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Java Programming Abstract classes and Interfaces.

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1 Java Programming Abstract classes and Interfaces

2 Classes A class is composed of A class defines a data type. public class Rectangle { private double width, height; public Rectangle(double width, double height) { this.width= width; this.height=height; } public double getArea() { return width * height; } public double getPerimeter() { return 2 * (width + height); } public double getAspectRatio() { return width / height; } public class Rectangle { private double width, height; public Rectangle(double width, double height) { this.width= width; this.height=height; } public double getArea() { return width * height; } public double getPerimeter() { return 2 * (width + height); } public double getAspectRatio() { return width / height; } data methods

3 Clients The class (or type) of an object defines – the data that is managed by the object – the methods we can apply on the object public void processRectangle(Rectangle r) { double x1 = r.getArea(); double x2 = r.getPerimeter(); double x3 = r.getAspectRatio(); String x4 = r.toString(); boolean x5 = r.equals(“Rectangle”); } public void processRectangle(Rectangle r) { double x1 = r.getArea(); double x2 = r.getPerimeter(); double x3 = r.getAspectRatio(); String x4 = r.toString(); boolean x5 = r.equals(“Rectangle”); } public class Rectangle { private double width, height; public Rectangle(double width, double height) { this.width= width; this.height=height; } public double getArea() { return width * height; } public double getPerimeter() { return 2 * (width + height); } public double getAspectRatio() { return width / height; } public class Rectangle { private double width, height; public Rectangle(double width, double height) { this.width= width; this.height=height; } public double getArea() { return width * height; } public double getPerimeter() { return 2 * (width + height); } public double getAspectRatio() { return width / height; }

4 Interface An interface is a collection of method signatures – access control – return type – method name – formal parameters – exceptions An interface must be implemented to – define the method bodies – define the data

5 Interface example public interface Shape { public double getArea(); public double getPerimeter(); public double getAspectRatio(); } public interface Shape { public double getArea(); public double getPerimeter(); public double getAspectRatio(); } collection of method signatures public class Rectangle implements Shape { private double width, height; public Rectangle(double width, double height) { this.width= width; this.height=height; } public double getArea() { return width * height; } public double getPerimeter() { return 2 * (width + height); } public double getAspectRatio() { return width / height; } public class Rectangle implements Shape { private double width, height; public Rectangle(double width, double height) { this.width= width; this.height=height; } public double getArea() { return width * height; } public double getPerimeter() { return 2 * (width + height); } public double getAspectRatio() { return width / height; } the methods are fully defined

6 Interfaces An interface cannot be instantiated – Shape s = new Shape(); – Shape s = new Rectangle(30, 10); An implementation must either – define all methods OR – be labeled as abstract Implementation denotes an is-a relationship – public class Rectangle implements Shape – means that “a Rectangle is-a Shape”

7 Implement three Shapes Rectangle – A box with width & height Isosceles Triangle – A triangle with two equal sides Ellipse – an oval with major/minor axis widthheight width height width height

8 Implementation public class Rectangle implements Shape { private double width, height; public Rectangle(double width, double height) { this.width= width; this.height=height; } public double getArea() { return width * height; } public double getPerimeter() { return 2 * (width + height); } public double getAspectRatio() { return width / height; } public class Rectangle implements Shape { private double width, height; public Rectangle(double width, double height) { this.width= width; this.height=height; } public double getArea() { return width * height; } public double getPerimeter() { return 2 * (width + height); } public double getAspectRatio() { return width / height; } public interface Shape { public double getArea(); public double getPerimeter(); public double getAspectRatio(); } public interface Shape { public double getArea(); public double getPerimeter(); public double getAspectRatio(); } widthheight // Constructing and naming a rectangle Shape x = new Rectangle(10,30); // Constructing and naming a rectangle Shape x = new Rectangle(10,30);

