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Newtons method

The slope or derivative of f(x) is computed at an initial guess, a, for the root of f x) = 0. The new value of the root, 6, is computed based on a first-order Taylor Series expansion of f x) about the initial guess, a, [Pg.80]

this method is iterative, but it only requires one initial guess. An important advantage of the Newton method is its rapid convergence. [Pg.80]

Two disadvantages are that it requires computation of the derivative and that it can be very sensitive to the initial guess (tends to blow up with poor initial guesses). [Pg.81]

Applying Newton s method to the same example gives [Pg.81]

pre-processing, set default tolerance and maximum iteration 7, number [Pg.82]


In these methods, also known as quasi-Newton methods, the approximate Hessian is improved (updated) based on the results in previous steps. For the exact Hessian and a quadratic surface, the quasi-Newton equation and its analogue = Aq must hold (where - g " and... [Pg.2336]

T. Schlick and M. L. Overton. A powerful truncated Newton method for potential energy functions. J. Comp. Chem., 8 1025-1039, 1987. [Pg.260]

P. Derreumaux, G. Zhang, B. Brooks, and T. Schlick. A truncated-Newton method adapted for CHARMM and biomolecular applications. J. Comp. Chem., 15 532-552, 1994. [Pg.260]

D. Xie and T. Schlick. Efficient implementation of the truncated Newton method for large-scale chemistry applications. SIAM J. Opt, 1997. In Press. [Pg.260]

Another option is a q,p) = p and b q,p) = VU q). This guarantees that we are discretizing a pure index-2 DAE for which A is well-defined. But for this choice we observed severe difficulties with Newton s method, where a step-size smaller even than what is required by explicit methods is needed to obtain convergence. In fact, it can be shown that when the linear harmonic oscillator is cast into such a projected DAE, the linearized problem can easily become unstable for k > . Another way is to check the conditions of the Newton-Kantorovich Theorem, which guarantees convergence of the Newton method. These conditions are also found to be satisfied only for a very small step size k, if is small. [Pg.285]

Quantum mechanical calculations are restricted to systems with relatively small numbers of atoms, and so storing the Hessian matrix is not a problem. As the energy calculation is often the most time-consuming part of the calculation, it is desirable that the minimisation method chosen takes as few steps as possible to reach the minimum. For many levels of quantum mechanics theory analytical first derivatives are available. However, analytical second derivatives are only available for a few levels of theory and can be expensive to compute. The quasi-Newton methods are thus particularly popular for quantum mechanical calculations. [Pg.289]

A transition structure is, of course, a maximum on the reaction pathway. One well-defined reaction path is the least energy or intrinsic reaction path (IRC). Quasi-Newton methods oscillate around the IRC path from one iteration to the next. Several researchers have proposed methods for obtaining the IRC path from the quasi-Newton optimization based on this observation. [Pg.154]

Peng, C. and Schlegel, H.B., Combining Synchronous Transit and Quasi-Newton Methods to Find Transition States , Israel Journal of Chemistry, Vol. 33, 449-454 (1993)... [Pg.65]

In HyperChem, two different methods for the location of transition structures are available. Both arethecombinationsofseparate algorithms for the maximum energy search and quasi-Newton methods. The first method is the eigenvector-following method, and the second is the synchronous transit method. [Pg.308]

The synchronous transit method is combined with quasi-Newton methods to find transition states. Quasi-Newton methods are very robust and efficient in finding energy minima. Based solely on local information, there is no unique way of moving uphill from either reactants or products to reach a specific reaction state, since all directions away from a minimum go uphill. [Pg.309]

Techniques used to find global and local energy minima include sequential simplex, steepest descents, conjugate gradient and variants (BFGS), and the Newton and modified Newton methods (Newton-Raphson). [Pg.165]

Then one can apply Newtons method to the necessaiy conditions for optimahty, which are a set of simultaneous (non)linear equations. The Newton equations one would write are... [Pg.486]

A simple way of finding the roots of an equation, other than by divine inspiration, symmetry or guesswork is afforded by the Newton method. We start at some point denoted x l along the x-axis, and calculate the tangent to the curve at... [Pg.234]

The most frequently used methods fall between the Newton method and the steepest descents method. These methods avoid direct calculation of the Hessian (the matrix of second derivatives) instead they start with an approximate Hessian and update it at every iteration. [Pg.238]

In a recent version, the LST or QST algorithm is used to find an estimate of the maximum, and a Newton method is then used to complete the optimization (Peng and Schlegel, 1993). [Pg.250]

The Synchronous Transit-Guided Quasi-Newton Method(s)... [Pg.251]

The modified Newton method [12] offers one way of dealing with multiple roots. If a new function is defined... [Pg.70]

Kinetic curves were analyzed and the further correlations were determined with a nonlinear least-square-method PC program, working with the Gauss-Newton method. [Pg.265]

