Showing papers in "Siam Review in 1972"

Weak Nonlinear Dispersive Waves: A Discussion Centered Around the Korteweg–De Vries Equation

Abstract: This article is concerned with wave propagation processes where some balance occurs in the competition between a nonlinear effect and a higher order derivative effect which might be of a dispersive or a dissipative nature. The Burgers equation and the Korteweg–de Vries equation, which are prototype scalar nonlinear dissipative and dispersive equations, are shown to be fundamental to this study, even when quite general systems of equations are involved. The role of the solitary wave solution is shown to be central to the study which is applied to gravity waves, plasma waves and to waves in lattices. Both steady state solutions and initial value problems are reviewed together with questions of stability, existence and uniqueness.

The Padé Table and Its Relation to Certain Algorithms of Numerical Analysis

Abstract: The algebraic theory of the Pade table of a formal power series is presented with a natural notation which indicates possible extensions to Laurent series. The theory is related to bigradient determinants, the epsilon and eta algorithms, and to a variant of the quotient-difference algorithm. Normality criteria for the Pade table, which provide existence theorems for the algorithms, are developed.

Basic Concepts Underlying Singular Perturbation Techniques

Abstract: In many singular perturbation problems multiple scales are used. For instance, one may use both the coordinate x and the coordinate $x^ * = \varepsilon ^{ - 1} x$. In a secular-type problem x and $x^ * $ are used simultaneously. This paper discusses layer-type problems in which $x^ * $ is used in a thin layer and x outside this layer. Assume one seeks approximations to a function $f(x,\varepsilon )$, uniformly valid to some order in $\varepsilon$ for x in a closed interval D. In layer-type problems one uses (at least) two expansions (called inner and outer) neither of which is uniformly valid but whose domains of validity together cover the interval D. To define “domain of validity” one needs to consider intervals whose endpoints depend on $\varepsilon $. In the construction of the inner and outer expansions, constants and functions of e occur which are determined by comparison of the two expansions “matching.” The comparison is possible only in the domain of overlap of their regions of validity. Once overlap is established, matching is easily carried out. Heuristic ideas for determining domains of validity of approximations by a study of the corresponding equations are illustrated with the aid of model equations. It is shown that formally small terms in an equation may have large integrated effects. The study of this is of central importance for understanding layer-type problems. It is emphasized that considering the expansions as the result of applying limit processes can lead to serious errors and, in any case, hides the nature of the expansions.

A Review of the Placement and Quadratic Assignment Problems

Abstract: The physical design of a computer requires the placement of logic packages which must be interconnected to form a backplane. In this paper' the placement problem is explored and a survey of algorithms is presented for the attack of it and the associated quadratic assignment problem. Also presented are known experimental results of the algorithms defined, conclusions and areas for future research. Although the problem of placement of modules has been emphasized, the techniques are useful for handling the general problem of finding optimal locations for a set of objects that must be interconnected.

Simplification and scaling

Abstract: An attempt is made to clarify two frequently used applied mathematical techniques. Section 1 begins with a description of the basic simplification procedure in which a term is neglected under the assumption that it is small, and the consistency of this assumption is later checked. Successful uses of the basic simplification procedure are illustrated. Wretched consistent approximations are presented, showing that the procedure can be misused. The situation is clarified by a discussion of the relation between the size of the residual and the goodness of the approximation in three simple problem classes. Section 2 discusses scaling : how to choose dimensionless variables in such a way that the relative size of the various terms in an equation is explicitly indicated by the magnitudes of the dimensionless parameters which precede them. Scaling is illustrated on a simple physical problem and on several known functions. It is pointed out that more than one scale may be necessary, and the connection with singula...

Boundary Layers in Linear Elliptic Singular Perturbation Problems

Abstract: This paper presents a survey of recent results obtained in the theory of singular perturbations for linear elliptic partial differential equations in two independent variables. The emphasis is on the constructive procedures and the various aspects of the boundary layer phenomenon.

The Galerkin Method for the Numerical Solution of Fredholm Integral Equations of the Second Kind

Abstract: A unified discussion of the Galerkin method is given for the approximate solution of Fredholm integral equations of the second kind and of similar linear operator equations. The numerical stability of the approximate solutions is also discussed. As an application of the Galerkin method, the classical Fredholm–Nystrom method is reviewed.

