Numerical Stability of Iterations for Solution of Nonlinear Equations and Large Linear Systems.

摘要

Some recent results are discussed concerning the problem of numerical stability of iterations for the solution of nonlinear equations F(x) = 0 and large linear systems Ax+g = 0 where A = A* is positive definite. For systems of nonlinear equations it is assumed that the function F depends on a so called data vector F(x) = F(x;d). One defines the condition number cond(F;d), numerical stability and well-behavior of iterations for the solution of F(x) = 0. Necessary and sufficient conditions for a stationary iteration to be numerically stable and well-behaved are presented. It is shown that Newton iteration for the multivariate case and secant iteration for the scalar case are well-behaved. For large linear systems the author presents the rounding error analysis for the Chebyshev iteration and for the successive approximation iterations. It is shown that these iterations are numerically stable and that the condition number of A is a crucial parameter.