Numerical Algorithms For Two Class Of Nonlinear Matrix Equations

Many integral differential equations arising from physics and engineering ma-terials field can be discretized to the corresponding nonlinear matrix equations.In general,it is difficult to find the exact solutions of these equations in theory,so the computation of numerical solutions is particularly important in practical ap-plications.With the rapid development of computer in recent years,the numerical solution of nonlinear matrix equation has become a very hot topic in computation-al mathematics field.Research of these numerical methods not only contributes to the development of equation theory itself,but also has a great importantly prac-tical significance.In this dissertation,we mainly study the numerical methods for finding solutions of the non-symmetric algebraic Riccati matrix equation in parti-cle transport theory and the nonlinear matrix equation in nano-device modeling field.In Chapter 2,we designs an adaptive mixed nonlinear block-splitting dou-ble Newton iteration for the algebraic Riccati matrix equation from the particle transport theory.This method mainly consists of two switches,one is for detecting whether the current state is in a critical case of low convergence,and the another aims automatically at converting the current iteration,when approaching or in the critical case,to double Newton iteration for acceleration.At the same time,we use the projection theory on the zero space and the range to establish the global convergence of this hybrid iterative method.Numerical experiments show that this iterative method can effectively calculate the minimum nonnegative solution when the algebraic Riccati matrix equation is near or in the critical case.In Chapter 3,we investigate the convergence rate of three predictor-corrector iterative methods for the algebraic Riccati matrix equation from the particle trans-port theory.The idea of three predictor-corrector iterations is to introduce differ-ent preprocessing step before the Newton iteration,resulting in various predictor-corrector iterations.By analyzing the whole iterative process deeply,we construct two different M matrices and prove that these predictor-corrector formats come from various regular splitting of the two M matrices.Based on the above findings,we establish the asymptotic convergence rates of the three predictor-corrector for-mats.In particular,we also theoretically prove that the new predictor-corrector iteration does not converge faster than the simple predictor-corrector iterative method.Our numerical experiment also verifies the convergence rate theorem that we establish.Chapter 4 discusses the numerical solution of nonlinear matrix equations in a class of nano-devices modeling.Aiming at the limitations of the original efficiency index of iterative method,a global efficiency index is proposed.In this way,we start with an infinite block tridiagonal matrix corresponding to the Green’s func-tion,transform the equation into a quadratic polynomial matrix equation and de-sign,via the cyclic reduction,a triple algorithm with cubic convergence rate.The relationship between the developed algorithm and Symplectic structure-preserving property is also discussed.Under proper solvable condition,the convergence and triple convergence rate of the tripling algorithm are established.We also design a low-rank structure tripling algorithm for large scale nonlinear matrix equation by using the low-rank structure derived from the discretizati,on of nano-devices.The iterative format of " kernel update" is established in detail,and a deep analysis of the complexity of the algorithm was carried out.Further error analysis shows that the low-rank tripling algorithm can transfer errors in the same way as the low-rank doubling algorithm.Numerical experiments also indicate that the new presented algorithm,compare with the low-rank doubling algorithm,is able to obtain smaller residuals of the equation with fewer iterations only at the expense of negligible computational time.