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Showing papers on "Piecewise linear function published in 1974"


Journal ArticleDOI
I. Tomek1
TL;DR: Two simple heuristic algorithms for piecewise-linear approximation of functions of one variable are described, which use a limit on the absolute value of error and strive to minimize the number of approximating segnents subject to the error limit.
Abstract: 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.

108 citations


Journal ArticleDOI
TL;DR: A computation scheme based directly on the dynamic programming formulation is proposed, a time-sharing computer program is discussed, and the results of an example problem are presented.
Abstract: A method for determining where to locate the inspection stations in a multistage production process with imperfect inspection is presented. Dynamic programming is used to establish that the optimal expected total cost function at every stage is piecewise linear and concave. While the optimal policy at every stage usually consists of one “inspect” region and one “do not inspect” region, this policy structure is found not to hold in general. A computation scheme based directly on the dynamic programming formulation is proposed, a time-sharing computer program is discussed, and the results of an example problem are presented.

82 citations


Journal ArticleDOI
TL;DR: In this paper, a method for finding optimal operating policies for a multiunit water resource system that extends over two river basins and serves multiple demands is presented, where the objective function for which consists of two parts, immediate economic losses within the month and the expected present value of future losses as a function of end-of-month storage levels in the reservoirs.
Abstract: A method for finding optimal operating policies for a multiunit water resource system that extends over two river basins and serves multiple demands is presented. The method was developed and tested for one of several systems that have been proposed for further development of the water resource in the urbanizing Piedmont Triad region of North Carolina. Monthly operating decisions are given by solutions of a piecewise linear programing problem, the objective function for which consists of two parts, immediate economic losses within the month and the expected present value of future losses as a function of end-of-month storage levels in the reservoirs. The latter function is estimated by imbedding the linear programing problem in a stochastic dynamic programing problem. An approximate solution technique for the larger problem is described, and computational experience is reported.

28 citations



Book ChapterDOI
01 Jan 1974
TL;DR: In this article, a geometric method to extend a quasiconformal mapping from R n to R n+1 was presented, which can be used to generalize the factorization to n ≥ 3.
Abstract: Publisher Summary This chapter presents a geometric method to extend a quasiconformal mapping from R n to R n+1 . For n = 1, the problem was solved by Ahlfors and Beurling, through an ingenious explicit integral formula. Ahlfors proved the result for n = 2. It seems quite difficult to generalize the factorization to n ≥ 3 and therefore, the chapter presents a method that is geometric and constructive. However, it depends heavily on approximations of arbitrary homeomorphisms by piecewise linear mappings, and it requires quite detailed information about the possibility of fitting together such mappings.

24 citations


Journal ArticleDOI
TL;DR: A set of piecewise-linear basis functions for signal or functional decomposition 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.
Abstract: 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.

19 citations


Book ChapterDOI
01 Jan 1974
TL;DR: In this article, an algorithm based on complementary pivoting was developed for solving F(x)=y, where F:R n→R n is convex and piecewise linear.
Abstract: An algorithm is developed for solving F(x)=y, where F:R n→R n is convex and piecewise linear. The algorithm is based upon complementary pivoting and proceeds by generating paths of solutions to F(x)=a+bz.

18 citations


Journal ArticleDOI
TL;DR: In this article, a generalized formulation of the imposed rotation method for the inelastic analysis of multistory frames is presented. But this method is not applicable to the analysis of multi-dimensional structures.
Abstract: The imposed rotation method for the inelastic analysis of frames rests on the interpretation of the actual bending moment distribution as the superposition of linear elastic moment responses to loads and to unknown plastic rotations regarded as imposed strains. The method given herein is a generalized formulation, covering second-order geometric effects and moment-axial force interaction. Finite element models of frames and piecewise linear moment-rotation laws for critical sections are assumed. Recent algorithms for solving quadratic programming and linear complementarity problems are shown to be efficiently applicable to the analysis of multistory frames. The safety factor with respect to local failure because of “brittle” flexural behavior turns out to be attainable by a suitably modified linear programming procedure.

