Newton’s Method in the Context of Gradients
نویسندگان
چکیده
This paper gives a common theoretical treatment for gradient and Newton type methods for general classes of problems. First, for Euler-Lagrange equations Newton’s method is characterized as an (asymptotically) optimal variable steepest descent method. Second, Sobolev gradient type minimization is developed for general problems using a continuous Newton method which takes into account a ‘boundary condition’ operator.
منابع مشابه
THE USE OF THE HE'S ITERATION METHOD FOR SOLVING NONLINEAR EQUATIONS USING CADNA LIBRARY
In this paper, we apply the Newton’s and He’s iteration formulas in order to solve the nonlinear algebraic equations. In this case, we use the stochastic arithmetic and the CESTAC method to validate the results. We show that the He’s iteration formula is more reliable than the Newton’s iteration formula by using the CADNA library.
متن کاملA Levenberg-Marquardt Method based on Sobolev gradients
We extend the theory of Sobolev gradients to include variable metric methods, such as Newton’s method and the Levenberg-Marquardt method, as gradient descent iterations associated with stepwise variable inner products. In particular, we obtain existence, uniqueness, and asymptotic convergence results for a gradient flow based on a variable inner product.
متن کاملComputing the Least Fixed Point of Positive Polynomial Systems
We consider equation systems of the form X1 = f1(X1, . . . , Xn), . . . , Xn = fn(X1, . . . , Xn) where f1, . . . , fn are polynomials with positive real coefficients. In vector form we denote such an equation system by X = f(X) and call f a system of positive polynomials, short SPP. Equation systems of this kind appear naturally in the analysis of stochastic models like stochastic context-free...
متن کاملar X iv : s ub m it / 00 06 49 6 [ cs . N A ] 1 6 M ar 2 01 0 Computing the Least Fixed Point of Positive Polynomial Systems ⋆
We consider equation systems of the form X1 = f1(X1, . . . , Xn), . . . , Xn = fn(X1, . . . , Xn) where f1, . . . , fn are polynomials with positive real coefficients. In vector form we denote such an equation system by X = f(X) and call f a system of positive polynomials, short SPP. Equation systems of this kind appear naturally in the analysis of stochastic models like stochastic context-free...
متن کاملar X iv : 1 00 1 . 03 40 v 2 [ cs . N A ] 1 2 Ja n 20 10 Computing the Least Fixed Point of Positive Polynomial Systems ⋆
We consider equation systems of the form X1 = f1(X1, . . . , Xn), . . . , Xn = fn(X1, . . . , Xn) where f1, . . . , fn are polynomials with positive real coefficients. In vector form we denote such an equation system by X = f(X) and call f a system of positive polynomials, short SPP. Equation systems of this kind appear naturally in the analysis of stochastic models like stochastic context-free...
متن کامل