Root Finding and Nonlinear Sets of Equations in Software Integration pdf417 2d barcode in Software Root Finding and Nonlinear Sets of Equations

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9. Root Finding and Nonlinear Sets of Equations generate, create barcode pdf417 none in software projects Developing with Visual Studio .NET To develop a better str PDF417 for None ategy, note that the Newton step (9.7.3) is a descent direction for f : rf x D .

F J / . J. F/ D F F <0 (9.7.5).

Thus our strategy is qu pdf417 2d barcode for None ite simple: We always rst try the full Newton step, because once we are close enough to the solution we will get quadratic convergence. However, we check at each iteration that the proposed step reduces f . If not, we backtrack along the Newton direction until we have an acceptable step.

Because the Newton step is a descent direction for f , we are guaranteed to nd an acceptable step by backtracking. We will discuss the backtracking algorithm in more detail below. Note that this method minimizes f only incidentally, either by taking Newton steps designed to bring F to zero, or by backtracking along such a step.

The method is not equivalent to minimizing f directly by taking Newton steps designed to bring rf to zero. While the method can nevertheless still fail by converging to a local minimum of f that is not a root (as in Figure 9.6.

1), this is quite rare in real applications. The routine newt below will warn you if this happens. The only remedy is to try a new starting point.

. 9.7.1 Line Searches and Backtracking When we are not close e Software pdf417 nough to the minimum of f , taking the full Newton step p D x need not decrease the function; we may move too far for the quadratic approximation to be valid. All we are guaranteed is that initially f decreases as we move in the Newton direction. So the goal is to move to a new point xnew along the direction of the Newton step p, but not necessarily all the way: xnew D xold C p; 0< 1 (9.

7.6) The aim is to nd so that f .xold C p/ has decreased suf ciently.

Until the early 1970s, standard practice was to choose so that xnew exactly minimizes f in the direction p. However, we now know that it is extremely wasteful of function evaluations to do so. A better strategy is as follows: Since p is always the Newton direction in our algorithms, we rst try D 1, the full Newton step.

This will lead to quadratic convergence when x is suf ciently close to the solution. However, if f .xnew / does not meet our acceptance criteria, we backtrack along the Newton direction, trying a smaller value of , until we nd a suitable point.

Since the Newton direction is a descent direction, we are guaranteed to decrease f for suf ciently small . What should the criterion for accepting a step be It is not suf cient to require merely that f .xnew / < f .

xold /. This criterion can fail to converge to a minimum of f in one of two ways. First, it is possible to construct a sequence of steps satisfying this criterion with f decreasing too slowly relative to the step lengths.

Second, one can have a sequence where the step lengths are too small relative to the initial rate of decrease of f . (For examples of such sequences, see [2], p. 117.

) A simple way to x the rst problem is to require the average rate of decrease of f to be at least some fraction of the initial rate of decrease rf p: f .xnew / f .xold / C rf .

xnew xold / (9.7.7).

Here the parameter sa tis es 0 < < 1. We can get away with quite small values of ; D 10 4 is a good choice. The second problem can be xed by requiring the rate of decrease of f at xnew to be greater than some fraction of the rate of decrease of f at xold .

In practice, we will not need to impose this second constraint because our backtracking algorithm will have a built-in cutoff to avoid taking steps that are too small. Here is the strategy for a practical backtracking routine: De ne g. / f .

xold C p/ (9.7.8).

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