9 Implementation public interface Shape { public double getArea(); public double getPerimeter(); public double getAspectRatio(); } public interface Shape { public double getArea(); public double getPerimeter(); public double getAspectRatio(); } public class IsocelesTriangle implements Shape { private double width, height; public IsocelesTriangle (double width, double height) { this.width= width; this.height=height; } public double getArea() { return width * height / 2; } public double getPerimeter() { return 2*Math.sqrt(width*width/4 + height*height) + width; } public double getAspectRatio() { return width / height; } public class IsocelesTriangle implements Shape { private double width, height; public IsocelesTriangle (double width, double height) { this.width= width; this.height=height; } public double getArea() { return width * height / 2; } public double getPerimeter() { return 2*Math.sqrt(width*width/4 + height*height) + width; } public double getAspectRatio() { return width / height; } width height // Constructing and naming a triangle Shape x = new IsocelesTriangle(10,30); // Constructing and naming a triangle Shape x = new IsocelesTriangle(10,30);

10 Implementation public interface Shape { public double getArea(); public double getPerimeter(); public double getAspectRatio(); } public interface Shape { public double getArea(); public double getPerimeter(); public double getAspectRatio(); } public class Ellipse implements Shape { private double width, height; public Ellipse(double width, double height) { this.width= width; this.height=height; } public double getArea() { return Math.PI * width * height / 4; } public double getPerimeter() { double a= width/2, b = height/2; double x = Math.max(a,b), y = Math.min(a,b); int digits = 53; double tolerance = Math.sqrt(Math.pow(a, digits)); double s = 0, m = 1; while(x-y>tolerance*y) { double y1 = Math.sqrt(x*y); double x1 = (x+y)/2; x = x1; y = y1; m *= 2; s += m * Math.pow(x-y,2); } return Math.PI * (Math.pow(a+b, 2)-s)/(x+y); } public double getAspectRatio() { return width / height; } public class Ellipse implements Shape { private double width, height; public Ellipse(double width, double height) { this.width= width; this.height=height; } public double getArea() { return Math.PI * width * height / 4; } public double getPerimeter() { double a= width/2, b = height/2; double x = Math.max(a,b), y = Math.min(a,b); int digits = 53; double tolerance = Math.sqrt(Math.pow(a, digits)); double s = 0, m = 1; while(x-y>tolerance*y) { double y1 = Math.sqrt(x*y); double x1 = (x+y)/2; x = x1; y = y1; m *= 2; s += m * Math.pow(x-y,2); } return Math.PI * (Math.pow(a+b, 2)-s)/(x+y); } public double getAspectRatio() { return width / height; } width height // Constructing and naming an ellipse Shape x = new Ellipse(10,30); // Constructing and naming an ellipse Shape x = new Ellipse(10,30);

