Gauss' principle of least constraint is solved in a sequential fashion via dynamic programming in this paper. The solution itself constitutes a new principle for constrained motion, which we may name the Bellman-Gauss principle for constrained motion.

Abstract : This is a preliminary paper in which the authors discuss a general framework for the treatment of pattern-recognition problems. They make precise the notion of a 'fuzzy' set. Then they show how this may be employed in a sequential experimental procedure to ascertain whether a symbol is a member of a particular set or not. The close relation between the problem of pattern recognition and interpolation is stressed. (Author)

If the matrixA is not of full rank, there may be many solutions to the problem of minimizing ‖Ax−b‖ overx. Among such vectorsx, the unique one for which ‖x‖ is minimum is of importance in applications. This vector may be represented asx=A+b. In this paper, the functional equation technique of dynamic programming is used to find the shortest solution to the least-squares problem in a sequential fashion. The algorithm is illustrated with an example.

Discrepancies between assumed dynamical models and observations are often handled by making further probabilistic assumptions, a tactic which has both strengths and weaknesses. A re-examination of filtering and smoothing is conducted, and an alternative multicriteria approach, which is probability free, is advanced. This approach involves vector minimization as a key ingredient, and it specializes to the well-known Kalman, Viterbi, Larson-Peschon, and Swerling filters.

This paper studies the dynamic properties of relative commodity prices, especially the Prebisch-Singer hypothesis on their secular decline, using a new family of unit root tests based on the Fourier approximation to the data's underlying trend. The approximation controls for low-frequency variations such as structural breaks or the long swings induced by hypothesized super cycles in the data. Regarding the extant literature, we find considerably more evidence against nonstationarity in relative commodity prices, and very limited support for the Prebisch-Singer hypothesis.

Suppose the prior theoretical beliefs concerning the generation of a time-series data set take two forms: a prior measurement specification that the data has been generated by a linear regression model; and a prior dynamic specification that the regression coefficients evolve only slowly over time, if at all. The objective is to understand the actual relationship between the observed data and the regressor variables. In particular, do the estimated regression coefficients display any systematic time-variation? Is time-constancy a reasonably satisfactory approximation?

An automatic method for obtaining the numerical solution of a class of nonlinear integral equations is described. An imbedding method is used to obtain the initial-value equations to be integrated. The derivatives required for the solution are computed automatically. The user need only enter the two known functions into the program. None of the derivatives associated with the imbedding method need be derived by hand. Examples with numerical results and a complete program listing are given.

-In a recent paper an algorithm FEED was introduced for the systematic exact evaluation of higher-order partial derivatives of functions of many variables. The present paper demonstrates that FEED can be applied to a much broader class of functions than envisioned previously. Specifically, FEED can be used to evaluate the higher-order partial derivatives of functions which are defined in terms of the derivatives of other functions, a task required in many applications. 1. I N T R O D U C T I O N In a recent paper [1] a "table algorithm" is introduced for the automatic evaluation of higher-order partial derivatives. Building on a key idea of Wengert [2], the evaluation of a complicated kth-order differentiable function F at a specified domain point x is decomposed into a sequence of simple evaluations of special functions of one or two variables. The evaluation of each special function is accomplished by calling a calculus subroutine which automatically computes and returns the distinct partial derivatives of the special function through order k, along with the special function value. The final stage in the sequence of special function evaluations yields the distinct partial derivatives of F at x through order k, along with the furrction value F(x) . The special functions considered in Ref. [1] for function decomposition are the two-variable functions u + v , u v , uv, u/v and u ~', (1) and arbitrary one-variable continuously differentiable functions, such as sin(u), cos(u), exp(u), log(u) and a u b + c (2) for arbitrary constants a, b and c. For example, as detailed in Section 2, evaluation of the function z = F ( x , y ) = x + log(xy) (3) and its distinct higher-order partial derivatives at a given domain point (x,y) is accomplished by calling, in order, calculus subroutines for the special functions

Every rectangular matrix has a Moore-Penrose generalized inverse. The purpose of this paper is to present several computational approaches to the determination of this generalized inverse, which plays a role in least-squares problems.

