Solving linear algebraic equations can be interesting

Here A is a square matrix of order n, whose elements are given real numbers ai3' with a determinant d(A)9 0] x and b denote column vectors, and the components of b are given real numbers. (Complex numbers would offer no essential difficulty.) It is desired to calculate the components of the unique solution x = A~b; here A~ is the inverse of A. Such problems arise in the most diverse branches of science and technology, either directly (e.g., the normal equations of the leastsquares adjustment of observations) or in an approximation to another problem (e.g., the difference-equation approximation to a selfadjoint boundary-value problem for a partial differential equation). These two are very frequent sources of numerical systems ; note that A>0 (i.e., A is symmetric and positive definite) in both examples. The order n is considered to range from perhaps 6 or 8 up to as large a number as can be handled. Stearn [ i l l ] , for instance, mentions the solution of a system of order 2300 by the U.S. Coast and Geodetic Survey. The accuracy demanded of an approximate solution £ varies widely; even the function which is to measure the accuracy of £ varies or is unknown. Some "customers" want to make the length \b— A%\ small; some, |^—^4 ô| ; others have apparently thought only in terms of getting A~b exactly. We all know that each component of the solution A~b can be expressed as a quotient of determinants by Cramer's rule. We have all evaluated determinants of orders 3, 4, and possibly 5, with a.-y integers; it is quite easy and rather boring. I therefore suspect that the average mathematician damns the practical solution of (1) as being both trivial and dull.