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TABLE 24-15 A Sampling of Methods De ned by LinkedList<T> (continued)

Here is an example that demonstrates the LinkedList<T> class:

The rst integral equals (1/25) log |x + 2|, the second integral equals (1/25) log |x 3|, and the third integral equals (1/5)/(x 3) In summary, we have found that x3 4x 2 log |x + 2| log |x 3| 1 1 dx = + C 25 25 5(x 3) 3x + 18

Equations (745), (748), and (749 to 752c) are the relations sought 74321 Spline Approximation of the Boundary The formulas introduced in Angeles et al (1990) are now applied to regions whose boundary is approximated by periodic parametric cubic splines Recalling expressions (726a and b), the (x, y) coordinates of one point on the boundary are represented as functions of parameter p, while xi and yi represent the Cartesian coordinates of the ith supporting point of the spline Moreover, integrals (742a to c) are approximated as A Ai

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// Demonstrate LinkedList<T> using System; using SystemCollectionsGeneric; class GenLinkedListDemo { static void Main() { // Create a linked list LinkedList<char> ll = new LinkedList<char>(); ConsoleWriteLine("Initial number of elements: " + llCount); ConsoleWriteLine(); ConsoleWriteLine("Adding 5 elements"); // Add elements to the linked list llAddFirst('A'); llAddFirst('B'); llAddFirst('C'); llAddFirst('D'); llAddFirst('E'); ConsoleWriteLine("Number of elements: " + llCount); // Display the linked list by manually walking // through the list LinkedListNode<char> node; ConsoleWrite("Display contents by following links: "); for(node = llFirst; node != null; node = nodeNext) ConsoleWrite(nodeValue + " "); ConsoleWriteLine("\n"); //Display the linked list by use of a foreach loop ConsoleWrite("Display contents with foreach loop: "); foreach(char ch in ll) ConsoleWrite(ch + " "); ConsoleWriteLine("\n");

Part II:

// Display the list backward by manually walking // from last to first ConsoleWrite("Follow links backwards: "); for(node = llLast; node != null; node = nodePrevious) ConsoleWrite(nodeValue + " "); ConsoleWriteLine("\n"); // Remove two elements ConsoleWriteLine("Removing 2 elements"); // Remove elements from the linked list llRemove('C'); llRemove('A'); ConsoleWriteLine("Number of elements: " + llCount); // Use foreach loop to display the modified list ConsoleWrite("Contents after deletion: "); foreach(char ch in ll) ConsoleWrite(ch + " "); ConsoleWriteLine("\n"); // Add three elements to the end of the list llAddLast('X'); llAddLast('Y'); llAddLast('Z'); ConsoleWrite("Contents after addition to end: "); foreach(char ch in ll) ConsoleWrite(ch + " "); ConsoleWriteLine("\n"); } }

1 O O O O Qx Qxi , Qy Qyi 1 1 n -1 1 n -1 n -1 1 n -1 Ixyi + Iyxi , IyO Iyi 2 1 1 1 n -1 n -1 n -1

Here is the output:

SOLUTION Since the denominator is a cubic polynomial, it must factor The factors of the constant term are 1 and 2 After some experimentation, we nd that x = 2 is a root and in fact the polynomial factors as x 3 + 2x 2 + x + 2 = (x + 2)(x 2 + 1)

Initial number of elements: 0 Adding 5 elements Number of elements: 5 Display contents by following links: E D C B A Display contents with foreach loop: E D C B A Follow links backwards: A B C D E Removing 2 elements Number of elements: 3 Contents after deletion: E D B Contents after addition to end: E D B X Y Z

(753a)

24:

Perhaps the most important thing to notice in this program is that the list is traversed in both the forward and backward direction by following the links provided by the Next and Previous properties The bidirectional property of doubly linked lists is especially important in applications such as databases in which the ability to move efficiently through the list in both directions is often necessary

(753b) (753c)

Thus we wish to write the integrand as the sum of a factor with denominator (x + 2) and another factor with denominator (x 2 + 1) The correct way to do this is x A Bx + C x = = + 2 2 + 1) 3 + 2x 2 + x + 2 x+2 (x + 2)(x x +1 x We put the right-hand side over a common denominator to obtain x x 3 + 2x 2 + x + 2 Identifying numerators leads to = A(x 2 + 1) + (Bx + C)(x + 2) x 3 + 2x 2 + x + 2

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