Newton process

The most costly operation in the Newton process is the Jacobian evaluation. In quasi-Newton methods, the same Jacobian is maintained during a few iterations, or approximations to J(x) are generated with increasing accuracy as the iterations proceed. 

These methods require the knowledge of the partial derivatives of S(a) or, what amounts to the same thing, of f(xu,a). A method, first proposed by Gauss, is based on the linearization of the model f(xu,a) this method could also be called a Newton process. a(fe) being the fcth approximation of a, it can be written as... 

The variables have been written without apostrophes. At the beginning of the Newton process, they have the old values, while the function F(X) supplies new constant values, driving the calculation. We demand that the corrected vector X f SX satisfies (11.17), that is, that the left-hand side of (11.19) is zero. This leaves... 

Because of the dependence on the axial parameters a and r this system of normal equations has a defect of the dimension two. By introduction of the linearized condition equations (19) and (20) this defect will be eliminated. The searched parameters Y result from a iterative Newton—process. Based on the start parameters Yo in a general i—th step of the iteration the actual solutions Yi = Yi-i + dYi will be calculated. After a new bnearization of A(Yi) a new least square solution will be determined. So the iteration of the unknowns reads as follows... 

It is simply Gaufi-Newton process of stationary solution of non-linear system. Unfortunately, the estimated points positions are in general stochastically biased quantities, because on the whole expected value (rfc) is the error of interpolative polynomial. But if we had assumed, that is a certain moment tk of synchronized observations in the whole network, then k = tk) would be an error of observation, for which the assumption E ek = 0 was adequate. Biased estimators are therefore the cost that we bear when we decide on kinematic network model with a system of non synchronic observations. 

Hence, the classical Newton process converges very rapidly if the initial guess belongs to the domain of attraction of a stationary point or if an iterate falls into this domain. Exactly that property... 

As mentioned above, the Newton process is a local procedure, i.e. convergence occurs only if the initial guess x° is chosen sufficiently closed to a stationary point (see theorem 3). When a minimization problem is under consideration, the applicability of the Newton method can essentially be extended if the characteristic of a minimizer is... 

Hessian matrix H(x) at a point x. The Newton process, however,... 

As explained at the beginning of this section, a quasi-Newton method is obtained when in the classical Newton process the Hessian matrix is approximated by estimate matrices which are computed by an update formula. Subsequently a few results pertaining to the behaviour of convergence of some quasi-Newton methods are presented. Furthermore, the applicability of these methods for locating minimizers and/or saddle points of energy functionals is discussed. 

Remark 3 In contrast to the classical Newton process, a quasi-Newton procedure is not self-correcting, i.e. the errors are accumulated, because a matrix k depends on all preceding matrices i=0(l)k-l. 

Convergence properties, comparable with those of the Newton process, cannot be expected for quasi-Newton methods, because a certain charge must be paid for the use of estimates to the Hessian matrix. But superlinear convergence, at least, should occur. 

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