Factorization of a characteristic polynomial by random search one of its trinomial divisors

In the paper the application of Monte Carlo approach for random search algorithms to finding a pair of conjugated roots of a characteristic polynomial (ChP) is presented. The proposed technique is oriented only to ChPs with real coefficients. The search is produced on the basis of random numbers with uniform distribution in a rather narrow bounded region of the complex plane. Two search methods are considered here: 1) Indirect method to finding a pair of complex conjugate roots; 2) Direct method which results are coefficients of the trinomial divisor of the ChP. Then the conjugated roots can be easily found as roots of the trinomial divisor. If high precision is necessary then the best random hit is used as good initial estimate for further specification of better results by the iterative Newton's method for two equations with two unknowns (the Jacobian is a 2×2 matrix). The specified techniques can be very easily and simply realized in the environment of MS Excel or MATLAB functions at minimum computing resource. The both proposed here methods can be used in the educational process providing good visualization of the study area. Irrespective of the high degree of ChP it is possible to direct the search process so that to find at once only those roots that are of interest in connection with the problem of stability of the corresponding electrical system. The offered algorithms allow creating effective software products.