Quasi-Newton methods algorithm

I he eigenvector-following (or Hessian mode) method implemented in HyperChem is based on an effieien t quasi-Newton like algorithm for loca tin g tran sitiori states, wh ieh can locate tran si-tion states for alternative rearran gern eri t/dissoeiation reactions, even when startin g from th e wron g regio n on th e poten tial en ergy surface. 

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. 

These methods utilize only values of the objective function, S(k), and values of the first derivatives of the objective function. Thus, they avoid calculation of the elements of the (pxp) Hessian matrix. The quasi-Newton methods rely on formulas that approximate the Hessian and its inverse. Two algorithms have been developed ... 

There Eire other Hessian updates but for minimizations the BFGS update is the most successful. Hessism update techniques are usually combined with line search vide infra) and the resulting minimization algorithms are called quasi-Newton methods. In saddle point optimizations we must allow the approximate Hessian to become indefinite and the PSB update is therefore more appropriate. 

Quasi-Newton methods may be used instead of our full Newton iteration. We have used the fast (quadratic) convergence rate of our Newton algorithm as a numerical check to discriminate between periodic and very slowly changing quasi-periodic trajectories the accurate computed elements of the Jacobian in a Newton iteration can be used in stability computations for the located periodic trajectories. There are deficiencies in the use of a full Newton algorithm, such as its sometimes small radius of convergence (Schwartz, 1983). Several other possibilities for continuation methods also exist (Doedel, 1986 Seydel and Hlavacek, 1986). The pseudo-arc length continuation was sufficient for our calculations. 

In quasi-Newton methods, a parametrized estimate of F or G is used initially, then updated at each iterative step. Saved values of q, p can be used in standard algorithms such as BFGS, described above. 

Quasi-Newton methods form an interesting class of algorithms that are theoretically closely related to nonlinear CG methods.6 95 96 They are found to perform very well in practice.6 100-102 109 110 QN research has been developing... 

In this paragraph, a Lesaint-Raviart method is presented. A Newton algorithm allowing fixed values of the viscoelastic extra-stress components outside the finite elements is used. A fixed-point algorithm on those exter extra-stress components is also involved. Tffis quasi-Newton method needs a storage requirement of the same size as that related to a classical decoupled method, but allows improved convergence [39]. 

There are a number of variations on the Newton-Raphson method, many of which aim to eliminate the need to calculate the full matrix of second derivatives. In addition, a family of methods called the quasi-Newton methods require only first derivatives and gradually construct the inverse Hessian matrix as the calculation proceeds. One simple way in which it may be possible to speed up the Newton-Raphson method is to use the same Hessian matrix for several successive steps of the Newton-Raphson algorithm with only the gradients being recalculated at each iteration. 

The BzzMinimizationQuasiNewton class is designed to solve unconstrained multidimensional minimization problems using the quasi-Newton method as the principal algorithm. If the algorithm does not converge even though... 

The BzzNonLinearSystem class is designed to solve nonlinear systems using a quasi-Newton method as the main algorithm and by availing of all the devices... 

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]. 

The conjugate gradient (CG) algorithm is one of the older methods and, strictly speaking, is not a quasi-Newton method. However, it is the method of choice for very large problems where the storage of the Hessian is not practicable. In the Fletcher-Reeves approach the search direction is given by... 

To relate the conjugate gradient algorithm to the quasi-Newton methods this formula can be expressed as an updating scheme for the inverse Hessian ... 

Cmbined methds. There are numerous other methods in the literature for finding transition states. However, the more common methods use simpler numerical algorithms in a more efficient way. The Berny optimization algorithm and the synchronous transit quasi-newton method (STQN) are good examples. 

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