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The purpose of this paper is to describe a new gradient method for maximizing general functions. After a brief discussion of various known gradient methods the mathematical foundation is laid for the new algorithm which rests on maximizing a quadratic approximation to the function on a suitably chosen spherical region. The method requires no assumptions about the concavity of the function to be maximized and automatically modifies the step size in the light of the success of the quadratic approximation to the function. The paper further discusses some practical problems of implementing the algorithm and presents recent computational experience with it. [ABSTRACT FROM AUTHOR]

The article comments on the paper "Syminv: An Algorithm for the Inversion of a Positive Definite Matrix by the Cholesky Decomposition," by Terry Seaks, published in volume 40 of "Econometrica." In his article Seaks describes an algorithm for the Cholesky decomposition of a positive symmetric matrix, citing J.H. Wilkinson's work. The author explains that the relevant conclusions in Wilkinson's paper are those for floating point calculation and that the numerical advantage of the Cholesky procedure depends on being able to accumulate inner products accurately.

The article reports on the computational solution of linear programming equations involving almost-block-diagonal matrices. The author focuses on dynamic programming methods that can be used to solve weakly coupled and multi-stage economic systems. He suggests that most economic and industrial problems can be reduced to the problem of maximizing the linear form. It is suggested that finding the mathematical solutions to multi-dimensional problems is important because of the wide range of values of the solution's parameters. Dantzig's simplex algorithm enables programming problems of this type to be solved using digital computers.