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APPLICATION: A Polynomial CLASS A polynomial is a mathematical function of the form: The greatest exponent, n, is called the degree of the polynomial. For example, p(x) = 7x 4 2 is a polynomial of degree 4. The simplest polynomials are constant polynomials such as p(x) = 6 (degree 0) and linear polynomials such as p(x) = 9x + 6 (degree 1). The unique zero polynomial p(x) = 0 is defined to have degree 1. In this section we present a Polynomial class whose instances represent mathematical polynomials and which supports the usual algebraic operations on polynomials. A polynomial can be regarded as a sum of distinct terms. A term is a mathematical function of the form t(x) = cx e, where c is any real number and e is any nonnegative integer. The number c is called the coefficient, and the number e is called the exponent. To define a class whose objects represent polynomials, we use a linked list of Term objects. For example, the polynomial p(x) = 3x 2 2x + 5 could be represented as a list of three elements, where the first element represents the term 3x 2, the second element represents the term 2x, and the third element represents the (constant) term 5. EXAMPLE 7.7 A Polynomial Class
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p(x) = a0 x n + a1 x n 1 + a2 x n 2 +
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public class Polynomial { private List<Term> list = new LinkedList<Term>(); public static final Polynomial ZERO = new Polynomial(); private Polynomial() { } // default constructor
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public Polynomial(double coef, int exp) { if (coef != 0.0) { list.add(new Term(coef, exp)); } } public Polynomial(Polynomial p) { // copy constructor for (Term term : p.list) { this.list.add(new Term(term)); } } public Polynomial(double... a) { for (int i=0; i<a.length; i++) { if (a[i] != 0.0) { list.add(new Term(a[i], i)); } } } public int degree() { if (list.isEmpty()) { return -1; } else { return list.get(list.size()-1).exp; } }
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public boolean isZero() { return list.isEmpty(); } public Polynomial plus(Polynomial p) { if (this.isZero()) { return new Polynomial(p); } if (p.isZero()) { return new Polynomial(this); } Polynomial q = new Polynomial(); ListIterator<Term> it = list.listIterator(); ListIterator<Term> itp = p.list.listIterator(); while (it.hasNext() && itp.hasNext()) { Term term = it.next(); Term pTerm = itp.next(); if (term.exp < pTerm.exp) { q.list.add(new Term(term)); itp.previous(); } else if (term.exp == pTerm.exp) { q.list.add(new Term(term.coef + pTerm.coef, term.exp)); } else { // (term.exp > pTerm.exp) q.list.add(new Term(pTerm)); it.previous(); } } while (it.hasNext()) { q.list.add(new Term(it.next())); } while (itp.hasNext()) { q.list.add(new Term(itp.next())); } return q; } public String toString() { if (this.isZero()) { return "0"; } Iterator<Term> it = list.iterator(); StringBuilder buf = new StringBuilder(); boolean isFirstTerm = true; while (it.hasNext()) { Term term = it.next(); double c = term.coef; int e = term.exp; if (isFirstTerm) { buf.append(String.format("%.2f", c)); isFirstTerm = false; } else { if (term.coef < 0) { buf.append(String.format(" - %.2f", -c)); } else { buf.append(String.format(" + %.2f", c)); }
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} if (e == 1) { buf.append("x"); } else if (e > 1) { buf.append("x^" + e); } } return buf.toString(); } private static class Term { private double coef; private int exp; public Term(double coef, int exp) { if (coef == 0.0 || exp < 0) { throw new IllegalArgumentException(); } this.coef = coef; this.exp = exp; } public Term(Term that) { // copy constructor this(that.coef, that.exp); } } }
Instead of using inheritance by extending a List class, this Polynomial class uses composition, declaring a list field at line 2. This design gives the user more control over the class by limiting it to only those methods actually defined in the class. Of course, those methods are mostly implemented by means of List methods. The list backing structure is declared to be a LinkedList<Term> collection. So each element of the list is a Term object. The list stores the polynomial s nonzero terms in increasing order of their exponents. The Term class is defined as an inner class at lines 102 117, that is, a static member class. It has two fields, coef and exp (for the coefficient and the exponent of the term), and two constructors. The second constructor (line 114) is a copy constructor, creating a duplicate of the term passed to it. It uses the this keyword to invoke the two-argument constructor defined at line 106. The Polynomial class has four constructors and four methods. The default constructor (also called the no-arg constructor ) defined at line 8 is declared to be private. This prevents it from being used outside of its class. Its purpose is to construct the Polynomial object that represents the zero polynomial, and it is invoked at line 3 to do that. To ensure that that object is unique, we prevent it from being constructed anywhere else. The constructor at line 8 creates a Polynomial object that represents a single term, such as 88.8x44. The constructor at line 14 is a copy constructor, which duplicates the object passed to it. The constructor at line 20 uses the Java var-args syntax to allow a variable number of arguments, in this case of type double. This is the same as a single argument of type double[]. That constructor creates a polynomial whose coefficients are the nonzero values in the array, each one generating a term of the form cn x n, where cn = a[n]. For example, the array {4, 0, 7, 0, 0, 0, 3} would produce the Polynomial object that represents 4 + 7x2 + 3x6. The degree() method at line 28 returns the polynomial s highest exponent. Since the terms are maintained in the list in increasing order of their exponents, the degree of the polynomial is the simply the exp field of the last element in the list. That element has index list.size()-1, so the expression at line 32 does the job.