Showing papers in "Siam Review in 1977"
TL;DR: In this paper, perturbation theory for the pseudo-inverse (Moore-Penrose generalized inverse), for the orthogonal projection onto the column space of a matrix, and for the linear least squares is surveyed.
Abstract: This paper surveys perturbation theory for the pseudo–inverse (Moore–Penrose generalized inverse), for the orthogonal projection onto the column space of a matrix, and for the linear least squares ...
393 citations
TL;DR: The methods of cyclic reduction and Fourier analysis are reviewed together with the FACR algorithm which combines the two methods and it is shown that the asymptotic operation count of the FACr algorithm for an $n \times n$ grid is O(n^2 \log _2 \ log _2 n) which compares with $O(n+1) for either cyclic reduce or FourierAnalysis used independently.
Abstract: The methods of cyclic reduction and Fourier analysis are reviewed together with the FACR algorithm which combines the two methods. It is shown that the asymptotic operation count of the FACR algori...
334 citations
TL;DR: A recent survey of dynamic optimal control models for advertising can be found in this paper, where the authors present an up-to-date survey of advertising capital models, sales-advertising response models, micromodels and control-theoretic empirical studies.
Abstract: The last ten years have seen a growing number of optimal control theory applications to the field of advertising. This paper presents an up-to-date survey of dynamic optimal control models in advertising that have appeared in the literature.The basic problem underlying these models is an optimal control problem to determine the optimal rate of advertising expenditures over time in a way that maximizes the present value of a firm’s net profit streams over a finite or infinite horizon. The profit depends on sales (or an appropriate surrogate), the state variable and the rate of advertising expenditures, the control variable. Sales, in turn, is related to advertising expenditures via a differential or difference equation termed a state equation.The models covered in this survey are organized under four headings: advertising capital models, sales-advertising response models, micromodels, and control-theoretic empirical studies. The discussion involves specifications, methods used, results and the economic sig...
236 citations
TL;DR: In this paper, the first bi-harmonic problem on general two-dimensional domains was solved using a mixed finite element method, where the continuous problem has been approximated by an appropriate mixed-finite element method.
Abstract: We describe in this report various methods, iterative and "almost direct," for solving the first biharmonic problem on general two-dimensional domains once the continuous problem has been approximated by an appropriate mixed finite element method. Using the approach described in this report we recover some well known methods for solving the first biharmonic equation as a system of coupled harmonic equations, but some of the methods discussed here are completely new, including a conjugate gradient type algorithm. In the last part of this report we discuss the extension of the above methods to the numerical solution of the two dimensional Stokes problem in p- connected domains (p $\geq$ 1) through the stream function-vorticity formulation.
234 citations
TL;DR: In this paper, a brief description of catastrophe theory and its applications is given, and a brief discussion of the applications of catastrophe theories in statistics and in geophysics (plate tectonics).
Abstract: We give a brief description of catastrophe theory, and of its applications; to my view, it is a fundamentally qualitative, interpretative theory, and, by itself, it has no ability to predict Examples are given of interpretations of singularities in statistics and in geophysics (plate tectonics)
79 citations
63 citations
TL;DR: One hundred years have elapsed since the publication of Routh's fundamental work on determining stability of constant linear systems as mentioned in this paper, and a survey is given of some of the remarkably wide range of problems which were solved since Rouths algorithm.
Abstract: One hundred years have elapsed since the publication of Routh’s fundamental work on determining stability of constant linear systems. A survey is given of some of the remarkably wide range of problems which were solved since by Routh’s algorithm.
62 citations
TL;DR: In this article, it was shown that the probability of finding the odd numbered roots is uniform, and that the bisection algorithm converges to the even numbered roots with probability zero.
