Gradient method definition

The final system of equations is S3munetric, positive definite, hence the solution of such a system can be obtained in an efficient way using standard matrix solvers like conjugate gradient method. 

The Levenberg-Marquardt method is able to move between Newton s method and the gradient method. This feature will be discussed later conversely, it is now important to consider this method for removing the problem of a nonpositive definite Hessian matrix. 

The conjugate gradient method is slower and less accurate than direct factorization when the positive definite matrix is dense as well as when it has a particular structure (i.e., a band matrix). 

Gradient methods are very efficient for minimizations however, to locate transition structures they must be modified or constrained to overcome the problems caused by the Hessian not being positive-definite. One approach is to partition the AT-dimensional optimization into a one-dimensional space for maximization, and an (N — l)-dimensional space for minimization. This partitioning in effect chooses a transition vector. The transition vector may be fixed, or may be allowed to vary in a restricted manner (e.g. according to a quadratic synchronous transit path). The search for a maximum in the... 

We now consider the origin of the excellent performance of the conjugate gradient method for a quadratic cost function, defined in terms of a symmetric, positive-definite matrix A and a vector b,... 

The conjugate gradient method for positive-definite matrices... 

The efficiency ofthe conjugate gradient method leads us to consider the existence ofmethods that solve linear systems in a similar manner but without requiring positive-definiteness. Such methods exist and are invoked in MATLAB by the keywords bieg, biegstab, and gmres. 

Previously, when solving the Poisson equation with Dirichlet boundary conditions, we obtained a matrix that was positive-definite and could be solved witti the conjugate gradient method. For this problem, however, we have a number of von Neumann boundary conditions, e.g. at the grid points, (x , = 0, ft), for which an approximation of the boundary... 

This makes it desirable to define other representations in addition to the electronically adiabatic one [Eqs. (9)-(12)], in which the adiabatic electronic wave function basis set used in the Bom-Huang expansion (12) is replaced by another basis set of functions of the electronic coordinates. Such a different electronic basis set can be chosen so as to minimize the above mentioned gradient term. This term can initially be neglected in the solution of the / -electionic-state nuclear motion Schrodinger equation and reintroduced later using perturbative or other methods, if desired. This new basis set of electronic wave functions can also be made to depend parametrically, like their adiabatic counterparts, on the internal nuclear coordinates q that were defined after Eq. (8). This new electronic basis set is henceforth refened to as diabatic and, as is obvious, leads to an electronically diabatic representation that is not unique unlike the adiabatic one, which is unique by definition. 

Steepest descent can terminate at any type of stationary point, that is, at any point where the elements of the gradient of /(x) are zero. Thus you must ascertain if the presumed minimum is indeed a local minimum (i.e., a solution) or a saddle point. If it is a saddle point, it is necessary to employ a nongradient method to move away from the point, after which the minimization may continue as before. The stationary point may be tested by examining the Hessian matrix of the objective function as described in Chapter 4. If the Hessian matrix is not positive-definite, the stationary point is a saddle point. Perturbation from the stationary point followed by optimization should lead to a local minimum x. ... 

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