Example 10 [8]

TaylorPhilllipsConsider the Hilbert matrix discussed in the module Problem Discretization using Approximation Theory. These matrices, which arise in simple polynomial approximation are notoriously ill conditioned and as For example, consider Thus, condition number can be computed as For n = 6, which is extremely bad.

Even for n = 3, the effects of rounding off can be quite serious. For, example, the solution of is If we round off the elements of to three significant decimal digits, we obtain then the solution changes to The relative perturbation in elements of matrix does not exceed 0.3%. However, the solution changes by 50%! The main indicator of ill-conditioning is that the magnitudes of the pivots become very small when Gaussian elimination is used to solve the problem.