11 Abstract Class public class Rectangle implements Shape { private double width, height; public Rectangle(double width, double height) { this.width= width; this.height=height; } public double getArea() { return width * height; } public double getPerimeter() { return 2 * (width + height); } public double getAspectRatio() { return width / height; } public class Rectangle implements Shape { private double width, height; public Rectangle(double width, double height) { this.width= width; this.height=height; } public double getArea() { return width * height; } public double getPerimeter() { return 2 * (width + height); } public double getAspectRatio() { return width / height; } public class IsocelesTriangle implements Shape { private double width, height; public IsocelesTriangle (double width, double height) { this.width= width; this.height=height; } public double getArea() { return width * height / 2; } public double getPerimeter() { return 2*Math.sqrt(width*width/4+height*height)+width; } public double getAspectRatio() { return width / height; } public class IsocelesTriangle implements Shape { private double width, height; public IsocelesTriangle (double width, double height) { this.width= width; this.height=height; } public double getArea() { return width * height / 2; } public double getPerimeter() { return 2*Math.sqrt(width*width/4+height*height)+width; } public double getAspectRatio() { return width / height; } public class Ellipse implements Shape { private double width, height; public Ellipse(double width, double height) { this.width= width; this.height=height; } public double getArea() { return Math.PI * width * height / 4; } public double getPerimeter() { double a= width/2, b = height/2; double x = Math.max(a,b), y = Math.min(a,b); int digits = 53; double tolerance = Math.sqrt(Math.pow(a, digits)); double s = 0, m = 1; while(x-y>tolerance*y) { double y1 = Math.sqrt(x*y); double x1 = (x+y)/2; x = x1; y = y1; m *= 2; s += m * Math.pow(x-y,2); } return Math.PI * (Math.pow(a+b, 2)-s)/(x+y); } public double getAspectRatio() { return width / height; } public class Ellipse implements Shape { private double width, height; public Ellipse(double width, double height) { this.width= width; this.height=height; } public double getArea() { return Math.PI * width * height / 4; } public double getPerimeter() { double a= width/2, b = height/2; double x = Math.max(a,b), y = Math.min(a,b); int digits = 53; double tolerance = Math.sqrt(Math.pow(a, digits)); double s = 0, m = 1; while(x-y>tolerance*y) { double y1 = Math.sqrt(x*y); double x1 = (x+y)/2; x = x1; y = y1; m *= 2; s += m * Math.pow(x-y,2); } return Math.PI * (Math.pow(a+b, 2)-s)/(x+y); } public double getAspectRatio() { return width / height; } Notice the similar code in these three implementations. Should aggregate into an abstract class. Notice the similar code in these three implementations. Should aggregate into an abstract class.

12 Abstract Class public class Rectangle extends AbstractShape { public Rectangle(double width, double height) { super(width, height); } public double getArea() { return width * height; } public double getPerimeter() { return 2 * (width + height); } public class Rectangle extends AbstractShape { public Rectangle(double width, double height) { super(width, height); } public double getArea() { return width * height; } public double getPerimeter() { return 2 * (width + height); } public class IsocelesTriangle extends AbstractShape { public IsocelesTriangle (double width, double height) { super(width,height); } public double getArea() { return width * height / 2; } public double getPerimeter() { return 2*Math.sqrt(width*width/4+height*height)+width; } public class IsocelesTriangle extends AbstractShape { public IsocelesTriangle (double width, double height) { super(width,height); } public double getArea() { return width * height / 2; } public double getPerimeter() { return 2*Math.sqrt(width*width/4+height*height)+width; } public class Ellipse extends AbstractShape { public Ellipse(double width, double height) { super(width, height); } public double getArea() { return Math.PI * width * height / 4; } public double getPerimeter() { double a= width/2, b = height/2; double x = Math.max(a,b), y = Math.min(a,b); int digits = 53; double tolerance = Math.sqrt(Math.pow(a, digits)); double s = 0, m = 1; while(x-y>tolerance*y) { double y1 = Math.sqrt(x*y); double x1 = (x+y)/2; x = x1; y = y1; m *= 2; s += m * Math.pow(x-y,2); } return Math.PI * (Math.pow(a+b, 2)-s)/(x+y); } public class Ellipse extends AbstractShape { public Ellipse(double width, double height) { super(width, height); } public double getArea() { return Math.PI * width * height / 4; } public double getPerimeter() { double a= width/2, b = height/2; double x = Math.max(a,b), y = Math.min(a,b); int digits = 53; double tolerance = Math.sqrt(Math.pow(a, digits)); double s = 0, m = 1; while(x-y>tolerance*y) { double y1 = Math.sqrt(x*y); double x1 = (x+y)/2; x = x1; y = y1; m *= 2; s += m * Math.pow(x-y,2); } return Math.PI * (Math.pow(a+b, 2)-s)/(x+y); } public abstract class AbstractShape implements Shape { protected double width, height; public AbstractShape(double width, double height) { this.width= width; this.height=height; } public double getAspectRatio() { return width / height; } public abstract class AbstractShape implements Shape { protected double width, height; public AbstractShape(double width, double height) { this.width= width; this.height=height; } public double getAspectRatio() { return width / height; }

