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Minimization or Maximization of Functions in Software Receive pdf417 2d barcode in Software Minimization or Maximization of Functions




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10. Minimization or Maximization of Functions use software pdf-417 2d barcode generation toinclude pdf 417 with software Basic Infromation about Micro QR Code fmin=f1; } else { xmi barcode pdf417 for None n=x2; fmin=f2; } return xmin; } };. 10.3 Parabolic Interpolation and Brent s Method in One Dimension We already tipped our pdf417 for None hand about the desirability of parabolic interpolation in the bracket routine of 10.1, but it is now time to be more explicit. A golden section search is designed to handle, in effect, the worst possible case of function minimization, with the uncooperative minimum hunted down and cornered like a scared rabbit.

But why assume the worst If the function is nicely parabolic near to the minimum surely the generic case for suf ciently smooth functions then the parabola tted through any three points ought to take us in a single leap to the minimum, or at least very near to it (see Figure 10.3.1).

Since we want to nd an abscissa rather than an ordinate, the procedure is technically called inverse parabolic interpolation. The formula for the abscissa x that is the minimum of a parabola through three points f .a/, f .

b/, and f .c/ is xDb 1 .b a/2 f .

b/ 2 .b a/ f .b/ f .

c/ f .c/ .b .

b c/2 f .b/ f .a/ c/ f .

b/ f .a/ (10.3.

1). as you can easily der ive. This formula fails only if the three points are collinear, in which case the denominator is zero (the minimum of the parabola is in nitely far away). Note, however, that (10.

3.1) is as happy jumping to a parabolic maximum as to a minimum. No minimization scheme that depends solely on (10.

3.1) is likely to succeed in practice. The exacting task is to invent a scheme that relies on a sure-but-slow technique, like golden section search, when the function is not cooperative, but that switches over to (10.

3.1) when the function allows. The task is nontrivial for several reasons, including these: (i) The housekeeping needed to avoid unnecessary function evaluations in switching between the two methods can be complicated.

(ii) Careful attention must be given to the endgame, where the function is being evaluated very near to the roundoff limit of equation (10.2.2).

(iii) The scheme for detecting a cooperative versus noncooperative function must be very robust. Brent s method [1] is up to the task in all particulars. At any particular stage, it is keeping track of six function points (not necessarily all distinct), a, b, u, v, w and x, de ned as follows: The minimum is bracketed between a and b; x is the point with the very least function value found so far (or the most recent one in case of a tie); w is the point with the second least function value; v is the previous value of w; and u is the point at which the function was evaluated most recently.

Also appearing in the algorithm is the point xm , the midpoint between a and b; however, the function is not evaluated there.. 10.3 Parabolic Interpolation and Brent s Method parabola through 1 2 3 parabola through 1 2 4 3 2 5 4. Figure 10.3.1.

Conver gence to a minimum by inverse parabolic interpolation. A parabola (dashed line) is drawn through the three original points 1,2,3 on the given function (solid line). The function is evaluated at the parabola s minimum, 4, which replaces point 3.

A new parabola (dotted line) is drawn through points 1,4,2. The minimum of this parabola is at 5, which is close to the minimum of the function..

You can read the code Software PDF 417 below to understand the method s logical organization. Mention of a few general principles here may, however, be helpful: Parabolic interpolation is attempted, tting through the points x, v, and w. To be acceptable, the parabolic step must (i) fall within the bounding interval .

a; b/, and (ii) imply a movement from the best current value x that is less than half the movement of the step before last. This second criterion ensures that the parabolic steps are actually converging to something, rather than, say, bouncing around in some nonconvergent limit cycle. In the worst possible case, where the parabolic steps are acceptable but useless, the method will approximately alternate between parabolic steps and golden sections, converging in due course by virtue of the latter.

The reason for comparing to the step before last seems essentially heuristic: Experience shows that it is better not to punish the algorithm for a single bad step if it can make it up on the next one. Another principle exempli ed in the code is never to evaluate the function less than a distance tol from a point already evaluated (or from a known bracketing point). The reason is that, as we saw in equation (10.

2.2), there is simply no information content in doing so: The function will differ from the value already evaluated only by an amount of order the roundoff error. Therefore, in the code below you will nd several tests and modi cations of a potential new point, imposing this restriction.

This restriction also interacts subtly with the test for doneness, which the method takes into account. A typical ending con guration for Brent s method is that a and b are 2 x tol apart, with x (the best abscissa) at the midpoint of a and b, and therefore fractionally accurate to tol. The calling sequence for Brent is exactly analogous to that of Golden in the previous section.

Indulge us a nal reminder that tol should generally be no smaller than the square root of your machine s oating-point precision..
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