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

10. Minimization or Maximization of Functions using software torender barcode pdf417 with web,windows application Microsoft Excel if (fu < pdf417 2d barcode for None fc) { shft3(bx,cx,u,u+GOLD*(u-cx)); shft3(fb,fc,fu,func(u)); } } else if ((u-ulim)*(ulim-cx) >= 0.0) { Limit parabolic u to maximum u=ulim; allowed value. fu=func(u); } else { Reject parabolic u, use default magni cau=cx+GOLD*(cx-bx); tion.

fu=func(u); } shft3(ax,bx,cx,u); Eliminate oldest point and continue. shft3(fa,fb,fc,fu); } } inline void shft2(Doub &a, Doub &b, const Doub c) Utility function used in this structure or others derived from it. { a=b; b=c; } inline void shft3(Doub &a, Doub &b, Doub &c, const Doub d) { a=b; b=c; c=d; } inline void mov3(Doub &a, Doub &b, Doub &c, const Doub d, const Doub e, const Doub f) { a=d; b=e; c=f; } };.

(Because of the housekeeping involved in moving around three or four points and their function values, the above program ends up looking deceptively formidable. That is true of several other programs in this chapter as well. The underlying ideas, however, are quite simple.

). 10.2 Golden Section Search in One Dimension Recall how t he bisection method nds roots of functions in one dimension ( 9.1): The root is supposed to have been bracketed in an interval .a; b/.

One then evaluates the function at an intermediate point x and obtains a new, smaller bracketing interval, either .a; x/ or .x; b/.

The process continues until the bracketing interval is acceptably small. It is optimal to choose x to be the midpoint of .a; b/ so that the decrease in the interval length is maximized when the function is as uncooperative as it can be, i.

e., when the luck of the draw forces you to take the bigger bisected segment. There is a precise, though slightly subtle, translation of these considerations to the minimization problem.

The analog of bisection is to choose a new point x, either between a and b or between b and c. Suppose, to be speci c, that we make the latter choice. Then we evaluate f .

x/. If f .b/ < f .

x/, then the new bracketing triplet of points is .a; b; x/; contrariwise, if f .b/ > f .

x/, then the new bracketing triplet. 10.2 Golden Section Search in One Dimension 2 4 1 5 4 6. Figure 10.2. PDF-417 2d barcode for None 1.

Successive bracketing of a minimum. The minimum is originally bracketed by points 1,3,2. The function is evaluated at 4, which replaces 2; then at 5, which replaces 1; then at 6, which replaces 4.

The rule at each stage is to keep a center point that is lower than the two outside points. After the steps shown, the minimum is bracketed by points 5,3,6..

is .b; x; c/ . In all cases, the middle point of the new triplet is the abscissa whose ordinate is the best minimum achieved so far; see Figure 10.

2.1. We continue the process of bracketing until the distance between the two outer points of the triplet is tolerably small.

How small is tolerably small For a minimum located at a value b, you might naively think that you will be able to bracket it in as small a range as .1 /b < b < .1 C /b, where is your computer s oating-point precision, a number like 10 7 (for float) or 2 10 16 (for double).

Not so! In general, the shape of your function f .x/ near b will be given by Taylor s theorem, f .x/ f .

b/ C 1 f 00 .b/.x 2 b/2 (10.

2.1). The second term wil Software barcode pdf417 l be negligible compared to the rst (that is, will be a factor smaller and will act just like zero when added to it) whenever s p 2 jf .b/j (10.2.

2) jbj jx bj < b 2 f 00 .b/ The reason for writing the right-hand side in this way is that, for most functions, the nal square root is a number of order unity. Therefore, as a rule of thumb, it is p times its central value, hopeless to ask for a bracketing interval of width less than a fractional width of only about 10 4 (single precision) or 10 8 (double precision).

Knowing this inescapable fact will save you a lot of useless bisections! The minimum- nding routines of this chapter will often call for a user-supplied argument tol, and return with an abscissa whose fractional precision is about tol (bracketing interval of fractional size about 2 tol). Unless you have a better estimate for the right-hand side of equation (10.2.

2), you should set tol equal to (or.
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