13 Abstract Class An abstract class – is not completely defined – may have methods – may have data – will usually have a collection of method signatures – cannot be instantiated AbstractShape s = new AbstractShape(3, 5); AbstractShape s = new Rectangle(3, 5);

14 public class ShapeDriver { public static Shape getRandomShape() { double width = Math.random() * 20; double height = Math.random() * 20; int type = (int) (Math.random() * 3); switch (type) { case 0: return new Ellipse(width, height); case 1: return new IsocelesTriangle(width, height); case 2: return new Rectangle(width, height); } throw new IllegalArgumentException(); } public static double totalArea(List shapes) { double totalArea = 0; for(Shape s : shapes) { totalArea += s.getArea(); } return totalArea; } public static void main(String[] args) { List shapes = new LinkedList<>(); for (int i = 0; i < 20; i++) { shapes.add(getRandomShape()); } System.out.println(totalArea(shapes)); } public class ShapeDriver { public static Shape getRandomShape() { double width = Math.random() * 20; double height = Math.random() * 20; int type = (int) (Math.random() * 3); switch (type) { case 0: return new Ellipse(width, height); case 1: return new IsocelesTriangle(width, height); case 2: return new Rectangle(width, height); } throw new IllegalArgumentException(); } public static double totalArea(List shapes) { double totalArea = 0; for(Shape s : shapes) { totalArea += s.getArea(); } return totalArea; } public static void main(String[] args) { List shapes = new LinkedList<>(); for (int i = 0; i < 20; i++) { shapes.add(getRandomShape()); } System.out.println(totalArea(shapes)); } public interface Shape { public double getArea(); public double getPerimeter(); public double getAspectRatio(); } public interface Shape { public double getArea(); public double getPerimeter(); public double getAspectRatio(); }

15 Generics A generic class is a class that is parameterized over types. public class Pair { private T1 first; private T2 second; public Pair(T1 first, T2 second) { this.first = first; this.second = second; } public T1 getFirst() { return first; } public T2 getSecond() { return second; } public void setFirst(T1 first) { this.first = first; } public void setSecond(T2 second) { this.second = second; } public class Pair { private T1 first; private T2 second; public Pair(T1 first, T2 second) { this.first = first; this.second = second; } public T1 getFirst() { return first; } public T2 getSecond() { return second; } public void setFirst(T1 first) { this.first = first; } public void setSecond(T2 second) { this.second = second; } T1 and T2 are type parameters. public class PairDriver { public static void main(String[] args) { Pair p1 = new Pair<>(“Kenny”, 1); Pair p2 = new Pair<>(2, 3.5); String x1 = p1.getFirst(); Integer x2 = p1.getSecond(); Integer x3 = p2.getFirst(); Double x4 = p2.getSecond(); Pair,Pair > p3 = new Pair<>(p1, p2); ??? x5 = p3.getFirst(); ??? x6 = p3.getSecond(); } public class PairDriver { public static void main(String[] args) { Pair p1 = new Pair<>(“Kenny”, 1); Pair p2 = new Pair<>(2, 3.5); String x1 = p1.getFirst(); Integer x2 = p1.getSecond(); Integer x3 = p2.getFirst(); Double x4 = p2.getSecond(); Pair,Pair > p3 = new Pair<>(p1, p2); ??? x5 = p3.getFirst(); ??? x6 = p3.getSecond(); } Type arguments are supplied when the class is used.

