If the nonlinear equation f (x) = 0f (x) = 0, f (x) f(x)f(x) is continuous and strictly monotonic on [a, b] [a, b], f (a) f (b) f (a).

The basic idea of dichotomy is to divide the root-containing interval into two parts step by step, and further search the root-containing interval by judging the sign of the function value at the end of the interval, so that the length of the root-containing interval is reduced to be small enough, thus obtaining the approximate value of the root that meets the given accuracy.

Iterative method, also known as trial and error method, is a process of recursively deriving new values from the old values of variables. Corresponding to iterative method, it is direct method (or one-time solution), that is, one-time solution.

Iterative algorithm is the basic method to solve problems with computers. It makes the computer execute a set of instructions (or some steps) repeatedly by using the characteristics of fast calculation speed and being suitable for repeated operation.

Every time this set of instructions (or these steps) is executed, a new value is deduced from the original value of the variable. Iterative methods are divided into exact iteration and approximate iteration. Typical iterative methods such as "dichotomy" and "Newton iterative method" belong to approximate iterative method.

The main research topic of iterative method is to construct a convergent iterative scheme for the discussed problem, and analyze its convergence speed and convergence range. Convergence theorems of iterative methods can be divided into the following three categories:

① Local convergence theorem: Assuming that the solution of the problem exists, it is concluded that the iterative method converges when the initial approximation is close enough to the solution;

(2) Semi-local convergence theorem: under the condition that the solution does not exist, the iterative method converges to the solution of the problem according to the conditions that the iterative method meets in the initial approximation;

③ Large-scale convergence theorem: Without assuming that the initial approximation is close enough to the solution, it is concluded that the iterative method converges to the solution of the problem.

Iterative method is widely used in solving linear and nonlinear equations, optimization calculation and eigenvalue calculation.