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Non-Linear Systems
Published in A. C. Faul, A Concise Introduction to Numerical Analysis, 2018
We return to the one-dimensional case for a comparison. Newton’s method and the secant method are related in that the secant method replaces the derivative in Newton’s method by a finite difference. f′(x(n))≈f(x(n))−f(x(n−1))x(n)−x(n−1).
Numerical Analysis
Published in Erchin Serpedin, Thomas Chen, Dinesh Rajan, Mathematical Foundations for SIGNAL PROCESSING, COMMUNICATIONS, AND NETWORKING, 2012
The convergence of Newton’s method is guaranteed only when the initial guess is close to the root. Starting guesses farther away from the root slow down the convergence considerably. Newton’s method displays convergence of a first order method when the iterates are far from the root. A bad initial guess can also lead to divergence of the iterations. A small value of f′ (x) can cause the iterates to move very far from the root leading to divergence or to convergence to another root. In the example presented earlier, an initial guess x0=3/2 leads to breakdown of the method since f′ (x0) = 0.
One-Dimensional Optimization Problem
Published in Nita H. Shah, Poonam Prakash Mishra, Non-Linear Programming, 2020
Nita H. Shah, Poonam Prakash Mishra
Newton’s method, also known as the Newton–Raphson method, is a method of finding roots. Using this concept we can find root of f′(x) and this point would be local minima of f(x). In case initial approximation is not close to x*, it may diverge.
New versions of Newton method: step-size choice, convergence domain and under-determined equations
Published in Optimization Methods and Software, 2020
Consider nonlinear equation written via the vector function . There exists the huge bunch of literature on solvability of such equations and numerical methods for their solution, see e.g. the classical monographs [4,22]. One of the most powerful methods is Newton method, going back to such giants as Newton, Cauchy, Fourier. The general form of the method is due to Kantorovich [14]; on history and references see [5,15,26,33]. The basic version of Newton method for (1) is applicable when is differentiable and is invertible (this implies m = n): The method converges under some natural conditions, moreover it can be used for obtaining existence theorems for the solution (see references cited above). Unfortunately Newton method converges only locally: it requires a good initial approximation (so called ‘hot start’). Convergence conditions can be relaxed for damped Newton method with .
Analytical Investigations of Reinforced Concrete Beam–Column Joints Constructed Using High-Strength Materials
Published in Journal of Earthquake Engineering, 2020
The Newton–Raphson method was applied to solve the nonlinear equations. In the regular Newton–Raphson method, the stiffness relation is evaluated in each iteration step. This method converges to the final answer within a few iterations as a quadratic convergence characteristic is shown by this method. Generally, the Newton–Raphson method needs a few iterations to converge, but each iteration step needs time to be done. On the other hand, the Quasi-Newton method (Secant method) uses the information of previous solution vectors and out-of-balance force vectors in order to come up with a better approximation of the solution. Iteration steps in this method are performed much faster compared to the regular Newton–Raphson method, which is desirable for the parametric investigation in this study. Hence, the Quasi-Newton iteration method is used to make a balance between the solution accuracy and the required computational effort.
Asphalt concrete master curve using dynamic backcalculation
Published in International Journal of Pavement Engineering, 2022
Gabriel Bazi, Tatiana Bou Assi
In numerical analysis, the Newton's method – also known as the Newton-Raphson method – is a root-finding algorithm which produces iteratively better approximations to the roots of a function. In optimisation, the Newton’s method is used to determine the stationary points of a function (minima or maxima) by finding the roots of the function’s first derivative. The Newton’s method uses the first few terms of the Taylor series (equation 9) of a real-valued function at .For finding the roots of a function, the linear approximation (first two terms) of the Taylor series are used and is set equal to zero (equation 10). The offset needed to improve the root for every iteration is shown in equation 11.For finding the stationary points (minima or maxima), the function’s first derivative is determined using the quadratic approximation (first three terms) of the Taylor series and set equal to zero. The offset , shown in equation 12, is calculated using the first and second derivatives.The process starts with an initial guess and a better approximation of the root is obtained after every iteration. This process is repeated as many times as necessary to get the desired accuracy. Figure 4 provides an illustration of how the root of a function f(x) = x3 + x−1 is better approximated after every iteration starting with an initial guess x0 = −0.6 and using equations 10 or 11 (Sauer 2012).