Solution of Modified Linear Simultaneous Equations: An Investigation Associated with the Analysis of Redundant Structures Available to Purchase

A PROBLEM arises in the matrix analysis of highly redundant structures, and no doubt often occurs in other engineering and scientific contexts, in which a set of linear simultaneous equations, which has already been solved for one particular set of values of the coefficients, has to be solved again when some of the coefficients are altered. In the structural context the problem is to find the effect of altering the sizes of some of the members of a structure. This alters the matrix of coefficients of the equations for the unknowns (the redundancies) in a rather special way, but in other problems the alterations in the matrix may be of a more general character. It was accordingly decided to investigate the general problem of matrix modification to sec if some method could be devised which did not involve the complete solution of the altered equations ab initio, but which made use of the work already done in solving the original equations. It seemed reasonable to suppose that the amount of work involved in finding the solution of the modified equations could be related to the number of altered coefficients, so that if only a few of the coefficients were altered the amount of additional work to obtain the new solution would be correspondingly small.