Convex Combination Algorithms For Solving Monotone Nonlinear Equations

Based on the MPRP algorithm and the CGD algorithm for solving nonlinearmonotone equations, we propose some convex combination algorithms to solvenonlinear monotone equations. We discuss the algorithms in four aspects as the staticconvex combination of the descent direction, the dynamic convex combination of thedescent direction, the static convex combination of the next iteration point and thedynamic convex combination of the the next iteration point, then prove the globalconvergences of all algorithms, finally we do some numerical experiments to verifythe effectiveness of the algorithms and compare with the CGD algorithm and theMPRP algorithm, the results show that the convex combination algorithms are someextent better than the original algorithms. At the same time, we find that the dynamicconvex combination algorithms are better than the static combination algorithms bythe numerical results of the two types of algorithms.In chapter1, we simply reviewe the contents of the conjugate gradient methodsand the montone nonlinear equations, introduce the CGD algorithm and the MPRPalgorithm for solving monotone nonlinear equations.In chapter2, we propose a static convex combination algorithm of the descentdirection for solving monotone nonlinear equations and prove the global convergenceof the algorithm, then verify the proposed algorithm by numerical experiments.In chapter3, after we consider the combination factors through dynamic selected,we propose a dynamic convex combination algorithm of the descent direction forsolving nonlinear monotone equations and compare the results of numericalexperiments with the results in chapter2. we discover that the results of numericalexperiments of the dynamic convex combination algorithm of the descent directionare superior than the results of numerical experiments of the static convexcombination algorithm of the descent directionIn chapter4, we mainly consider the static convex combination of the nextiteration point and strictly prove the global convergence of the algorithm, then testthe algorithm by numerical experiments. The numerical experiments show that thealgorithm is an effective algorithm.In chapter5, similar to chapter3, we consider the combination coefficients fromstatic into dynamic, we propose a dynamic convex combination algorithm of the next iteration point and test the algorithm by numerical experiments, we find that thealgorithm is better than the algorithm in chapter4by numerical experiments.