Monotone discrete Newton iterations and elimination
The paper studies the Newton method for the solution of nonlinear algebraic systems with a Jacobian which is an M-matrix. For this case, it is well known that with two properly choosen starting guesses it is possible to obtain two monotone sequences of the Newton iterations which approach the exact solution quadratically and componentwise from below and from above, respectively. These results are extended to the case of the discrete Newton method using a finite difference approximation of the Jacobian. It is also shown that the quadratic convergence can be achieved in this case if the stepsize for the approximation of the Jacobian goes to zero. All these properties are studied once again in the case when some unknowns are eliminated directly from the nonlinear system. Numerical experiments illustrate the theoretical results and show a suitable choice of the stepsize for the discrete Newton method.
scientific article; zbMATH DE number 825384