Newton-Raphson sequence

A wide variety of summaries that (a) start at the median and (b) make an adjustment that would be the first step of a Newton-Raphson sequence tending to the minimum of some more reasonable potential are known to be broadly effective in the simple case.1 For how complicated a situation will taking just one step do so well As yet we have little evidence. 

A Newton/Raphson iteration yields the value for Bj. An average molecular weight yields the value of Aj. In reference to Figure 1, the calibration sequence is now completed. Block data storage incorporates and E. ... 

A sequence of Newton-Raphson iterations is obtained by solving equation (4 4) redefining the zero point, p0, as the new set of parameters recalculating g and H and returning to equation (4 4). Such a procedure converges quadratically, that is, the error vector in iteration n is a quadratic function of the error vector in iteration n-1. This does not necessarily mean that the NR procedure will converge fast, or even at all. However, close to the stationary point we can expect a quadratic behaviour. We shall return later to a more precise definition of what close means in this respect. 

Berna, T. J. Westerberg, A. W., "Polynomial, Chao-Seader and Newton Raphson - The Use of Partially Ordered Pivot Sequences" DEC 06-l-79 Dept, of Chem. Eng., Carnegie-Mellon University, Pittsburgh, Penn. 15213 (January 1979). 

The time step, At, is used to switch the method from being a relaxation method to a global Newton method. When the time step is small, e.g., if = 0.1, then the changes in the independent variables are small. The method performs like a damped Newton-Raphson method, where the steps are small but in the direction of the solution and without any oscillation. When the value of At is large, i.e., At = 1000, the method performs like a Newton-Raphson method. The value of At at each column trial determines the speed and stability of the method, The units of the time step are the same as the flows to and from the column. The calculation sequence of the Ketchum method is as follows ... 

This alternate form of the equations produces a faster convergence as shown in an example given by Wigley (41) and also converges more rapidly than Newton-Raphson. EQTemploys an additional control on the continued fraction method which generates monotone sequences (43,44). Its chief virtues are strict error bounds and increased stability with respect to a range of analyses of aqueous solutions used as input. 

Iteration and convergence method explicit equations Monotone sequences and secant method Newton- Raphson Free ion molali-ties by difference Newton- Raphson conti nued fraction Newton- Raphson Newton-Raphson conti nued fraction conti nued fraction for anions only conti nued fraction conti nued fraction conti nued fraction brute force... 

The convergence of the successive substitution method is slow, i.e., it may require many iterations for the sequence to converge. The Newton-Raphson method has a faster rate of convergence, which is given as follows ... 

Formulas for bL x and 6, 2 are given by Eqs. (6-24) and (6-25), respectively, and Bx and B2 denote the specified values. The desired values of and 02 may be found by use of the Newton-Raphson method.5 After the 0 s have been determined, the corrected 6, s and d, s needed to initiate the next trial on the system are readily computed. Details pertaining to the sequence of calculations are presented below. 

After the rjfs that made each Rj = 0 had been found, they were used together with the most recent set of 7 s and 0/s to evaluate the elements of J and f of Eq. (8-45). This sequence of calculations in the application of the Newton-Raphson method is called procedure 1. 

From the definition of the vectors X0 of R, observe that each vector of the sequence Xt, X2,Xk, X + u. .. generated by the Newton-Raphson method on the basis of any X0 of R is also a member of R. 

Calculation of the inverse Hessian matrix can be a potentially time-consuming operation that represents a significant drawback to the pure second derivative methods such as Newton-Raphson. Moreover, one may not be able to calculate analytical second derivatives, which are preferable. The quasi-Newton methods (also known as variable metric methods) gradually build up the inverse Hessian matrix in successive iterations. That is, a sequence of... 

Since the bubble enriched velocity is zero at the boundary of each element, the nodal values of are in fact the required solution. The unknown does not need to be solved. Here Kn is a function of due to the shear-rate dependent viscosity. This makes the system of equations non-linear. One may simply solve the equations using a Picard iteration where Kn is evaluated using the value obtained from the previous step and so a sequence of linear problems is solved. Alternatively, some authors (e.g., Pichelin and Coupez 1998) have used the Newton-Raphson iteration. 

Quasilinearization is a technique where nonlinear differential equations are solved by obtaining a sequence of solutions to related linear equations. The method somewhat resembles a generalized Newton-Raphson method. The most important development in the quasilinearization technique was the use of the maximum operation to prove that the representation of the original nonlinear equations by a sequence of linear equations converges to the nonlinear equation. This result is due to Bellman, who also used the concept in his development of dynamic programming (Bellman, 1957). 

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