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Figure B1: The function g(x) and the linear approximations to g(x) at x1 and x2 as determined by Newton s method in Example B3 improved estimate of the zero of g Due to the nonzero higher order terms (hot s), the result may not immediately yield a zero of g, but iteration of xk+1 = xk G 1 g(xk ) k results in quadratic convergence of xk toward a (simple) zero of g Example B3 Figure B1 shows the rst three iterations of eqn (B58) for g(x) = cos(x) x2 with a starting condition of x = 05 The gure shows the function g(x), the linear approximation at x = x1 , and the linear approximation at x = x2 The sequence of estimates of the zero of g(x) is as follows: k 1 2 3 4 5 6 xk 050000000000000 092420692729320 082910575599742 082414613172820 082413231240991 082413231230252 yk 062758256189037 -025169067707687 -001188097378152 -000003292121442 -000000000025583 000000000000000 (B58)

Virtually every control has events associated with it. To view a control s events, click the lightning bolt button. To switch back to properties click the Properties button.

17:

The height and width of the list in rows and characters If either is zero, then the widget resizes to incorporate all of the list elements Defines the selection mode of the list; one of single, browse, multiple, or extended

The convergence is faster near the nal result due to the accuracy of the Taylor approximation of eqn (B57) improving as x approaches the location of the zero of the g

get(INDEX) delete(INDEX [, LAST]) see(INDEX) selectionFrom(INDEX) selectionTo(INDEX) selection(FROM, TO) selectionClear( ) selectionPresent( ) curselection( ) Table 17-9

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Gets the string, starting at INDEX Deletes the row at INDEX, or the rows between INDEX and LAST Brings the element INDEX into the current view Selects all the rows from INDEX to the end of the list

Selects all the rows from the beginning of the list to INDEX Selects the rows starting at FROM and ending at TO Clears the selection Returns true if there is an active selection A list of the index values of all the selected items

Figure B2: Illustration of the scenario for Example B4 The position of two satellites P1 and P2 and measurements of the ranges v1 and v2 are available

Every method of the MenuButton widget supports the now familiar index format, although the index refers to the individual menu item: number The index of the menu item, starting at zero for the first item When the menu is configured for tear-off, the first entry is a separator automatically inserted by the widget Indicates the last entry Where the location cursor is currently active Indicates that none of the menu options are active A pattern to be matched against all entries This only matches exactly; regular expressions are supported

Properties and methods for the MenuButton widget are shown in Table 17-10

B142

If true, shows a small diamond to the right of the menu The state of the menu one of normal, active, or disabled

menu command(OPTIONS) separator(OPTIONS) radiobutton(OPTIONS) checkbutton(OPTIONS) cascade(OPTIONS) add(TYPE, OPTIONS) delete(INDEX1 [, INDEX2]) insert(INDEX1, TYPE, OPTIONS)

If you prefer to work in Source view rather than Design view, you can enlist IntelliSense to help you find both the properties and events for any given control. As you press the spacebar, the list of members for the control will be displayed. As you type,

Returns the underlying menu associated with this menu button Creates a standard menu item using the properties in OPTIONS A separator A radio button menu item using the properties in OPTIONS A check button menu item using the properties in OPTIONS Inserts a new cascading (hierarchical) menu using the properties in OPTIONS Adds a new menu of TYPE with OPTIONS Deletes the menu item INDEX1 or the items from INDEX1 to INDEX2 Inserts a menu item of TYPE with OPTIONS into the location INDEX1

If we have a situation where y = f (x) for x Rn and y Rm where m n, then in general no exact solution will exist As in Newton s method we can use iteration to nd the solution that minimizes y f (x) Starting at an estimate xk the Taylor s approximation to f (x) near xk is f (x) = f (xk ) + Fk (x xk ) (B59)

entryconfigure(INDEX, OPTIONS) Changes the properties of the menu item according to OPTIONS pointed to by INDEX entrycget(INDEX) Gets the configuration options for the menu item at INDEX

Table 17-10

f where Fk = x x=x is the Jacobian matrix for f (x) evaluated at xk The k approximate cost function is then

17:

The configurable options supported for the methods in Table 17-10 work like all other properties and are listed in Table 17-11 Note that because you can have hierarchical menus, individual items can use further methods from Table 17-10

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