Linear Equation Solvers

In one type the 6,are fed into the machine in such a way as to drive the unknowns Xj to their correct values. In the second type, the xf are not driven by power supplied from the constant inputs but reach an equilibrium situation corresponding to the solution by a process of adjustment. We are concerned in this article with the operating conditions for this last type of machine when the adjusting process is determined by a linear operator with constant coefficients. We suppose that each a,-,can run independently through a real range which contains the origin. Thus, the determinant may be zero and any matrix can be represented by a suitable choice of scale for the unknowns and the £>,. Adjusting machines which are stable even when the determinant is zero may be designed. For instance, a block diagram is given in the author's book1 for such a machine. Another example is the set-up described by Goldberg and Brown2 which will insure stability when a certain type of feedback is used. However, in each case the coefficient network is duplicated. In the present article, we point out that if an adjusting type of machine is to operate successfully whenever the determinant A is not zero, then the square of the determinant must enter the indicial equation of the equations of motion for the machine. This necessary condition for successful operation rules out any linear feedback which does not involve using the a,-,twice. This result generalizes certain aspects of the necessity argument indicated in Goldberg and Brown. In Sees. 2 and 3 below, we describe precisely the type of machine we are concerned with. These machines may function continuously or in discrete steps. In Sees. 4 and 5 we obtain necessary and sufficient conditions that the machine should operate successfully in all cases where a solution is uniquely determined. These conditions are analogous to stability conditions for a linear network. In the case of a continuous machine, this analogy is readily established; in the case of a discrete step machine, these operational conditions are obtained by considering certain parts of the theory of linear difference equations. In Sec. 6 we prove the mathematical theorem upon which our result is based. It is shown in Sec. 7 that an adjusting machine with a linear feedback network, which is independent of the coefficients of the equations, will not always operate successfully. Section 8 contains the mathematical basis for Sec. 5 which is concerned with discrete step machines.