract: Aiming at a realistic description of highly excited states in semiconductors the derivation of kinetic equations is reformulated where emphasis is laid on the consideration of many-body effects without perturbation expansion arguments. By the variational derivation technique a set of equations for the Green's functions and the self-energy is obtained, which is formally closed and should be an appropriate starting point for any kind of iteration or approximation. The connection of this technique with the diagram technique given by Keldysh and the translation technique for thermodynamic Green's functions according to Kadanoff and Baym is demonstrated. The general equations are then exactly transformed to difference and sum coordinates, enabling an adequate approximation in the case of slowly varuing (in space and time) external fields in terms of local quantities. In linear approximation with respect to the drift operator D a generalized Boltzmann equation is derived, which clearly exhibits many-body effects in all drift and collision contributions.

Journal ArticleDOI
TL;DR: In this paper, it was shown that the linear problem defined by S and N can be regarded as having a linear optimal algorithm if we allow the range of φ to be extended in a natural way.
Abstract: LetF1 andF2 be normed linear spaces andS:F0 →F2 a linear operator on a balanced subsetF0 ofF1. IfN denotes a finite dimensional linear information operator onF0, it is known that there need not be alinear algorithmφ:N(F4) →F2 which is optimal in the sense that ‖φ(N(f)) −S(f‖ is minimized. We show that the linear problem defined byS andN can be regarded as having a linear optimal algorithm if we allow the range ofφ to be extended in a natural way. The result depends upon imbeddingF2 isometrically in the space of continuous functions on a compact Hausdorff spaceX. This is done by making use of a consequence of the classical Banach-Alaoglu theorem.

Journal ArticleDOI
TL;DR: In this paper, the inversion of a finite Laplace transform has been studied and a nonlinear algorithm with finite error is presented, where the error of the algorithm can be arbitrarily small if appropriate information is used.
Abstract: We exhibit linear problems for which every linear algorithm has infinite error, and show a (mildly) nonlinear algorithm with finite error. The error of this nonlinear algorithm can be arbitrarily small if appropriate information is used. We illustrate these examples by the inversion of a finite Laplace transform, a problem arising in remote sensing.

Journal ArticleDOI
TL;DR: In this article, a linear approximation for surface-wave radiation by two adjacent slender bodies is derived and compared with a three-dimensional numerical method, assuming that the distance between the bodies is on the order of their lengths, and the far field disturbance due to each body is obtained by distributing wave sources and dipoles on its centreline and solving a pair of coupled integral equations for their strengths and moments respectively.
Abstract: A linear approximation for surface-wave radiation by two adjacent slender bodies is derived and compared with a three-dimensional numerical method. The approximation incorporates slender-body theory for a single body and accounts for wave interaction between the bodies. It is assumed that the distance between the bodies is on the order of their lengths. The far-field disturbance due to each body is obtained by distributing wave sources and dipoles on its centreline and solving a pair of coupled integral equations for their strengths and moments respectively. The hydrodynamic added-mass and damping coefficients are then calculated from simple expressions involving the source strengths and the hydrodynamic coefficients of each body separately. Wave exciting forces are also calculated from a far-field reciprocity relation. The approximation performs well even when the separation distance is comparable to the characteristic transverse dimension of each body.

Journal ArticleDOI
James D Emery1
TL;DR: In this paper, the distance between a piecewise linear approximation and a general curve is given for the approximate computation of the metric on general curves, and an algorithm has been successfully implemented in Fortran.
Abstract: An algorithm is given for computing a metric on piecewise linear curves. A bound for the distance between a piecewise linear approximation and a general curve allows the approximate computation of the metric on general curves. The algorithm has been successfully implemented in Fortran.

Journal ArticleDOI
Christian Ronse1
TL;DR: Generalisation du theoreme de Santalo aux fonctions lineaires and affines as mentioned in this paper, applied au traitement numerique des signaux, is a generalisation of Santalo's theorem.
Abstract: Generalisation du theoreme de Santalo aux fonctions lineaires et affines. Application au traitement numerique des signaux


Journal ArticleDOI
TL;DR: In this paper, the controllability properties of both the nonlinear and linear systems depend on certain Lie brackets of the vector field under consideration, which suggests that a linear approximation based on Lie bracket matching should be constructed at x(0).
Abstract: The following problem is examined: given a nonlinear control system dot-x(t) = f(x(t)) + the sum to m terms(i=1) u sub i (t)g sub i (x(t)) on R(n) and a point x(0) in R(n), approximate the system near x(0) by a linear system. One approach is to use the usual Taylor series linearization. However, the controllability properties of both the nonlinear and linear systems depend on certain Lie brackets of the vector field under consideration. This suggests that a linear approximation based on Lie bracket matching should be constructed at x(0). In general, the linearizations based on the Taylor method and the Lie bracket approach are different. However, under certain mild assumptions, it is shown that there is a coordinate system for R(n) near x(0) in which these two types of linearizations agree. The importance of this agreement is indicated by examining the time responses of the nonlinear system and its linear approximation and comparing the lower order kernels in Volterra expansions of each.

Journal ArticleDOI
TL;DR: In this paper, the null-field approach is modified so that the Q matrix (which in a straightforward manner gives the transition matrix) is obtained as a series instead of an integral.
Abstract: The direct and inverse scattering problems in two‐dimensional acoustics at a fixed frequency are considered. It is shown how the null‐field approach can be modified so that the Q matrix (which in a straightforward manner gives the transition matrix) is obtained as a series instead of an integral. For an obstacle which is a perturbation of a circle this series form gives an approximate, very explicit, expression for the transition matrix. A few numerical examples are given to show the utility and limitations of this approximation. For the inverse problem, the series form of Q gives a system of nonlinear polynomial equations which are solved by the imbedding method. Some numerical examples show that quite accurate results are obtained by this method in cases where the system of equations can be kept small. The linear approximation of the system of polynomial equations yields a method that works surprisingly well and which is also promising for the more difficult three‐dimensional and vector problems.

