THE CONVERGENCE OF 2 NEWTON-LIKE METHODS FOR SOLVING BLOCK NONLINEAR EQUATIONS AND A CLASS OF R-POINT (R-ORDER A-STABLE ONE-BLOCK METHODS(1)ST)

In this paper, for a Newton-like method for solving block nonlinear eq uations arising in the numerical solution of stiff. ODEs y' = f(y), wh ich involves a smaller quantity of computation, we prove that it is co nvergent and the convergence is independent of the stiffness of fly), and give the error estimate. Furthermore, we present a modified Newton -like method involving an even smaller quantity of computation in cert ain cases, and prove that the modified method is convergent and the co nvergence is independent of the stiffness of f(y) for constant coeffic ient linear ODEs. Secondly, for any positive integer r, we discuss and construct a class of r-point (r + 1)st-order A-stable one-block metho ds suitable for the solution of stiff ODEs. Finally, we put forward an implementation strategy combining this one-block method and the r-poi nt rth-order A-stable one-block method of Zhao Shuangsuo and Zhang Guo feng (1997). The numerical tests show that the strategy is efficient. (C) 1997 Elsevier Science B.V.