Quarterly of Applied Mathematics 

About: Quarterly of Applied Mathematics is an academic journal published by Brown University. The journal publishes majorly in the area(s): Boundary value problem & Nonlinear system. It has an ISSN identifier of 0033-569X. Over the lifetime, 3574 publications have been published receiving 113942 citations. The journal is also known as: QAM.

A method for the solution of certain non – linear problems in least squares

Abstract: The standard method for solving least squares problems which lead to non-linear normal equations depends upon a reduction of the residuals to linear form by first order Taylor approximations taken about an initial or trial solution for the parameters.2 If the usual least squares procedure, performed with these linear approximations, yields new values for the parameters which are not sufficiently close to the initial values, the neglect of second and higher order terms may invalidate the process, and may actually give rise to a larger value of the sum of the squares of the residuals than that corresponding to the initial solution. This failure of the standard method to improve the initial solution has received some notice in statistical applications of least squares3 and has been encountered rather frequently in connection with certain engineering applications involving the approximate representation of one function by another. The purpose of this article is to show how the problem may be solved by an extension of the standard method which insures improvement of the initial solution.4 The process can also be used for solving non-linear simultaneous equations, in which case it may be considered an extension of Newton's method. Let the function to be approximated be h{x, y, z, • • • ), and let the approximating function be H{oc, y, z, • • ■ ; a, j3, y, ■ • ■ ), where a, /3, 7, • ■ ■ are the unknown parameters. Then the residuals at the points, yit zit • • • ), i = 1, 2, ■ • • , n, are

Sampling-based approaches to calculating marginal densities

Abstract: Stochastic substitution, the Gibbs sampler, and the sampling-importance-resampling algorithm can be viewed as three alternative sampling- (or Monte Carlo-) based approaches to the calculation of numerical estimates of marginal probability distributions. The three approaches will be reviewed, compared, and contrasted in relation to various joint probability structures frequently encountered in applications. In particular, the relevance of the approaches to calculating Bayesian posterior densities for a variety of structured models will be discussed and illustrated.

On a routing problem

Abstract: : Given a set of N cities, with every two linked by a road, and the times required to traverse these roads, we wish to determine the path from one given city to another given city which minimizes the travel time. The times are not directly proportional to the distances due to varying quality of roads, and v varying quantities of traffic. The functional equation technique of dynamic programming, combined with approximation in policy space, yield an iterative algorithm which converges after at most (N-1) iterations.