Algorithm of Finding All Real Roots Based on Solution Space Compression

Abstract

Finding the approximate solution of linear or nonlinear equations is a mathematical problem, which is often encountered in engineering. As it is known, Galois E. had proven that over five orders algebraic equation was no root-finding formula. The solutions of high-order equation are obtained by some of iterative algorithms, e.g. Binary method and Newton method. To Binary method and Newton method, the function must be continuous, monotone and differentiable. The function must be only simple root in solution space, too. A few of solution subspace have been obtained by dividing solution space according to Binary method's idea. In each subspace, real root will be searched for by algorithm proposed. If subspace has simple root or multi roots, the subspace will be reserved. If no, the subspace will be discarded. The subspace that has solution will be divided again. With increasing of iterations, solution space will be compressed, and converge at the solution of equations. Solution space compression will be finished until all real roots are found. The proposed algorithm is applicable to solve equation's simple root and multi roots problem. As such, it is fit for linear equations and nonlinear equations. Eventually, some test cases illustrate this approach is very available.