Publisher Summary This chapter presents Newton's and related methods. Newton's method in its modern form is given by the iteration starting with an initial guess x0, and with m determined by some stopping criterion. The iteration process is terminated when accuracy is considered good enough. This is probably the most widely known method for solving equations, although it is not the most efficient, nor the most robust. It is not guaranteed to converge from an arbitrary starting point x0, so it is usually used in conjunction with some other method that is globally convergent. This should give an x0 close enough to the root C for Newton's method to converge. The chapter also describes considerable number of methods, which are usually more efficient than Newton's itself. They are based on the evaluation of f and/or f' at some point(s) other than xi. Also included are some that involve multiplying f and/or fˊ by a power of xi or a constant, as well as some further miscellaneous ones.