Though well-known for its simplicity and efficiency, Newton’s method applied to a complex polynomial can fail quite miserably, even on a relatively large open set of initial guesses. In this work, we present some ana...Though well-known for its simplicity and efficiency, Newton’s method applied to a complex polynomial can fail quite miserably, even on a relatively large open set of initial guesses. In this work, we present some analytic and numerical results for Newton’s method applied to the complex quartic family where is a parameter. The symmetric location of the roots of?allows for some easy reductions. In particular, when λ is either real or purely imaginary, standard techniques from real dynamical systems theory can be employed for rigorous analysis. Classifying those λ-values where Newton’s method fails on an open set leads to complex and aesthetically intriguing geometry in the λ-parameter plane, complete with fractal-like figures such as Mandelbrot-like sets, tricorns and swallows.展开更多
Tensor complementarity problem (TCP) is a special kind of nonlinear complementarity problem (NCP). In this paper, we introduce a new class of structure tensor and give some examples. By transforming the TCP to the sys...Tensor complementarity problem (TCP) is a special kind of nonlinear complementarity problem (NCP). In this paper, we introduce a new class of structure tensor and give some examples. By transforming the TCP to the system of nonsmooth equations, we develop a semismooth Newton method for the tensor complementarity problem. We prove the monotone convergence theorem for the proposed method under proper conditions.展开更多
基金Project supported by Key Industrial Projects of Major Science and Technology Projects of Zhejiang(No.2009C11023)Foundation of Zhejiang Educational Committee(No.Y200907886)Major High-Tech Industrialization Project of Jiaxing(No.2009BY10004)
文摘Though well-known for its simplicity and efficiency, Newton’s method applied to a complex polynomial can fail quite miserably, even on a relatively large open set of initial guesses. In this work, we present some analytic and numerical results for Newton’s method applied to the complex quartic family where is a parameter. The symmetric location of the roots of?allows for some easy reductions. In particular, when λ is either real or purely imaginary, standard techniques from real dynamical systems theory can be employed for rigorous analysis. Classifying those λ-values where Newton’s method fails on an open set leads to complex and aesthetically intriguing geometry in the λ-parameter plane, complete with fractal-like figures such as Mandelbrot-like sets, tricorns and swallows.
文摘Tensor complementarity problem (TCP) is a special kind of nonlinear complementarity problem (NCP). In this paper, we introduce a new class of structure tensor and give some examples. By transforming the TCP to the system of nonsmooth equations, we develop a semismooth Newton method for the tensor complementarity problem. We prove the monotone convergence theorem for the proposed method under proper conditions.
基金’The author gratefully acknowledges the partial supports of the Chinese National Science The author gratefully acknowledges the partial supports of the Chinese National Science Foundation Grant (10071050)the Science Foundation Grant (02ZA14070)Shang
