Cauchy Method Research Articles

On the Convergence of Some Feasible Direction Algorithms for Nonlinear Programming

Abstract : Recently Goldstein has considered the problem of determining conditions on the selection of directions in feasible direction algorithms for unconstrained maximization problems which assure the gradient vanishes in the limit. This paper extends his work (when specialized to Euclidian spaces) in various ways, notably by considering constrained maximization problems and by allowing a wider choice in the selection of the step size. For example we allow the ''optimal'' step size among others. We give conditions on the directions which assure convergence to a stationary point in feasible direction algorithms for maximizing a real valued continuous function on a closed set. We also apply this result to establish convergence to a stationary point of several known methods (e.g., Cauchy's method of steepest ascents, the Newton-Raphson method, and ufrank and Wolfe's method), a simple variant of one of Zoutendijk's methods that eliminates the need for his ''anti-zigzagging'' procedures, and some new second order methods that require quadratic programs to be solved at each stage. (Author)

Geometrical Acoustics' Theory of Waves from a Point Source in a Temperature- and Wind-Stratified Atmosphere

This paper is concerned with the development of practical numerical methods of estimating the amplitude and travel times of received pulses generated by a small source in a temperature- and wind-stratified atmosphere. The basic theory of Blokhintzev [J. Acoust. Soc. Am. 18, 322–334 (1946)] is shown to hold for propagation in the atmosphere even when gravitational forces are included, if wave periods are sufficiently short. The “ray-tracing” equations are derived from Blokhintzev's eikonal equation using Cauchy's method of characteristics and the boundary conditions on these equations appropriate to a point source are discussed. The numerical solution of the ray-tracing equations by integration over height is discussed and an integration approximation is introduced for handling the singularities at ray turning points. The method is illustrated by numerical examples.

Cauchy's method of minimization

Cauchy's method of minimization

Linear Ordinary Differential Equations with Constant Coefficients: Identification of Boole's Integral with that of Cauchy

We consider an equation with constant coefficientswhere a≠0 and f(x) is continuous in a suitable interval. Suppose that the symbolic polynomial P(D) has been fully decomposed into its (real or complex) linear factors, so that the equation may be writtenwhere b1, …, bq are distinct, and m1+…+mq = n. The Complementary Function being now known, we may write down a particular integral of (1) by Cauchy's method.