Showing papers on "Approximation algorithm published in 1972"
TL;DR: An approximation algorithm is presented which uses an iterative method to produce polygons with a small—but not minimum—number of vertices that lie on the given curve that justifies the abandonment of the minimum-vertices criterion.
1,323 citations
TL;DR: In this article, the problem of designing a stable recursive digital filter to have an arbitrarily prescribed frequency response may be considered as an approximation problem using the minimum p -error criterion, which is successfully solved using the Fletcher-Powell algorithm.
Abstract: The problem of designing a stable recursive digital filter to have an arbitrarily prescribed frequency response may be considered as an approximation problem. Using the minimum p - error criterion, a new problem of minimizing a function of n variables results, which is successfully solved using the Fletcher-Powell algorithm. An important theorem guaranteeing the existence of a stable optimum for a large class of synthesis problems is stated, and necessary modifications to the Fletcher-Powell algorithm to assure stability are considered. Finally a number of results of the application of this method are given.
265 citations
TL;DR: Three new methods of performing square rooting rapidly which utilize multiplication and no division are presented, considered for convergence rate, efficiency, and implementation.
Abstract: With the increasing availability of high-speed multiplication units in large computers it is attractive to develop an iterative procedure to compute division and square root, using multiplication as the primary operation. In this paper, we present three new methods of performing square rooting rapidly which utilize multiplication and no division. Each algorithm is considered for convergence rate, efficiency, and implementation. The most typical and efficient one of the already-known algorithms which utilizes multiplication, here called the N algorithm, is introduced for the purpose of comparison with the new algorithms. The effect and importance of the initial approximation is considered. (One of the algorithms, here called the G algorithm, is described in detail with the emphasis on its high efficiency.)
81 citations
43 citations
25 Oct 1972
TL;DR: An improved polynomial-time algorithm is found for a problem of scheduling on multiprocessing systems, by treating this as a bin packing problem.
Abstract: In this paper we consider polynomial-time algorithms for bin packing and their applications. The previously studied FIRST FIT and BEST FIT algorithms are shown to be special cases of a more generalized class of algorithms which all have similar worst case behavior. Linear time algorithms are then introduced which, though "faster" than FIRST FIT and BEST FIT, have the same, or better, worst case behavior. Finally, an improved polynomial-time algorithm is found for a problem of scheduling on multiprocessing systems, by treating this as a bin packing problem.
29 citations
TL;DR: In this paper, the best approximation to h(Ç) in the L(T) norm from among the functions in the form (m, r, c) was characterized. But the main objective was not to characterize the approximation, but rather to establish criteria for evaluating {at}, {bj}, {ct} and {rjt} achieving
Abstract: i = 0 \\ i = 0 \\ k = l 1 = 0 \\ where the knots a, /? and rjkeT are of multiplicity n, m and /ik respectively, and the total multiplicity of the interior knots is stipulated to be YJ=i fa — > The functions (2) display each rjt as a simple knot with the terms involving the knots a and jS omitted. The class of functions of the form (3) are designated as ^ m , r . Our main objective is to characterize the best approximation to h(Ç) in the L(T) norm from among the functions in ^„,m>^. Formally stated, we wish to establish criteria for evaluating {at}, {bj}, {ct} and {rjt} achieving
22 citations
TL;DR: A decision-directed scheme for estimating the mean is expressed as a stochastic approximation algorithm and the algorithm is shown to converge, but not to the true value, by means of the theory of stochastically approximation.
Abstract: A decision-directed scheme for estimating the mean is expressed as a stochastic approximation algorithm. The algorithm is shown to converge, but not to the true value, by means of the theory of stochastic approximation. A modification of the algorithm that converges to the true value is presented.
12 citations
TL;DR: It is shown that this method of determining the best approximation by an alternating family on an interval is by approximating on finite subsets of the interval can fail to converge, particularly in the case of polynomial rational approximation and exponential approximation when the best analogy is degenerate.
Abstract: A method of determining the best approximation by an alternating family on an interval is by approximating on finite subsets of the interval. In this note we show that this method can fail to converge, particularly in the case of polynomial rational approximation and exponential approximation when the best approximation is degenerate.
7 citations
01 Jan 1972
TL;DR: An application is made of a descent algorithm to the problem of finding the vector x of dimension p to minimise ‖f(x)‖ where the dimension of f is n and it is shown that the nature of the algorithm depends strongly on the choice of norm.
Abstract: An application is made of a descent algorithm (essentially a modification of Newton’ s method) to the problem of finding the vector x of dimension p to minimise ‖f(x)‖ where the dimension of f is n. It is shown that the nature of the algorithm depends strongly on the choice of norm. For example, the Gauss-Newton method is in general first order and can be divergent from a point arbitrarily close to the solution unless ‖f‖ is sufficiently small. However the corresponding algorithm in the maximum norm gives second order convergence in the usual case in which the extremal deviation occurs at just p + 1 points provided a further algebraic condition related to the Haar condition is satisfied. The first case can be interpreted as data dependent, the second as model dependent.
7 citations
01 Dec 1972
TL;DR: Stochastic approximation is used to illustrate the connection between several recently proposed recursive algorithms for adaptation of array processing structures and the manner in which incorporation of a priori statistical information leads from one algorithm to another.
Abstract: Stochastic approximation is used to illustrate the connection between several recently proposed recursive algorithms for adaptation of array processing structures. The development illustrates the manner in which incorporation of a priori statistical information leads from one algorithm to another. This link establishes a hierarchy among the algorithms. Sufficient conditions are established to ensure convergence of the algorithms and the conditions are shown to be universal in the sense that they are independent of the particular algorithm implemented. The signal processing problem considered throughout the paper is one of estimating (in real time) the time history of a stochastic, propagating signal that is immersed in additive, propagating and nonpropagating noise. The propagating noise is sometimes called interference.
3 citations
TL;DR: In this article, the problem of approximating a given element of a linear space by a family of elements, depending on a parameter, as well as possible, is transformed into a maximin.
Abstract: The present paper is concerned with the problem of approximating a given element of a linear space by a family of elements, depending on a parameter, as well as possible. Normlike convex functionals are used as measures for the quality of approximation. By means of quasilinearization of the convex approximation measure the approximation problem is transformed into a maximin. or programming problem, which is sometimes dealt with much easier. From the maximim-formulation a dual problem, replacing the primal approximation problems, is derived with the aid of a maximin-theorem of Ky Fan. New resultats on linear Chebyshev approximation with restricted parameters are obtained in this manner.
TL;DR: Two simple examples are given which show that the approximation problem may not be well-conditioned (not even continuous) in this case and suggest that by a slight modification in the formulation of the problem the inherent ill-conditioning can be eliminated.
Abstract: The dependence of a best approximation on the domain of definition has not been studied so far for problems involving approximations by non-Chebychev spaces. Two simple examples are given which show that the approximation problem may not be well-conditioned (not even continuous) in this case. The examples also suggest that by a slight modification in the formulation of the problem the inherent ill-conditioning can be eliminated. This leads to the consideration of best e-approximations.