Solving equations exactly

The problem of the solution of a given se t of lin ear equations Ax = b on a high-speed digital computer has been studied intensively, and there are a large number of method s, more or less sati sfactory , for carrying out such a solution. Nevertheless occasions arise when existing methods are inadequate, either because the solutions are required exactly, or because the coefficie nt matrix A is "ill-conditioned." A notorious exam ple of the latter is furnished by the Hilbert matrices A = H" given by