Some Research On Method For Solving Nonlinear Equation

In terms of the objective world, the most natural phenomena in its essence is nonlinear, it is more difficult than the linear problem to solve, the advent of the computer, has greatly increased the human ability to solve the complex nonlinear problems, makes numerical calculation and simulation widely used.Numerical calculation has its limits, however, especially in the treatment of infinite domain, multiple solution and singularity, etc, is very difficult. And analytical approximate method has great advantage in dealing with these problems. Therefore, analytical approximation method and numerical method, have important scientific value, the two complement each other, has been highly appreciated by international academic circles.In this thesis, two aspects were considered:First, differential equation:Based on the NDLT-HPM (nonlinear distribution Laplace-homotopy perturbation algorithm), this thesis proposes a modified NDLT-HPM (referred to as MNDLT-HPM) by introducing parameters, and the introduction of the parameters makes the solution more flexible, and can regulate and control the convergence of series solution domain, overcomes the limitations of NDLT-HPM that the series solution may be not convergent and series solution can converge to the accurate solution when embedding parameter p= 1, thus obtaining accurate enough approximate analytical solution.Second, algebraic equation:The first part:With two kinds of classical iterative format weighted and a kind of five-order iterative format was constructed by Thiele type continued fractions. Besides, based on three kinds of three-order iterative formats proposed by Kou jisheng, three kinds of twelve-order iterative formats were constructed by increasing one step iteration. The second part:Because of the difficulty of solving the roots of some nonlinear equations, on the premise of meeting certain precision, consider solving the scope of the roots is feasible. In this thesis, nonlinear equation and polynomial were connected by the Sequential Number Theoretic Optimization (SNTO) in number theory, besides, the roots of polynomial and matrix eigenvalues were linked by friend matrix, so the range of nonlinear equation roots could be estimated by the study of matrix eigenvalues.