01 Jan 1986
TL;DR: In this paper, the authors compare two algorithms for obtaining estimates of the center and radius of a sphere from measurements on the surface of the sphere, and recommend that, for parameter estimation, the second (linear approximation) method should be applied first, with the parameter estimates it generates, then used as starting values for at least one iteration of more than the first (maximum likelihood) method.
Abstract: In this paper, we compare two algorithms for obtaining estimates of the center and radius of a sphere from measurements on the surface of the sphere. The first method is the maximum-likelihood solution of the problem which leads to the use of nonlinear least squares. The second method is an approximation of the first method but is computationally simpler, requiring only a linear least-squares solution. We show that the second method has problems estimating the parameters when the center of the sphere is far from the origin of the coordinate system, and the radius of the sphere is small. In many cases which would occur in practice, however, this method gives reliable estimates. The first method estimates reliably in all these cases. We also show that, although estimates of the precision of the parameter estimates can be found with each method, the second method tends to overestimate the variability of the parameters in cases that could easily occur in practice, while the first method again gives reliable estimates. We recommend that, for parameter estimation, the second (linear approximation) method should be applied first, with the parameter estimates it generates, then used as starting values for at least one iteration of more » the first (maximum likelihood) method. We further recommend that estimates of the variance of the parameters be derived using the techniques for the first (maximum likelihood) method. « less

Proceedings ArticleDOI
07 Apr 1986
TL;DR: This work proposes a Maximum Entropy method to reconstruct the object from either the Fourier domain data or directly from the original diffracted field measurements, giving a new definition for the entropy of an object considered as a function of R2to C.
Abstract: In diffraction tomography, the generalized Radon theorem relates the Fourier Transform (FT) of the diffracted field to the two-dimensional FT of the diffracting object. The relation stands on algebraic contours, which are semi-circles in the case of Born or Rytov first order linear approximations. We propose a Maximum Entropy method to reconstruct the object from either the Fourier domain data or directly from the original diffracted field measurements. To do this, we give a new definition for the entropy of an object considered as a function of R2to C. To take into account the presence of noise, a χ2statistics is added to the entropy measure. The objective function thus obtained is minimized using variational techniques and a conjugate-gradient iterative method. The computational cost and practical implementation of the algorithm are discussed. Some simulated results are given which compare this new method with the classical ones.

Journal ArticleDOI
TL;DR: In this article, a simplified numerical method is described for computing the radiation integral of a reflector antenna which is associated with either the aperture field method (AFM) or the induced current method (ICM).
Abstract: A simplified numerical method is described for computing the radiation integral of a reflector antenna which is associated with either the aperture field method (AFM) or the induced current method (ICM). The method involves linear approximation of the phase function and absorption of the approximation error into amplitude function. Then the integrand is rewritten as a slowly varying function multiplied by the phase exponential. The slowly varying function is expanded in terms of a polynomial for the radial direction and trigonometric functions for the circumferential direction. Expansion coefficients are determined by matching at points on the aperture. The matching points are chosen in a manner similar to that of a two-dimensional numerical integration scheme.

Journal ArticleDOI
TL;DR: In this article, a linear system reduction by impulse response energy approximation is analyzed and limitations are discussed, and a numerical example is included to show that in general the claims made for the procedure are not true.
Abstract: Linear system reduction by impulse response energy approximation is analyzed and limitations are discussed. To show that in general the claims made for the procedure are not true, a numerical example is included.

Journal ArticleDOI
TL;DR: In this paper, a linear approximation of the nonsymmetric Jordan-Thiry theory is presented, where the Lagrangian for the scalar field connected to the ''gravitational constant'' is computed up to the second order of approximation.
Abstract: This paper is devoted to a linear approximation of the nonsymmetric Jordan-Thiry theory. We deal basically with the Langrangian for the scalar field Ψ connected to the «gravitational constant». We compute this Lagrangian up to the second order of approximation inhμν =gμν -ημν. We prove that, in the zeroth and first orders of approximation in the electromagnetic case, the Lagrangian vanishes. We consider the spin content of the theory and we find a condition in which the theory is completely ghost and tachyon free.

Patent
16 Jun 1986
TL;DR: In this paper, the relative coordinates between the older two points and between newer two points out of three points in a time series coordinate point string of the linear graphic are rotated so that the opening angle between the relative coordinate is turned in a prescribed direction by a relative coordinate conversion part 4.
Abstract: PURPOSE:To attain real time processing for the linear approximation and encoding of a linear graphic by rotating the relative coordinates between the older two points and between newer two points out of three points in a time series coordinate point string of the linear graphic, and if necessary, executing interpolation processing or the like and reading out the opening angles of three points while referring a table. CONSTITUTION:Three points in the time series coordinate point string of the linear raphic through a data input part 2 are selected by a three-point selecting part 3 through a pen or the like, the relative coordinates between older two points and between newer two points are determined and the relative coordinates are rotated so that the opening angle between the relative coordinates is turned in a prescribed direction by a relative coordinate conversion part 4. When the relative coordinates are threshold or more, the relative coordinates are interpolated by a virtual coordinate interporation part 5 so that the length of an arm formed by the relative coordinates is an upper limit value or less. The table of an angle calculation part 6 is refereed by the interpolated relation coordinates to determine the opening angle between the relative coordinates. The start and end points of linear approximation are encoded by a final sampling point output part 7. Thus, the linear approximation and encoding can be processed with real time even if a microprocessor having no floating point calculating circuit is used.