Convergence Newton’s method

In practical implementations Newton s method converges with any prescribed accuracy e only if... 

Convergence of Newton s method. We are now in a position to find out the conditions under which Newton s method converges. With this aim, the differences... 

Numerical calculations for 7 = 5/3 (a = 1.5) showed that the itera.-tions within the framework of Newton s method converge even if the steps r are so large that the shock wave runs over two-three intervals of the grid W/j in one step r. Of course, such a large step is impossible from a computational point of view in connection with accuracy losses. Thus, the restrictions imposed on the step r are stipulated by the desired accuracy rather than by convergence of iterations. 

Will Newton s method converge for these functions ... 

After the desired value of 6 has been obtained, (/ ,)co may be computed by use of Eq. (2-25). (Note, Newton s method converges to the positive root of g(9), provided 6 — 0 is taken to be the first trial value see Prob. 2-11). For the case where the dew-point temperature of the distillate is specified instead of the distillate rate D, the g function has the form shown in Prob. 2-13. 

Whenever Newton s method converges, its convergence rate is quadratic. 

Sufficiently close to the optimizer, Newton s method converges quadratically. To see what this means, let Xjt be a sequence of points converging to x (obtained, for example, by a sequence of Newton steps)... 

Before leaving this simple example, we notice that for the function Eq. (23) Newton s method converges cuhically towards the optimizer x = 0, see Eq. (25). This occurs since the Gaussian is an even function in x around x which implies that its third derivative vanishes at x. ... 

When Newton s method converges, i.e. for large values of n, quantities of the order of — x can be neglected in comparison to unity in the relationship (A1.9), then it will converge rapidly in the vicinity of the solution. Towards the end of the iteration process, the number of valid digits is practically doubled with every iteration step. 

Quadratic convergence is often the fastest reference behavior for large-scale functions. Tensor methods based on fourth-order approximations to the objective function can achieve more rapid convergence but they are restricted to problems of less than about 100 variables. A recent proposal has also shown that Newton s method converges faster than quadratic if higher-order derivatives are used. ... 

Owing to the special structure of the Cl energy function, Newton s method converges even faster than quadratically for Cl energies. According to Exercise 11.4.3, the third derivative at c = Cex with C at the origin becomes... 

The calculation procedure for Newton s method is almost the same as that of Gauss-Newton method with the exception that the vector of corrections to the parameters is calculated from Eq. (7.173). If O is quadratic with respect to b (that is, linear regression), Newton s method converges in only one step. Like all other methods applying Newton s technique for the solution of the set of nonlinear equations, a relaxation factor may be used along with Eq. (7.173) when correcting the parameters. 

This means that if we are very close to the solution, Newton s method converges quadrat-ically. For example, assume that we are sufficiently close to a solution for this quadratic convergence to hold and that et = 10 . Then, the sequence of errors in the next few iterations is approximately... 

Figure 2.9 overlays upon this contour plot of the 2-norm, a trajectory of solution estimates obtained from Newton s method with an initial guess of (2,2). Newton s method converges in seven iterations to the desired accuracy however, the first step carries the estimate too far into a region of increasing 2-norm. Such a step is not very helpful for finding a solution. 

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