16 Comparable & Comparator Java has two commonly used interfaces – Comparable This interface imposes a total ordering on the objects of each class that implements it. This ordering is referred to as the class's natural ordering. The compareTo method this object with the specified object for order. Returns a negative integer, zero, or a positive integer as this object is less than, equal to, or greater than the specified object. – Comparator A comparison function, which imposes a total ordering on some collection of objects. The compare method compares its two arguments for order. Returns a negative integer, zero, or a positive integer as the first argument is less than, equal to, or greater than the second. public interface Comparable { public int compareTo(T e); } public interface Comparable { public int compareTo(T e); } public interface Comparator { public int compare(T e1, T e2); } public interface Comparator { public int compare(T e1, T e2); }

17 Example Can the AbstractShape class implement the Comparable interface? – We define the natural ordering of shapes by keying on the area. public abstract class AbstractShape implements Shape, Comparable { protected double width, height; public AbstractShape(double width, double height) { this.width = width; this.height = height; } public double getAspectRatio() { return width / height; } public int compareTo(Shape e) { double diff = getArea() – e.getArea(); if(diff < 0) { return -1; } else if(diff > 0) { return 1; } else { return 0; } public abstract class AbstractShape implements Shape, Comparable { protected double width, height; public AbstractShape(double width, double height) { this.width = width; this.height = height; } public double getAspectRatio() { return width / height; } public int compareTo(Shape e) { double diff = getArea() – e.getArea(); if(diff < 0) { return -1; } else if(diff > 0) { return 1; } else { return 0; }

18 Why the interface? public static AbstractShape getSmallest(AbstractShape[] shapes) { if(shapes.length == 0) throw new NoSuchElementException(); AbstractShape smallest = shapes[0]; for(int i=1; i { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://images.slideplayer.com/11/3278469/slides/slide_18.jpg", "name": "Why the interface.", "description": "public static AbstractShape getSmallest(AbstractShape[] shapes) { if(shapes.length == 0) throw new NoSuchElementException(); AbstractShape smallest = shapes[0]; for(int i=1; i

19 Why the interface? public static Comparable getSmallest(Comparable[] items) { if(items.length == 0) throw new NoSuchElementException(); Comparable smallest = items[0]; for(int i=1; i { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://images.slideplayer.com/11/3278469/slides/slide_19.jpg", "name": "Why the interface.", "description": "public static Comparable getSmallest(Comparable[] items) { if(items.length == 0) throw new NoSuchElementException(); Comparable smallest = items[0]; for(int i=1; i

20 Generic Methods Methods can introduce type parameters public T randomChoice(T x1, T x2) { if(Math.random() <.5) { return x1; } else { return x2; } public T randomChoice(T x1, T x2) { if(Math.random() <.5) { return x1; } else { return x2; } Generic type parameter String s = randomChoice(“a”, “b”); Double x = randomChoice(1.0, 2.3); Integer y = randomChoice(3,5); Shape u = new Rectangle(10,30); Shape v = new Rectangle(30, 50); Shape t = randomChoice(u, v); String s = randomChoice(“a”, “b”); Double x = randomChoice(1.0, 2.3); Integer y = randomChoice(3,5); Shape u = new Rectangle(10,30); Shape v = new Rectangle(30, 50); Shape t = randomChoice(u, v);

21 Generic Methods Can we write a generic method to accept to elements of some type and return the smallest element? public T smallest(T x1, T x2) { if(x1 < x2) { return x1; } else { return x2; } public T smallest(T x1, T x2) { if(x1 < x2) { return x1; } else { return x2; } public T smallest(T x1, T x2) { if(x1.compareTo(x2) < 0) { return x1; } else { return x2; } public T smallest(T x1, T x2) { if(x1.compareTo(x2) < 0) { return x1; } else { return x2; } This doesn’t work since the “compareTo” method is not supported on objects that don’t implement Comparable. This doesn’t work since the “<“ operator is not supported on object types.

