6. Safe Starting Regions for Iterative Methods
Ramon E. Moore · Society for Industrial and Applied Mathematics eBooks · 1979
If an iterative method for solving a linear system converges at all, it generally converges from any initial approximation. For example, if ‖E‖<1 , then x(k+1)=b+Ex(k) produces a sequence {x(k)} converging to the solution of x=b+Ex from any x(0) . The situation for nonlinear systems is quite different. In general, an iterative method for solving a nonlinear system of equations will converge to a solution only from initial approximations which are fairly close to a solution. While a linear system with real coefficients can have, at most, one isolated solution, a nonlinear system can have any number of isolated solutions, depending on the particular nonlinear system. By an “isolated solution” we mean a solution which has a neighborhood containing no other solutions. Clearly, even in one dimension, nonlinear equations (for instance polynomial equations) can have several isolated solutions. This can cause difficulties for iterative methods at points in between. There are many sources of difficulties for iterative methods for nonlinear systems (see the example following Theorem 5.6 for instance). A practical problem is that of finding a safe starting point from which an iterative method will converge to a solution of a nonlinear system. For some systems it may be easy; for others it is extremely difficult. In this chapter we will consider some applications of interval analysis to the design of search procedures for finding safe starting points for iterative methods.