Problems:

CS100 Homework 6 (Spring 2023)#

There will be four problems in total: one set of objective questions and three programming problems.

(May 4 22:22) Problem 2 has been added.

(May 9 02:38) Problem 3 has been added.

(May 9 07:12) Problem 4 has been added.

Unit Test: Inheritance and Operator Overloading#

This problem is to answer questions.

👉 Go to GitHub for the HTML page file

Dynamic Array 3.0#

This problem is based on the Dynarray you wrote in Homework 6 Problem 2. Before adding anything new to it, your Dynarray should meet all the requirements in that problem first.

In this task, your Dynarray should support the following new things.

Subscript operator#

The Dynarray should support the subscript operator, so that we can use a[i] instead of a.at(i) to access the i-th element.

Let a be an object of type Dynarray or const Dynarray. The behavior of a[i] should be exactly the same as a.at(i) , except that the subscript operator does not perform bounds checking. That is, no exception should be thrown if i >= a.size().

Relational operators#

The Dynarray should support the six relational operators: <, <=, >, >=, == and !=. These operators perform lexicographical comparison of two Dynarray.

Lexicographical comparison is an operation with the following properties:

Two ranges are compared element by element.
The first mismatching element defines which range is lexicographically less or greater than the other.
If one range is a prefix of another, the shorter range is lexicographically less than the other.
If two ranges have equivalent elements and are of the same length, then the ranges are lexicographically equal.
An empty range is lexicographically less than any non-empty range.
Two empty ranges are lexicographically equal.

Since we use C++17, you still have to define all six of them. It is often good practice to implement operator< and operator== first, and define the rest in terms of them.

Note that in homework 8, we will make this Dynarray a class template Dynarray<T>, and we should always minimize the requirements on unknown types when we do generic programming. Since C++17 does not have compiler-generated comparison operators, we suggest that your implementation depend only upon the operator< and operator== of the element type.

You are free to choose to define them as either members or non-members.

Output operator#

We want to print a Dynarray directly using operator<<. For example,

int arr[] = {1, 2, 3, 5};
Dynarray a(arr, arr + 4);
Dynarray b;
std::cout << a << '\n' << b << std::endl;

The output is as follows.

[1, 2, 3, 5]
[]

In details:

Elements are separated by a comma ( , ) followed by a space.
The printed content starts with [ and ends with ']' . If the dynamic array is empty, just print an empty pair of brackets [] .

OJ tests#

There are three subtasks on OJ. Subtask $i$ will be run only if all the testcases of subtask $i-1$ are passed.

Subtask 1 is a compile-time check. Subtask 2 contains all the testcases from Homework 5 Problem 3 and Homework 6 Problem 2. Subtask 3 contains the new testcases specific for this problem.

Polynomial#

In this task, you will design and implement a class Polynomial representing a real polynomial of the following form:

P(x)=\sum_{i=0}^na_ix^i,\quad a_i,x\in\mathbb R

where $a_0,\cdots,a_n$ are the coefficients and $a_n\neq 0$ . The degree of a polynomial is the highest degree of the terms with non-zero coefficients, denoted by

\deg(P)=n.

In particular, the following corner cases are defined.

$a_0x^0=a_0$ even if $x=0$ .
A polynomial has at least one coefficient. When $n=0$ , the polynomial becomes a constant function $P(x)=a_0$ whose degree is $0$ , even if $a_0=0$ .

There are a number of operations that can be performed on polynomials, including addition, subtraction, multiplication, etc. To ensure that the coefficient of the highest-order term is always non-zero, we may need a function adjust to remove the unnecessary trailing zeros.

class Polynomial {
 private:
  // `m_coeffs` stores the coefficients.
  // Note: This is not the unique correct implementation.
  // For example, you may separate the constant term from others,
  // and store the constant term using another variable.
  std::vector<double> m_coeffs;

  static auto isZero(double x) {
    static constexpr auto eps = 1e-8;
    return x < eps && -x < eps;
  }

  // Remove trailing zeros, to ensure that the coefficient of the term with
  // the highest order is non-zero.
  // Note that a polynomial should have at least one term, which is the
  // constant. It should not be removed even if it is zero.
  // If m_coeffs is empty, adjust() should also insert a zero into m_coeffs.
  void adjust() {
    // YOUR CODE HERE
  }

  // Other members ...
};

Considering potential floating-point errors, it is suggested to use the function isZero(x) to determine whether x is zero, which is effectively $|x|<\epsilon$ for some very small $\epsilon>0$ .

The coefficients, of course, should be stored in a sequential container, and std::vector is a perfect choice.

