Quasi-Newton vector

The vector p is called the quasi-Newton vector at the point x. If M =H(x), then p is the Newton vector. If M =I then p corresponds to... 

As mentioned above, update formulae have been developed to avoid a frequent evaluation of the Hessian matrix. They modify a given matrix by using quantities which have been employed before in the procedure (gradient and quasi-Newton vector). If a matrix (approximating H(x )) is available, then by an update formula a sequence of matrices M may be generated which can be used instead... 

By considering that p = -M g(x ) (quasi-Newton vector ), the above equation may be rewritten as follows... 

Notice, the right-hand term is essentially determined by the ratio of the gradient vector and the quasi-Newton vector. 

The assumptions of theorem 8 may be fulfilled. If the vector p is equal to the quasi-Newton vector, i.e. 

Thus a quasi-Newton method converges superlinearly if and only if the quasi-Newton vector converges in magnitude and direction to the Newton vector. Consequentlyy the gradient method (i.e. for all k in... 

If x <0 then the calculated quasi-Newton vector p is inconsistent... 

Descent methods are specific (quasi-)Newton methods which look for minimizers only. They differ from the general (quasi-)Newton methods in the line search step which is added to ensure that the procedure makes a sufficient progress in the direction to a minimizer, particularly in the case when the initial guess is far away from a solution. Line search means that at a point x the energy functional E is minimized along the (quasi-)Newton vector p, i.e. a positive value is determined such that... 

This formula is exact for a quadratic function, but for real problems a line search may be desirable. This line search is performed along the vector — x. . It may not be necessary to locate the minimum in the direction of the line search very accurately, at the expense of a few more steps of the quasi-Newton algorithm. For quantum mechanics calculations the additional energy evaluations required by the line search may prove more expensive than using the more approximate approach. An effective compromise is to fit a function to the energy and gradient at the current point x/t and at the point X/ +i and determine the minimum in the fitted function. 

The quasi-Newton methods estimate the matrix = H-1 by updating a previous guess of C in each iteration using only the gradient vector. These methods are very close to the quasi-Newton methods of solving a system of nonlinear equations. The order of convergence is between 1 and 2, and the minimum of a positive definite quadratic function is found in a finite number of steps. 

Alternatively, so-called secant methods can be used to approximate the Jacobian matrix with far less effort (Westerberg et al., 1979). These provide a superlinear rate of convergence that is, they reduce the errors less rapidly than the Newton-Raphson method, but more rapidly than the method of successive substitutions, which has a linear rate of convergence (i.e., the length of the error vector is reduced from 0.1, 0.01, 10 , 10 , 10 , ...). These methods are also referred to as quasi-Newton methods, with Broyden s method being the most popular. 

In general, the error e tic-q-i+j, 0) is a non-linear function of the parameter vector 0. Therefore, the above problem is a well-known nonlinear least squares problem (NLSP) that may be solved by various optimisation algorithms such as the Levenberg-Marquardt algorithm [2], the quasi-Newton method or the Gauss-Newton (GN) algorithm [3]. 

When only the Davidson part of the quasi-Newton step is used, we observe nearly identical convergence if all vectors are retained in the eigenvector subspace. With truncation of the subspace, the convergence of the Davidson method degrades somewhat relative to the quasi-Newton method. Thus, at the equilibrium geometry, the Davidson method converges to Eh in 15 iterations with two vectors in the subspace, compared with the 12 iterations for the quasi-Newton method. 

It should be realized that DUS by itself does not constitute an optimization algorithm. On its own, DUS does not produce an amplitude vector that cannot be written as a linear combination of those already generated. Rather, DUS provides an improved mechanism for utilizing the information contained in the quasi-Newton corrections (13.4.10). As we shall see in Section 13.4.4, the acceleration achieved by the DIIS procedure is often quite dramatic, significantly reducing the number of iterations needed for convergence. 

To avoid the numerous and costly function evaluations required by the finite difference method, various quasi-Newton methods have been developed in which the approximation ftl of the Jacobian is constructed from the recent values of the function vector /( ). /(Jc ),. We describe here the popular 5wy Jenkwefhoii(Broy den,... 

In the study of the performance of Newton s method for a simple 2-D system, we have seen that far away from the solution, the update steps are large and lie in erratic directions. The efficiency and robustness of Newton s method (and quasi-Newton variations such as that of Broyden) are improved dramatically through use of a reduced-step algorithm, in which only a fraction of the update vector is accepted. The full update vector is generated by solving the linear system... 

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