Week 4: Exercises

1. Vector

Expand the Vector class that we begin writing in the lecture:

class Vector {
    double[] v;
    
    public Vector(params double[] v) {
        this.v = v;
    }

    …
}

Add these elements:

• a read-only property dims that returns the number of dimensions of the vector

• a read-only property isZero that is true if the vector is zero

• an operator + that adds two vectors

• an operator * that multiplies a scalar times a vector, yielding a new vector

• an operator * that computes the dot product of two vectors, yielding a scalar

• a method angle that computes the angle between two vectors in radians

2. Polynomials

Design and implement a C# class Polynomial representing a polynomial of a single variable. Your class should include the following:

• a constructor that takes a variable number of arguments of type double, yielding a polynomial with those coefficients. For example,

      new Polynomial(1.0, 4.0, 5.0)

  should yield the polynomial x² + 4x + 5.

• an overloaded operator + that adds two polynomials

• an overloaded operator - that subtracts two polynomials

• an overloaded operator * that multiplies two polynomials, or a polynomial and a scalar

• a method that evaluates a polynomial for a given value of x

• a method that returns the first derivative of a polynomial

• a method that finds an approximate zero of a polynomial, i.e. a value x for which the polynomial evaluates to zero

• a ToString() method that yields a string such as "x^2 + 4x + 5"

• a static method Parse() that parses a string such as "x^2 + 4x + 5", yielding a polynomial.

3. Queue

Write a class Queue implementing a queue of integers. A queue should have an unlimited size. The Queue class should have these members:

• Queue()

• bool isEmpty()

• void enqueue(int i)

• int dequeue()

• bool tryDequeue(out int i)

All operations should run in O(1).

4. Double-Ended Queue

Write a class Deque implementing a double-ended queue of integers:

Your class should have these members:

• Deque()

• bool isEmpty()

• void enqueueFirst(int i)

• int dequeueFirst()

• void enqueueLast(int i)

• int dequeueLast()

All operations should run in O(1). In dequeueFirst() and dequeueLast(), you may assume that the queue is not empty.