Abstract: A student of elementary probability may be amused by an application of probability to numerical analysis. Let f be a continuous function on the closed interval [a, b] such that f(a)f(b) 0, a E (Ck, bk), so let [ak+1, bk+1]=[Ck, bk]. Ibk-akl= 2-kIbo-aol, so that the bisection algorithm is guaranteed to converge to some root of f on [a, b]. If f has more than one root on [a, b], a problem in [1, p. 35] asks which root the bisection algorithm usually locates. If f has n distinct, simple roots x1
59 citations
58 citations
56 citations
TL;DR: In this paper, a semilocal convexity of integrals and of integrands is discussed and used in obtaining sufficient theorems for global minima, and the question of the possible use of such convex...
Abstract: A so-called semilocal convexity of integrals and of integrands is discussed in 1, then used in 2 in obtaining sufficiency theorems for global minima. The question of the possible use of such convex...
TL;DR: In this article, procedures are outlined which systematically isolate certain nonlinear evolution equations which fit into the above methodology, such as the partial differential nonlinear Schrodinger and a differential-difference nonsmooth Schroffinger equation, and an analogy to Fourier analysis is brought out.
Abstract: An important recent advance in nonlinear wave motion has been the discovery of a method of solution to a class of nonlinear evolution equations. The technique relies on a relation between the evolution equation, and an associated linear eigenvalue (scattering) problem. The initial value solution is found by the method of inverse scattering. In this paper, procedures are outlined which systematically isolate certain nonlinear evolution equations which fit into the above methodology. As examples of the ideas, the partial differential nonlinear Schrodinger and a differential-difference nonlinear Schrodinger equation are considered in detail. Some of the other physically important evolution equations are enumerated and an analogy to Fourier analysis is brought out.
TL;DR: This paper examines recent work on the complexity of combinatorial algorithms, highlighting the aims of the work, the mathematical tools used, and the important results.
Abstract: This paper examines recent work on the complexity of combinatorial algorithms, highlighting the aims of the work, the mathematical tools used, and the important results. Included are sections discussing ways to measure the complexity of an algorithm, methods for proving that certain problems are very hard to solve, tools useful in the design of good algorithms, and recent improvements in algorithms for solving ten representative problems. The final section suggests some directions for future research.
TL;DR: In this paper, a description of the qualitative theory of ODEs which can be used to analyze oscillatory systems in biology is given, i.e., the existence and stability of periodic solutions of n-dimensional nonlinear systems of ordinary differential equations.
Abstract: A description is given of the qualitative theory of ordinary differential equations which can be used to analyze oscillatory systems in biology, i.e., the existence and stability of periodic solutions of n-dimensional nonlinear systems of ordinary differential equations. The theory is organized around the biological topics. References to interacting chemical systems are also included.
TL;DR: In this article, the authors used the invariant imbedding method to approximate various explicit and implicit free boundary problems for a linear one-dimensional diffusion equation with a sequence of free boundary problem for ordinary differential equations.
Abstract: Frequently, diffusion processes require the determination of a free surface from overprescribed boundary data. A commonly used constructive solution technique for such problems is the method of straight lines. This paper illustrates the steps involved in the solution process. Specifically, the method of lines is used to approximate various explicit and implicit free boundary problems for a linear one-dimensional diffusion equation with a sequence of free boundary problems for ordinary differential equations. It is shown that these equations have solutions which can be readily obtained with the method of invariant imbedding. It also is established for a model problem that the approximate solutions converge to a unique (almost) classical solution as the discretization parameter goes to zero.
TL;DR: In this paper, the authors present a review of integral equations arising in the course of solving differential problems by inverting differential operators and describing phenomena by models which require summations (integrations) over space or time or both.
Abstract: This review was presented, in part, as one of two introductory expositions in a SIAM Symposium on “Applications and Numerical Solutions of Integral Equations” October 11, 1971, at Madison, Wisconsin. Integral equations arise in two principal ways: (i) in the course of solving differential problems by inverting differential operators, and (ii) in describing phenomena by models which require summations (integrations) over space or time or both. Typical examples of both types are described. These include integral equations occurring in geophysical exploration, both by the gravimetric and the seismic methods; those of electrostatic and magnetostatic potential theory; of heat flow; integrodifferential and integral equations encountered in diffusion and radiation, where stochastic models are basic; the equations for lift on an airplane wing, both the infinite case (two-dimensional conformal mapping) and the more difficult finite case with nonuniform cross section; the equations of renewal theory and of biologic...
