Algorithms for non-linear simultaneous equations
In Chapter 5, we introduced triangular factorization and forwards and backwards substitution as a direct algorithm for solving linear equations. In a finite number of steps (assuming infinite precision arithmetic) we could solve the linear equations exactly. We have already remarked in Section 126.96.36.199 that no direct algorithm exists for general non-linear simultaneous equations.
Our approach to solving non-linear simultaneous equations will be iterative. We will start with an initial guess of the solution and try to successively improve on the guess. We will continue until the current iterate becomes sufficiently close to the solution according to a suitable criterion. In general, we will not ...