Modified Newtons Method

In these methods, the Jacobian matrix is either evaluated analytically or approximated numerically. Since the Jacobian is recalculated at each iteration, the linear system (7.38) must also be solved. 

Several expedients and precautions are required to scale down the drawbacks of Newton s method. 

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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). 

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

Modified Newton methods require calculation of second derivatives. There might be cases where these derivatives are not available analytically. One may then calculate them by finite differences (Edgar and Himmelblau, 1988 Gill et al. 1981 Press et al. 1992). The latter, however, requires a considerable number of... 

When f is nonlinear, as it nearly always is, then an iteration is required to determine y +i. For stiff problems, the iterative solution is usually accomplished with a modified Newton method. We seek the solution yn+i to a nonlinear system that may be stated in residual form as... 

Program a modified Newton method to solve the problem, seeking the solution near x 0.5. Explore the performance of the algorithm (including failure to converge) beginning with initial iterates of xo = 1 and xo = 3. 

Many variations of the correction method we have proposed can be used. Among these are the use of the same Jacobian for several iterations and the use of modified Newton methods such as Marquardt s method (8). We have tried Marquardt s method on some of these problems without observing any significant improvement, but this is only a tentative evaluation. Improved methods for generating starting conditions would be helpful. 

The modified Newton methods evaluate the Hessian either analytically or by a numerical approximation in x and solve the linear system with a direct method that exploits matrix symmetry. 

The gradient approaches zero in the neighborhood of the solution. If it is inaccurate, numerical problems may arise either in the evaluation of the second-order derivatives (modified Newton methods) or in updating the Hessian (quasi-Newton methods). 

The main difference between the modified Newton and quasi-Newton methods is in the evaluation of the Hessian the modified Newton methods approximate the matrix using local information in the neighborhood of xj the quasi-Newton methods update the matrix using gradient values evaluated at each iteration. 

The only remaining open issue is to analyze the strategies to limit the Newton method s shortcomings (as already proposed for the modified Newton methods) ... 

While an in-depth one-dimensional search was executed in the old programs based on these algorithms to assure the theoretical basis of the method, nowadays it is well known that a limited search, such as the one described for the modified Newton methods, is computationally more high performance. 

It is worth noting that the rank-1 formula (3.147) was recently revalued for its quadratic termination, even though the one-dimensional searches are nonexact. To implement it satisfactorily, you need to make sure that the matrices are preserved as positive definite. If the one-dimensional search is inefficient, it is opportune to adopt an alternative method. In line with the previous discussions, it is possible to use either a heuristic method or a modified Newton method. 

For modified Newton methods, it is sufficient to have a positive definite Hessian. 

The first class comprises the modified Newton methods where the Jacobian is evaluated analytically or numerically approximated in x,. 

However, the modified Newton methods still have two of the shortcomings of Newton s method when a single processor is available ... 

Therefore, it is not advisable to implement a quasi-Newton method without a modified Newton method. 

Quasi-Newton methods are less predisposed to parallel computations than modified Newton methods. 

Since these underdimensioned nonlinear systems will also be solved using a modified Newton method, it is essential to tackle the subproblem of the solution of underdimensioned linear systems. 

A comparison of modified Newton methods for unconstrained optimization. Computer Journal, 14, 293—294. 

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