18 citations


01 Mar 1974
TL;DR: In this article, a mixed-integer linear programming (MPSX) formulation is developed for minimizing delay to traffic in a signal controlled road network, where offsets, splits of green time and a common cycle time for the network are considered as decision variables simultaneously.
Abstract: A mixed-integer linear programming formulation is developed for minimizing delay to traffic in a signal controlled road network. Offsets, splits of green time and a common cycle time for the network are considered as decision variables simultaneously. The traffic flow pattern is modeled as a periodic platoon, and a link performance function is derived in the form of a piecewise linear convex surface representing the delay incurred by these platoons. Stochastic effects are accounted for by a saturation deterrence function representing the expected overflow queue on each link and are included as an additive component in the objective function. Computational results, using the MPSX system, are given for an arterial with 11 signals in Waltham, Mass., and a portion of the UTCS network in Washington, D.C. containing 20 nodes, 63 links and 21 loops. /Author/

14 citations


Journal ArticleDOI
TL;DR: Run length encoding seems to be superior for pictures of high detail and significant increase in compaction ratios is not achieved till one uses polygonal approximations of boundaries, if then.

12 citations


Journal ArticleDOI
TL;DR: In this article, two semidiscrete collocation approximations using smooth cubic splines are developed as approximates to the solution of two-point linear parabolic boundary value problems.
Abstract: Two semidiscrete collocation approximations using smooth cubic splines are developed as approximations to the solution of two-point linear parabolic boundary value problems.L∞-convergence results are presented for these two approximations as well as the piecewise linear Galerkin approximation. Several computational examples are given to illustrate the convergence results and demonstrate the applicability of the method.

Journal ArticleDOI
TL;DR: In this paper, the Green's function solutions for a shear dislocation with symmetry in the direction of infinite length have been integrated analytically over the direction in which the displacement solution is reduced from a surface integral over the fault plane to a line integral of temporal convolutions.
Abstract: Summary The Green's function solutions for a shear dislocation with symmetry in the direction of infinite length have been integrated analytically over the direction of infinite length. The displacement solution is reduced from a surface integral over the fault plane to a line integral of temporal convolutions over the width of the fault. If the time history function of fault displacement is any piecewise linear function the convolutions can be integrated analytically, reducing the solutions to line integrals. Numerical results and plots are presented for a simple example. The two-dimensional solutions give exact first motions at points ‘over’ the fault. Also the solutions can be used to test two-dimensional finite difference and finite element computer programs.

Journal ArticleDOI
TL;DR: In this article, a general expression for the probability density function of the output of a cross correlator, the inputs of which are assumed to consist of clipped sine waves of similar frequency plus uncorrelated, stationary Gaussian noise, was derived.
Abstract: A general expression is derived for the probability density function of the output of a cross correlator, the inputs of which are assumed to consist of clipped sine waves of similar frequency plus uncorrelated, stationary Gaussian noise. The correlator output is shown to be a piecewise linear function of the random phase difference between the two input processes; hence, the density function for the correlator output is obtained by a relatively simple transformationfrom the probability density function of the random phase difference.


Journal ArticleDOI
TL;DR: A hybrid technique combining the rapid gradient generation capability of the method of gradients with the rapid convergence characteristics of finite-dimensional, variable-metric function minimization algorithms is presented.

Journal ArticleDOI
TL;DR: In this paper, the authors use the theory of bifurcation and piecewise linear approximation to the characteristic with a falling segment, in the qualitative investigation of a system which is of practical interest.

Journal ArticleDOI
TL;DR: In this article, a method for computing the square root of the sum of two squared variables by a piecewise linear approximation is presented, which is obtained by restricting the number of 1s in the hardwired coefficients and by using well suited intervals for the linear approximation.
Abstract: A method for calculating the square root of the sum of two squared variables by a piecewise linear approximation is presented. Simplicity is obtained by restricting the number of 1s in the hard-wired coefficients and by using well suited intervals for the linear approximation. To obtain a small minimax error, a constant gain is introduced and optimised along with the coefficients.

Journal ArticleDOI
TL;DR: In this article, an extension of a recent algorithm is proposed for the rapid determination of steadystate oscillations in large classes of piecewise-linear (PWL) systems, which can be directly incorporated at marginal cost into a computer-oriented state-space approach for generating the time-domain response of such systems.
Abstract: An extension of a recent algorithm is proposed for the rapid determination of steady-state oscillations in large classes of piecewise-linear (PWL) systems. The modified algorithm has been successfully applied and can be directly incorporated at marginal cost into a computer-oriented state-space approach for generating the time-domain response of such systems.

Journal ArticleDOI
TL;DR: In this paper, a consistent finite element model for a circular wheel was developed based on triangular and quasi-triangular domains and a piecewise linear displacement field, and the minimum stress-rate principle of plasticity was used to obtain the solution of this two-dimensional continuum problem with internal unloading.