TL;DR: In this paper, the concepts of abstract convexity are developed and discussed, and the ideas are then applied to the behavioral, social and physical sciences, in particular to the study of color vision, decision theory, mathematical economics and quantum mechanics.
Abstract: This expository article develops the concepts of abstract convexity. Basic representation theorems are proved and discussed. The ideas are then applied to the behavioral, social and physical sciences. In particular, it is shown how convexity can be used in the study of color vision, decision theory, mathematical economics and quantum mechanics.
TL;DR: In this article, a comparison is made between the uniformly valid asymptotic representations which can be developed for the solution of a singular perturbation boundary value problem involving a linear second order differential equation by using both the technique of matched ASM and the method of multiple scales.
Abstract: A comparison is made between the uniformly valid asymptotic representations which can be developed for the solution of a singular perturbation boundary value problem involving a linear second order differential equation by using both the technique of matched asymptotic expansions and the method of multiple scales. Next, there is a discussion of some of the subtle features as well as the relative advantages, limitations, logical extensions, and typical applications of these two methods of obtaining uniformly valid asymptotic representations when applied to slightly more general singular perturbation problems which arise from investigations of various phenomena in the natural sciences. Finally, a parameter identification example relevant to biological population dynamics is presented to illustrate the fact that the mere knowledge of these singular perturbation techniques can be a powerful analytical tool.
TL;DR: By considering suitable distributions of the Dirac delta function and its derivatives on lines and curves, solutions are obtained in closed form for various boundary value problems in mathematical physics in this article, where fields considered are electrostatics, magnetostatics, potential theory, hydrodynamics, elasticity, scattering of low frequency electromagnetic and acoustic waves.
Abstract: By considering suitable distributions of the Dirac delta function and its derivatives on lines and curves, solutions are obtained in closed form for various boundary value problems in mathematical physics The fields considered are electrostatics, magnetostatics, potential theory, hydrodynamics, elasticity, scattering of low frequency electromagnetic and acoustic waves Not only do we succeed in obtaining solutions which are not known so far, but we find that the present method enables us to recover the known solutions also in a very simple fashion
TL;DR: In this article, the problem of proving the validity of a formal approximation as an asymptotic approximation in singular perturbation problems is studied in a general formulation, and various ramifications of the prob...
Abstract: The problem of proving the validity of a formal approximation as an asymptotic approximation in singular perturbation problems is studied in a general formulation. Various ramifications of the prob...
TL;DR: The Mathematics Clinic at Claremont as mentioned in this paper is the only one of its kind so far as we are aware, which trains students of applied mathematics for immediate employment in industry or government.
Abstract: This paper discusses a new device for training students of applied mathematics for immediate employment in industry or government. The Mathematics Clinic at Claremont is the only one of its kind so far as we are aware. A description is given of the organization and operation of the Clinic, and two projects which were tackled through the Clinic are discussed in some detail.
TL;DR: In this paper, the problem of determining polynomials of minimal degree satisfying the data is solved by consider the data and the objective of determining the polynomial satisfying the objective.
Abstract: Given an incidence matrix E, the general problem of interpolating data on E is investigated. The objective of determining the polynomials of minimal degree satisfying the data is solved by consider...
TL;DR: A survey of the methods which utilize known information about a desired solution of a system of nonlinear algebraic equations is given in this paper, where the relationship between the various methods is pointed out.
Abstract: A survey of the methods which utilize known information about a desired solution of a system of nonlinear algebraic equations is given. The relationship between the various methods is pointed out. It is shown that the Levenberg–Marquardt–Tikhonov smoothing and damping techniques are particular cases of the minimization of a general functional.