CHAPTER 24
APPLICATIONS OF SYSTEMS OF EQUATIONS, MATRICES AND DETERMINANTS

229. Introduction. The numerical treatment of problems in this field is not entirely straightforward, and requires particular care even when we are handling problems in quite a moderate number of unknowns, e.g., for about 10. This whole subject is at present under active investigation in view of the development of high speed automatic digital computing machines, by means of which it is possible to contemplate the solution of problems in which the number of unknowns is of the order of hundreds. These investigations are in various directions. One of these is a re-examination of old methods and includes their history, classification and unification, together with detailed studies of special methods and attempts at the eradication of the superstitions and the justification of the suspicions which are still rife in the rather primitive field of numerical analysis. A second is the devising and examination of newer methods, such as the gradient or finite iteration methods. A third direction is the study of methods appropriate for special systems, such as those which arise by the discretization of differential equations.

We shall show some of the advantages and disadvantages of the classical methods, indicate some of the newer methods, and show some of the difficulties which can arise. This will be done mainly by discussions of special numerical examples.

For up-to-date accounts of this aspect of the subject, reference should be made to Proceedings of a Symposium on Simultaneous Linear Equations and the Determination of Eigenvalues.1 The

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