Some Poised and Nonpoised Problems of Interpolation

Abstract: Taylor's formula may be looked upon as a one-point interpolation formula with remainder. In 1906, Birkhoff used this simple observation to make a unified approach to the various interpolation and quadrature formulas of numerical analysis. Later, in 1931, Polya gave a simple criterion to determine whether a given two-point interpolation problem is uniquely solvable (or poised). Interest in these questions was revived recently in 1966 by I. J. Schoenberg who generalized the result of Polya in one direction and also introduced the idea of using an incidence matrix to describe an interpolation problem. Earlier P. Taran had initiated the study of “lacunary” interpolation on specially chosen abscissas and had shown its relevance to problems of convergence and approximation. These problems of P. Taran have a unique solution only for some specially chosen abscissas and so are not poised. Here we give a survey of the recent progress in this field with some applications and open problems.

Steepest Ascent for Large Scale Linear Programs

Abstract: Many structured large scale linear programming problems can be transformed into an equivalent problem of maximizing a piecewise linear, concave function subject to linear constraints. The concave problem can be solved in a finite number of steps using a steepest ascent algorithm. This principle is applied to block diagonal systems yielding refinements of existing algorithms. An application to the multistage problem yields an entirely new algorithm.

Computation of Pseudoinverse Matrices Using Residue Arithmetic

Abstract: This paper discusses the use of residue (single and multiple modulus) arithmetic in computing the Moore-Penrose pseudoinverse of a matrix. The process can significantly reduce roundoff error. Proper selection of the modulus (or moduli) is critical. Necessary and sufficient criteria for modulus selection are developed.

Asymptotic Behavior of Integrals

Abstract: An account is given of some developments in the asymptotic evaluation of integrals of a single variable. After a discussion of Laplace integrals and quadrature formulas, estimates are provided of the errors in Laplace-type integrals and the method of steepest descents. Then Fourier transforms and the method of stationary phase are considered; integrands which are generalized functions are included and there is also a brief description of integrals of convolution type. Finally, uniformly valid formulas for the coalescence of two saddle points or of a saddle point and singularity are derived.

The Poincaré–Lighthill Perturbation Technique and Its Generalizations

Abstract: The known generalizations of the Poincare-Lighthill method of strained coordinates are investigated and compared. Some new conditions for its applicability are conjectured and some of its limitations are shown.

Computation of Hankel Transforms

Abstract: This paper deals with the numerical computation of the Hankel transform. Different methods of calculating this transform are presented with computer times and limitations on calculation for each method.

Solving the Matrix Equation $\sum _{\rho = 1}^r f_\rho (A)Xg_\rho (B) = C$

Abstract: The equation of the title is attacked via Taylor's formula for matrix functions. This approach contrasts with that of Lancaster [1], who used methods of contour integration suggested by functional analysis. The paper is designed as a supplement to his, being organized around a number of generalizations of his theorems. Solutions of the target equation are expressed in terms of component matrices of A and B and in terms of contour integrals.

Almost Periodic Functions Invented for Specific Purposes

Abstract: This is an expository survey of new definitions of almost periodic functions which were developed from researches on almost periodic differential equations. Some of these are not well known but are very useful for showing the almost periodicity of solutions of differential equations.

Improvement of Rayleigh–Ritz Eigenfunctions

Abstract: A common method of obtaining approximate eigenfunctions of a linear boundary value problem, $H\psi = E\psi $ and homogeneous linear boundary conditions, is the Rayleigh–Ritz method. We consider here the case where $\psi $ is expanded as $\psi = \sum _1^N {c_n w_n } $ and the $w_n $ are solutions of a “standard” problem, with the same boundary conditions. The eigenvalue E and coefficients $c_n $ are conventionally determined from the condition $\delta [(\psi ,H\psi ) - E(\psi ,\psi )] = 0$. We show here (without rigorous proofs) that, for large N, most of the error of the $ u$th Rayleigh–Ritz function $\psi _ u $ at the point $x_1 $ is removed by the addition of the correction term $\Delta \psi _ u (x_1 ) \equiv (\psi _ u (x),(H - E_ u )g(x,x_1 ))$. Here $E_ u $ is the approximate Rayleigh–Ritz eigenvalue and $g(x,x_1 )$ is a function, largely arbitrary, except for a specified singular behavior at $x = x_1 $ (an appropriate kink in one dimension, a $[4\pi ({\bf r} - {\bf r}_1 )]^{ - 1} $ singularity ...

Numerical Solutions in the Simplest Problem of the Calculus of Variations

Abstract: This is an expository paper dealing with the numerical techniques for solving the simplest problem in the calculus of variations Both first-variation methods and second-variation methods are reviewedConcerning first-variation methods, a direct approach and an indirect approach are given They both cause a reduction of the integral at each iterationConcerning second-variation methods, two possibilities are explored: (i) reduction of the integral, and (ii) reduction of the performance index In case (i), two viewpoints are employed: (a) minimization of the sum of the first variation and the second variation, and (b) minimization of the first variation for a given second variation In case (ii), two viewpoints are employed : (a) quasi-linearization of the Euler equation, and (b) descent process on the performance index