Patent
Donald L. Duttweiler1
22 Nov 1974
TL;DR: In this article, an improved non-precision component function generator is proposed to mitigate nonlinear distortion and to provide a less expensive digital signal processing arrangement, which includes two serially connected integrators which are advantageously switched in a complementary fashion.
Abstract: An analog-to-digital counting encoder or a digital-to-analog counting decoder for a pulse code modulation signal typically requires a function generator using precision components to generate a piecewise linear comparison signal corresponding to a segment companding law. As a result, nonproportional component change introduces nonlinear distortion in the encoded or decoded signal. To mitigate nonlinear distortion and to provide a less expensive digital signal processing arrangement, the invention resides in an improved nonprecision component function generator. The function generator includes two serially connected integrators which are advantageously switched in a complementary fashion to provide the comparison signal.

Journal ArticleDOI
TL;DR: Using a generalized LANGRANGE function for a linear hyperbelie programming problem, the author of as mentioned in this paper construets a dual problem, which becomes a piecewise linear convex programming problem.
Abstract: Using a generalized LANGRANGE function for a linear hyperbelie programming problem the author of the present paper construets a dual problem, which becomes a piecewise linear convex programming problem. For such dual problems, propositions on duality are proved containing in particular all propostions of the duality theory of linear programming.

Posted Content
TL;DR: In this article, the problem of determining the optimal location of plants, and their respective production and distribution levels, in order to meet demand at a finite number of centers, is investigated, where the possible locations of plants are restricted to a finite set of sites, and the demands are allowed to be random.
Abstract: This paper investigates the problem of determining the optimal location of plants, and their respective production and distribution levels, in order to meet demand at a finite number of centers. The possible locations of plants are restricted to a finite set of sites, and the demands are allowed to be random. The cost structure of operating a plant is dependent on its location and is assumed to be a piecewise linear function of the production level, though not necessarily concave or convex. The paper is organized in three parts. In the first part, a branch and bound procedure for the general piecewise linear cost problem is presented, assuming that the demand is known. In the second part, a solution procedure is presented for the case when the demand is random, assuming a linear cost of production. Finally, in the third part, a solution procedure is presented for the general problem utilizing the results of the earlier parts. Certain extensions, such as capacity expansion or reduction at existing plants, and geopolitical configuration constraints can be easily incorporated within this framework.



01 Jan 1974
TL;DR: The NCUBE computer implementation of this concept provides an easily derived form of piecewise linear discrimination that permits the separation of some types of data groups that are not linearly separable.
Abstract: Cluster analysis involves the unsupervised grouping of data. The process provides an automatic procedure for generating known training samples for pattern classification. NCUBE, the clustering algorithm presented, is based upon the concept of imposing a gridwork on the data space. The NCUBE computer implementation of this concept provides an easily derived form of piecewise linear discrimination. This piecewise linear discrimination permits the separation of some types of data groups that are not linearly separable.

Book ChapterDOI
01 Jan 1974
TL;DR: A multicategory pattern classifying machine which uses contextual information is developed and the convergence proof of the learning algorithm is shown and some results of computer simulations on hand-printed letters are presented.
Abstract: A multicategory pattern classifying machine which uses contextual information is developed. In the machine proposed here recognition is based on the information of succeeding patterns as well as the features of the given pattern. It is believed that the ability to recognize is substantially improved by using contextual information, as suggested by Raviv and by Edwards and Chambers. The machine consists of piecewise linear weighting devices and maximum selectors. The learning is performed by adjusting N weighting vectors and N × N additive weighting coefficients. In this paper we show the convergence proof of the learning algorithm and present some results of computer simulations on hand-printed letters. The performance of the machine compared favorably with that of other methods.

Journal ArticleDOI
01 Dec 1974
TL;DR: It has been shown that a fast, stable and accurate algorithm results from using Bergeron's method of characteristics for distributed parameter lines and a semi-implicit Runge?Kutta method for solution of the differential equations which describe the dynamics of linear/nonlinear lumped parameters.
Abstract: A numerical technique for computing switching transients in multiphase nonlinear systems is presented. The nonlinear terminal characteristics of the lumped parameters have been considered in totality without approximating them as piecewise linear characteristics It has been shown that a fast, stable and accurate algorithm results from using Bergeron's method of characteristics for distributed parameter lines and a semi-implicit Runge—Kutta method for solution of the differential equations which describe the dynamics of linear/nonlinear lumped parameters.