22 Generic Methods Can we write a generic method to accept elements of some type and return the smallest element? Notation to put an upper-bound on a methods generic parameter – TYPENAME extends UPPERBOUND Examples: – – > – public > T smallest(T x1, T x2) { if(x1.compareTo(x2) < 0) { return x1; } else { return x2; } public > T smallest(T x1, T x2) { if(x1.compareTo(x2) < 0) { return x1; } else { return x2; } This works since T has an “upper bound” of Comparable. This means that whatever T is, it is a sub- class of Comparable. String x1 = smallest(“a”, “b”); Integer x2 = smallest(15, 3); Double x3 = smallest(2, -18); String x1 = smallest(“a”, “b”); Integer x2 = smallest(15, 3); Double x3 = smallest(2, -18);

23 Generics and Subtyping Consider the following example. What are the conformance rules for generic classes? Pair p1 = new Pair (“a”, “b”); p1.setFirst(4); // IS THIS VALID? p1.setSecond(“c”); // IS THIS VALID? Pair p2 = new Pair (“a”, 3); p2.setFirst(4); // IS THIS VALID? p2.setSecond(“c”); // IS THIS VALID? p1 = p2; // IS THIS VALID? p1.setFirst(4); p1.setSecond(“c”); Pair p1 = new Pair (“a”, “b”); p1.setFirst(4); // IS THIS VALID? p1.setSecond(“c”); // IS THIS VALID? Pair p2 = new Pair (“a”, 3); p2.setFirst(4); // IS THIS VALID? p2.setSecond(“c”); // IS THIS VALID? p1 = p2; // IS THIS VALID? p1.setFirst(4); p1.setSecond(“c”);

24 Generics and Conformance Conformance rules – If A is a non-generic super-class of B then objects of type B conform to A Shape s = new Rectangle(10,30); Number x = new Double(3.5); – If A is a generic super-class of B, then objects of B type conform to A only if each generic parameter is an exact match. List x = new LinkedList ; List y = new LinkedList ;

25 Bounded Type Parameters When a method declares a parameterized type, the actual parameters must match exactly. public Object pickOne(TwoOfAKind pair) { if(Math.random() <. 5) { return pair.getFirst(); } else { return pair.getSecond(); } public Object pickOne(TwoOfAKind pair) { if(Math.random() <. 5) { return pair.getFirst(); } else { return pair.getSecond(); } public class TwoOfAKind { private T first; private T second; public TwoOfAKind (T first, T second) { this.first = first; this.second = second; } public T getFirst() { return first; } public T getSecond() { return second; } public void setFirst(T first) { this.first = first; } public void setSecond(T second) { this.second = second; } public class TwoOfAKind { private T first; private T second; public TwoOfAKind (T first, T second) { this.first = first; this.second = second; } public T getFirst() { return first; } public T getSecond() { return second; } public void setFirst(T first) { this.first = first; } public void setSecond(T second) { this.second = second; } TwoOfAKind p1 = new TwoOfAKind (“a”, “b”); TwoOfAKind p2 = new TwoOfAKind (1, ”c”); Object x = pickOne(p1); Object y = pickOne(p2); TwoOfAKind p1 = new TwoOfAKind (“a”, “b”); TwoOfAKind p2 = new TwoOfAKind (1, ”c”); Object x = pickOne(p1); Object y = pickOne(p2);

26 Generics and Wildcards Wildcards allow us to write truly generic functions. – ? denotes ANY TYPE public Object pickOne(TwoOfAKind pair) { if(Math.random() <. 5) { return pair.getFirst(); } else { return pair.getSecond(); } public Object pickOne(TwoOfAKind pair) { if(Math.random() <. 5) { return pair.getFirst(); } else { return pair.getSecond(); } TwoOfAKind p1 = new TwoOfAKind (“a”, “b”); TwoOfAKind p2 = new TwoOfAKind (1, ”c”); Object x = pickOne(p1); Object y = pickOne(p2); TwoOfAKind p1 = new TwoOfAKind (“a”, “b”); TwoOfAKind p2 = new TwoOfAKind (1, ”c”); Object x = pickOne(p1); Object y = pickOne(p2);

