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listIterator(); listIterator(int index); remove(int index); set(int index, Object object); subList(int start, int stop);

where yu(0)=yo. Because numerical techniques are to be used, we will restrict our attention to investigating yu(t) at sampling instants nT where T is the sampling period. Evaluating the differential equation at t = n T , we have

CHAP. 8]

From calculus we know that the derivative of a function y,(t) at t = nT is simply the slope of the function at t = n T . This slope may be approximated by the relationship

7.2 7.3

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(a) Insert this approximation into the sampled differential equation above and find a difference equation that relates y(n) = y,(nT) and x ( n ) = x,(nT), and specify the appropriate initial conditions. ( b ) With x,(t) = u ( t ) and y,(O-) = I, numerically solve the differential equation using the difference equation approximation obtained above.

public Set keySet(); public Collection values(); public Set entrySet(); public interface Entry { Object getKey(); Object getValue(); Object setValue(Object value); boolean equals(Object o); int hashCode(); } boolean equals(Object o); int hashCode(); }

[ya(nT)

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The AbstractList class implements the methods of the List interface, including the 10 index methods listed above in the answer to Question 7.1 which are not in the AbstractCollection class. The AbstractSequentialList class is designed to serve as a base class for linked list classes. It specifies the two abstract methods listIterator() and size() which must be implemented by any concrete subclass. The ListIterator class extends the Iterator class in a way that is analogous to the way the AbstractSequentialList class extends the AbstractList class. (See the answer to Question 7.2 above.) Ordinary Iterator objects are unidirectional iterators that iterate on array lists; ListIterator objects are bidirectional iterators that iterate on linked lists. Instances of the ArrayList class use contiguous, indexed, direct access (array) storage. Instances of the LinkedList class use linked (sequential) access storage. So array lists provide faster access, while linked lists provide faster modifications (insertions and removals). There isn t much difference between an ArrayList object and a Vector object: They both provide direct indexed access. As part of the Java Collections Framework, the ArrayList class was introduced more recently, in Java 1.2, and so it is probably more preferred. The Vector class has about twice as many methods, but many are redundant and consequently a bit confusing. An ArrayList object should be preferred when frequent lookups are expected. A LinkedList object should be preferred when frequent additions and/or removals are expected. (See the answer to Question 7.6 above.)

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- ya(nT - T)] + crya(nT) = .r,(nT)

THE HashMap CLASS As the class hierarchy in Figure 8.1 shows, Java defines four implementations of its Map interface: the AbstractMap class, the HashMap class, the TreeMap class, and the WeakHashMap class. EXAMPLE 8.1 A German-English Dictionary

If we let y(n) = ya(nT) and x(n) = xa(nT),

void loadRandomLetters( LinkedList list, int n) { list.clear(); while (0 < n--) list.add("" + (char)('A' + (int)(Math.random()*26))); } void printForward(LinkedList list) { for (ListIterator itr=list.listIterator(); itr.hasNext(); ) System.out.println(itr.next()); } void printBackward(LinkedList list) { ListIterator itr=list.listIterator(list.size()); while (itr.hasPrevious()) System.out.println(itr.previous()); } void exchange(LinkedList list, int i, int j) { Object ithObj = list.get(i); Object jthObj = list.get(j) ; list.set(i,jthObj); list.set(j,ithObj); }

With this becomes ~ ( n - ay(n )

This program uses the HashMap class to build a German-English dictionary:

1) = a T x ( n )

The solution to the generalized Josephus Problem:

~ ( 0 = yo )

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

We must now derive the initial condition on y(n) at time n = - 1 from the initial condition at n = 0. From the difference equation, we have y(0) - ay(-I) = aTx(0) With y(0) = 1 and x(0) = 1, the initial condition becomes

public class Josephus public static final public static final public static final { int SOLDIERS = 11; int SKIP = 2; String ALPHA = "ABCDEFGHIJKLMNOPQRSTUVWXYZ";

With x,(t) = u(t) or x(n) = u(n),

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