Scalar Functions of a Vector

Abstract

Frequently in industry or commerce one encounters a function ψ of several variables, x1, x2, … , xn and one wishes to determine certain of its properties. Since the independent variables x i comprise the elements of the vector x, ψ may be regarded as a scalar function of a vector and written ψ(x). A common problem is the calculation of those values of the variables x i that cause ψ to attain either its greatest or least possible value, where the x i s may or may not be required to satisfy certain ancillary conditions (constraints). This problem, the general optimisation problem, cannot be solved exactly and various iterative methods for its solution have been proposed. The basis of most of these methods is the approximation of the general function ψ(x) by a simpler function o(x) which is accurate for values of x close to a particular value, x0 say. The accuracy of this approximation may be expressed formally by saying that │ ψ(x) − o(x) │ < δ provided ║ x − x0 ║ < ∊, where δ and ∊ are small positive constants and ║ • ║ denotes any convenient vector norm. The optimisation problem is then solved for the simpler function o(x) and this solution is used to obtain an improved approximation to the solution of the more general optimisation problem. Although general mathematical programming is beyond the scope of this work, the approximating functions are almost invariably expressed in terms of matrices and their properties thus come within our terms of reference. Before investigating these, however, we define two important functions that may be derived from the general function ψ(x).