27 Generics and Wildcards The wildcard can be constrained. If A is the name of some class then – ? extends A the ? stands for some class that is either class A or a SUB CLASS OF A A is an upper-bound public Comparable pickOne(TwoOfAKind pair) { if(Math.random() <. 5) { return pair.getFirst(); } else { return pair.getSecond(); } public Comparable pickOne(TwoOfAKind pair) { if(Math.random() <. 5) { return pair.getFirst(); } else { return pair.getSecond(); } TwoOfAKind p1 = new TwoOfAKind (“a”, “b”); TwoOfAKind p2 = new TwoOfAKind (1, ”c”); Object x = pickOne(p1); Object y = pickOne(p2); TwoOfAKind p1 = new TwoOfAKind (“a”, “b”); TwoOfAKind p2 = new TwoOfAKind (1, ”c”); Object x = pickOne(p1); Object y = pickOne(p2);

28 Generics and Wildcards The wildcard can be constrained. If A is the name of some class then – ? super A the ? stands for some class that is either class A OR A SUPER CLASS OF A A is a lower-bound public Object pickOne(TwoOfAKind pair) { if(Math.random() <. 5) { return pair.getFirst(); } else { return pair.getSecond(); } public Object pickOne(TwoOfAKind pair) { if(Math.random() <. 5) { return pair.getFirst(); } else { return pair.getSecond(); } TwoOfAKind p1 = new TwoOfAKind (“a”, “b”); TwoOfAKind p2 = new TwoOfAKind (1, 3.5); Object x = pickOne(p1); Object y = pickOne(p2); TwoOfAKind p1 = new TwoOfAKind (“a”, “b”); TwoOfAKind p2 = new TwoOfAKind (1, 3.5); Object x = pickOne(p1); Object y = pickOne(p2);

29 Generic Interface Example public interface Function { public Y apply(X x); } public interface Function { public Y apply(X x); } public class Square implements Function { public Double apply(Double x) { return x * x; } public class Square implements Function { public Double apply(Double x) { return x * x; } public class isEven implements Function { public Boolean apply(Integer x) { return x % 2 == 0; } public class isEven implements Function { public Boolean apply(Integer x) { return x % 2 == 0; } public class Redness implements Function { public Integer apply(Color color) { return color.getRed(); } public class Redness implements Function { public Integer apply(Color color) { return color.getRed(); } The interface describes one function Each of these non-abstract classes defines that function.

30 Roots The root of a function f(x) is the value of x such that f(x) = 0. – A function may have multiple roots. Problem: given a continuous function f and interval [a, b], find a root in [a, b] if one exists. – Find a root of f(x) = x*x - 1 in [0, 10]. x = 1 – Find a root of f(x) = sin(x) in [-Pi,Pi]. x = -Pi, Pi, 0

31 Bisection method

32 Finding the root of a function public static double root(Function f, double a, double b, double tol) { int NMAX = 50; int n = 1; while(n <= NMAX) { double c = (a + b)/2; double fc = f.apply(c); if(fc == 0 || (b-a)/2 < tol) { return c; } n = n + 1; if(Math.signum(fc) == Math.signum(f.apply(a))) { a = c; } else { b = c; } throw new IllegalArgumentException(); } public static double root(Function f, double a, double b, double tol) { int NMAX = 50; int n = 1; while(n <= NMAX) { double c = (a + b)/2; double fc = f.apply(c); if(fc == 0 || (b-a)/2 < tol) { return c; } n = n + 1; if(Math.signum(fc) == Math.signum(f.apply(a))) { a = c; } else { b = c; } throw new IllegalArgumentException(); }

33 Finding the root of a function public static double squareRoot(final double value) { return root( new Function () { public Double apply(Double x) { return x * x - value; } }, 0, Math.max(value, 1), 1e-12); } public static double squareRoot(final double value) { return root( new Function () { public Double apply(Double x) { return x * x - value; } }, 0, Math.max(value, 1), 1e-12); }


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