A Modified Damped Newton Method And Its Acceleration

Damped Newton method and Newton method hae the advantages like fast convergence and simple iteration. So it has attracted many people’s attention. But it has some disadvantages. For example, it need to calculate the second derivative matrix and its inverse. So 2()k?f x must be nonsingular and positive definite. Otherwise, the algorithm can not generate new iteration point and iteration cann’t go on. In this paper, we are aimed at the above shortcomings of the damped Newton method. The damped Newton method was revised. We obtained a new iterative method(modified damped Newton method). In other words, we will use a matrix()kQ x +aI to replace 2()k?f x of the damped Newton formula. The corresponding iterative formula becomes[ ]11()()k k k k kx x lQ x aI f x-+= - + ?, where()kQ x is a matrix, I is a unit matrix andkl is positive constant. Thus the modified damped Newton iterative direction become into[ ]1()()k k kp Q x aI f x-= - + ?. Hence, for any given initial value, when the inverse of two order derivative matrix in the formula of damped Newton method does not exist or not the positive definite. This kind of modified damped Newton method can continue to iterate. The optimal point or near the optimal point.In this paper, we start from the search direction and illustrates that the search direction of the new algorithm 1[()]()k k kp M x f x-= - ? is a declined direction. According to the convexity of the objective function and its Taylor expandtionary formula, one can get the next iteration point k1x+ is a good approximation of optimum point *x. Then from that local and global aspects, the convergence of the modified damped Newton mothod is analyzed. We know that the modified damped Newton mothod under certain conditions has at least two order convergence. The end of the third chapter of the numerical test of the modified damped Newton mothod is given. The calculation results are compared with calculation results of the Newton mothod. The results shows that the convergence speed of the modified damped Newton mothod is faster than the Newton mothod.The fourth chapter of this paper, we also accelerate the modified damped Newton method and obtain a faster convergence algorithm——the modified damped Newton method after accelerated, called the JS method, and the convergence is analyzed by numerical examples. The results show that the JS method converges speed is faster than the modified damped Newton method.