An effective predictor-corrector process for large scale linear system of equations
The author treats the old problem of seriously ill-conditioned systems of linear algebraic equations \(A x = b, \enspace A \in \mathbb R^{m \times m}, \enspace x,b \in \mathbb R^m\), i.e. systems, whose condition number cond\((A) = \enspace \mid \mid A \mid \mid \enspace \mid \mid A^{-1} \mid \mid \) is very large. To improve the accuracy of the solution by direct methods commonly iterative refinement methods of \textit{R. S. Martin, G. Peters}, and \textit{J.-H. Wilkinson} [Numer. Math. 7, 362--383 (1965; Zbl 0135.37402)] are used. But e.g., for linear systems with a Hilbert matrix of order \(m \geq 12\) Wilkinson's refinement iteration gives solutions without any significant figures using double precision arithmetic. The Wilkinson refinement algorithm for ill-conditioned linear system is related to the explicit Euler method for the solution of stiff ordinary differential equations \[ \frac{dz}{dx} = A^{-1}(b - A z), \enspace z(0) = 0, \] where \(z = A^{-1}b\) is a unique stationary point. Starting from the knowledge that the explicit Euler method is not appropriate for stiff systems the author introduces a one-step second derivative extended backward differentiation formula of order four combined with Enright's second derivative linear method that computes the asymptotically stable equilibrium point of the dynamic system which is the solution of the linear system. The accuracy of the predictor-corrector process is validated by Hilbert matrix examples up to the order of 4000.
ill-conditioned linear system of algebraic equations
stiff system of ordinary differential equations
scientific article; zbMATH DE number 5072700
scientific article