An automatic method for obtaining the numerical solution of a certain class of nth-order nonlinear optimal control problems is described. The derivatives required for the solution of the two-point boundary value problem are computed automatically. The user of the program need only input the Hamiltonian function by calling the appropriate Fortran subroutines. This paper extends the methods described by the authors in previous papers in that it increases the order of the system that can be handled by the program from second order to the general nth order.

In a recent paper, Sukhanov derived a new method for transforming a nonlinear two-point boundary-value problem into an initial-value problem. Sukhanov's equations involve only the solution of ordinary differential equations and not partial differential equations. An earlier paper by the authors presented their interpretation of Sukhanov's method. An alternative method is presented in this paper. Numerical results are given.

In this paper, we take advantage of the availability of higher-order derivatives through the table method (see Ref. 1) and suggest a simple variant of the Lagrangian method for constrained optimization. Our method, and the software that we currently have can be used to minimize functions with many variables subject to an arbitrary number of constraints.

This paper shows that the generalized Newton algorithm [GN(r)], developed by Kalaba and Tishler (Ref. 1), can be described as a fixed-point algorithm. In addition to specifying sufficient conditions for convergence of the GN(r), we show that, forr=1, 2, 3, its rate of convergence increases with the order of the derivatives which are used.

An automatic method for obtaining the numerical solution for the simplest problem in the calculus of variations is described. The nonlinear two-point boundary-value Euler-Lagrange equation is solved using the Newton-Raphson method for obtaining successive approximations of the solution. The derivatives required for the solution of the problem are computed automatically using the table method. The user of the program need only input the integrand of the objective function in the calculus-of-variations problem and specify the boundary conditions. None of the derivatives usually associated with the Euler-Lagrange equation and the Newton-Raphson method need be calculated by hand. An example is given with numerical results. The automatic solution of the simplest problem in the calculus of variations in this paper is considered to be the first step in the automatic solution of more general optimal-control problems.

This paper describes a computer program for an optimization algorithm using first- up to the rth-order derivatives to find the optimum of r-continuously-differentiable functions of many variables. The algorithm, developed by Kalaba and Tishler [6,7], obtains the exact values of the derivatives required for the optimization from the “table algorithm” presented in [4] and [5]. The optimization algorithm described here collapses to the well-known Newton–Raphson algorithm when only first- and second- order derivatives are used.

An automatic method for obtaining the numerical solution of first-order nonlinear optimal-control problems is described. The nonlinear two-point boundary-value problem is solved using the gradient method for obtaining successive approximations of the solution. The derivatives required for the solution of the problem are computed automatically using the table method. The user of the program need only input the integrand of the cost functional and the Hamiltonian and specify the initial conditions and the terminal time. None of the derivatives usually associated with Pontryagin's maximum principle and the gradient method need be calculated by hand. Examples are given with numerical results.

This paper presents a general optimization algorithm using first-order torth order derivatives to find the optimum of anr-continuously differentiable function of many variables. This algorithm collapses to the Newton-Raphson algorithm when only first- and second-order derivatives are used. The computation of the required higher-order derivatives are readily available through thetable algorithm. The generalized CES production function is used as an example.

It is well known that the method of quasi-linearization is quadratically convergent for nonlinear two-point boundary value and system identification problems. In order to avoid the storage of successive approximations, it is sometimes convenient to integrate the original differential equations instead of storing the solution of the linearized equations. This reduces the rate of convergence, but it...

A formal algorithm is given for the systematic exact evaluation of higher order partial derivatives of functions of many variables. The algorithm improves upon Wengert's method in two key respects. Applications are envisioned wherever gradients, Jacobians, Hessians, and power series expansions could be employed.