As seen in Chapter 2 a suitable measure of the discrepancy between a model and a set of data is the objective function, S(k), and hence, the parameter values are obtained by minimizing this function. Therefore, the estimation of the parameters can be viewed as an optimization problem whereby any of the available general purpose optimization methods can be utilized. In particular, it was found that the Gauss-Newton method is the most efficient method for estimating parameters in nonlinear models (Bard. 1970). As we strongly believe that this is indeed the best method to use for nonlinear regression problems, the Gauss-Newton method is presented in detail in this chapter. It is assumed that the parameters are free to take any values. [Pg.49]

In this chapter we are focusing on a particular technique, the Gauss-Newton method, for the estimation of the unknown parameters that appear in a model described by a set of algebraic equations. Namely, it is assumed that both the structure of the mathematical model and the objective function to be minimized are known. In mathematical terms, we are given the model... [Pg.49]

Minimization of S(k) can be accomplished by using almost any technique available from optimization theory. Next we shall present the Gauss-Newton method as we have found it to be overall the best one (Bard, 1970). [Pg.50]

More elaborate techniques have been published in the literature to obtain optimal or near optimal stepping parameter values. Essentially one performs a univariate search to determine the minimum value of the objective function along the chosen direction (Ak ) by the Gauss-Newton method. [Pg.52]

Formulation of the Solution Steps for the Gauss-Newton Method Two Consecutive Chemical Reactions... [Pg.53]

Equations 4.14 and 4.15 are used to evaluate the model response and the sensitivity coefficients that are required for setting up matrix A and vector b at each iteration of the Gauss-Newton method. [Pg.54]


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2N Newton methods

2N Newton-Raphson methods

A Global Newton-like Method

Block-diagonal Newton-Raphson method

Broyden’s quasi-Newton method

By Newton-Raphson method

Computational methods Newton-Raphson

Constrained Gauss-Newton Method for Regression of Binary VLE Data

Damped Newton method

Energy minimisation methods Newton-Raphson

Equivalence of Gauss-Newton with Quasilinearization Method

Equivalence to Gauss-Newton Method

Estimating the Jacobian and quasi-Newton methods

Finding roots, Newton method

Formulation of the N(r 2) Newton-Raphson Method

Function Newton s method

Gaufi-Newton method

Gauss-Newton Method for Algebraic Models

Gauss-Newton Method for Partial Differential Equation (PDE) Models

Gauss-Newton method

Gauss-Newton method, nonlinear least-squares

General functions Newton-Raphson methods

Global Newton methods

Global Newton’s method

Globalizing the convergence of Newtons Method

Inexact Newton methods

Interpolation Newton method

Kinetic Newton methods

Level shifting Newton-Raphson methods

Modified Newton methods

Modified Newton methods matrix

Modified Newton-Raphson Method

Molecular mechanics Newton-Raphson method

Multidimensional Newtons Method

N Newton Methods

Newton eigenvector method

Newton inversion method

Newton iteration method

Newton line search methods

Newton method regularized

Newton methods for large problems

Newton s method

Newton-Euler method

Newton-Raphson iterative method

Newton-Raphson method Almost Band Algorithm

Newton-Raphson method convergence

Newton-Raphson method equilibrium calculations

Newton-Raphson method in multidimensions

Newton-Raphson method multivariable

Newton-Raphson method tolerance

Newton-Raphson methods characteristics

Newton-Raphson methods extrapolations

Newton-Raphson methods minima

Newton-Raphson methods optimization techniques

Newton-Raphson methods saddle points

Newton-Raphson root-squaring method

Newton-like method

Newton-like method global

Newton-raphson method

Newton-type method

Newton/sparse matrix methods

Newtons Method and Parallel Computations

Newtons Method for Simultaneous Nonlinear Equations

Newtons method for a single equation

Newtons method for multiple nonlinear equations

Newtons method in two variables

Newton—Raphson method composition

Newton—Raphson method structure

Newton’s gradient method

Newton’s iteration method

Newton’s method convergence

Nonlinear Gauss-Newton method

Nonlinear modified Newton methods

Nonlinear quasi-Newton methods

Numerical methods Newton-Raphson method

Optimization quasi-Newton methods

Pseudo-Newton-Raphson methods

Quasi-Newton convergence method

Quasi-Newton methods

Quasi-Newton methods BFGS optimization

Quasi-Newton methods algorithm

Quasi-Newton methods examples

Quasi-Newton methods procedures

Quasi-Newton methods updating Hessian matrix

Quasi-Newton methods with unit Hessian

Robust reduced-step Newton method

Second Derivative Methods The Newton-Raphson Method

Shooting Newton-Raphson method

Solving linear equations (Newtons method)

System of implicit non-linear equations the Newton-Raphson method

The 2N Newton-Raphson Method

The Gauss-Newton Method

The Gauss-Newton Method - Nonlinear Output Relationship

The Gauss-Newton Method for Discretized PDE Models

The Gauss-Newton Method for PDE Models

The Newton method

The Newton-Raphson Method

The Newton-Raphson method applied to solutions

The regularized Newton method

The trust-region Newton method

Truncated Newton method

Truncated Newton-Raphson optimization method

Trust-region Newton method

Trust-region Newton optimization method

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