Constructors and copy control#

The class Polynomial should meet the following requirements.

Default-constructible. A Polynomial can be default-initialized to the constant function $P(x)=0$ .
Copyable, movable and destructible. A Polynomial should have a copy constructor, a move constructor, a copy assignment operator, a move assignment operator and a destructor. These functions should perform the corresponding operations directly on the member m_coeffs. Before starting to define them, think about the following questions:
- Will the compiler generate these functions for you if you do not define them?
- What are the behaviors of the compiler-generated functions? Do they meet the requirements?
Note: Direct moving of the member m_coeffs will possibly make the moved-from polynomial have no coefficients (its m_coeffs will possibly be empty), which is not in a valid state if we require all Polynomials to have at least one coefficient. But considering that this is not the concentration of this problem, we only require that the moved-from object can be safely assigned to or destroyed, which means that the compiler-generated move operations suffice. The move operations of Polynomial are not required to be noexcept here. You are free to choose any reasonable design.

Constructible from a pair of InputIterators. This function has been implemented for you, as long as your adjust() is correct.

template <typename InputIterator>
Polynomial(InputIterator begin, InputIterator end)
    : m_coeffs(begin, end) { adjust(); }

This allows the following ways of constructing a Polynomial:

double a[] = {1, 2, -1, 3.5};
std::vector b{2.5, 3.3, 0.0};
std::list c{2.7, 1.828, 3.2}; // not a vector, but have iterators
std::deque<double> d; // empty
Polynomial p1(a, a + 4); // 1 + 2x - x^2 + 3.5*x^3
Polynomial p2(b.begin(), b.end()); // 2.5 + 3.3x, the trailing zero removed
Polynomial p3(c.begin(), c.end()); // 2.7 + 1.828x + 3.2*x^2
Polynomial p4(d.begin(), d.end()); // 0

If the iterator range [begin, end) is empty, i.e. begin == end, this constructor initializes the Polynomial to be P(x)=0.

Constructible from a const std::vector<double>, which contains the coefficients.
```
std::vector a{2.5, 3.3, 4.2, 0.0, 0.0};
std::vector<double> b; // empty
Polynomial p1(a); // 2.5 + 3.3x + 4.2*x^2.
Polynomial p2(b); // 0
Polynomial p3(std::move(a)); // Move it, instead of copying it.
```
Note that if the argument passed in is a non-const rvalue, it should be moved instead of being copied. Moreover, implicit conversion from a std::vector<double> to Polynomial through this constructor should be disabled.

Basic operations#

Let p be of type Polynomial and cp be of type const Polynomial. The following operations should be supported.

Both p.deg() and cp.deg() return the degree of the polynomial. Any reasonable integral return type is allowed, but it should not be too small.
For an integer i, both p[i] and cp[i] return the coefficient of the term $x^i$ as a read-only number. Any reasonable return type is allowed. The behavior is undefined if i is not in the valid range [0, deg()].
For an integer i and a real number c, p.setCoeff(i, c) sets the coefficient of the term $x^i$ to c. The behavior is undefined if i < 0. If i > deg(), it has the same effect as adding a monomial $cx^i$ to it, which means that after this operation,
- p.deg() becomes i.
- p[j] is zero for every j in (d, i), where d is the degree of p before this operation.
Note that allowing the user to modify the coefficients may result in trailing zeros, which your code must handle correctly. This is why we don’t allow p[i] to return the non-const reference to the i-th coefficient.
std::vector<T>::resize may help you.
operator== and operator!= should be defined. Two polynomials a and b are equal if and only if a - b is the zero constant function $P(x)=0$ . Equivalently, a == b if and only if
- a.deg() == b.deg(), and
- isZero(a[i] - b[i]) holds for every integer i in [0, a.deg()].
For a number x0 (suppose it is a double), both p(x0) and cp(x0) return the value of the polynomial at $x=x_0$ , i.e. $P\left(x_0\right)$ .

Arithmetic operations#

The arithmetic operators (as well as the corresponding compound assignment operators) that need to be supported are as follows.

-a (the unary negation operator), a + b, a += b, a - b, a -= b.
a * b, a *= b.

Derivatives and integrals#

For a polynomial p,

p.derivative() should return the derivative of p, i.e. $\displaystyle\frac{\mathrm d p(x)}{\mathrm dx}$ . Note that this should still be a polynomial.
p.integral() should return the integral of p, defined as
$\int_0^xp(t)\mathrm dt.$
Note that this should still be a polynomial.

Notes#

For each member function, should it be const, or non-const, or having the “const vs non-const overloads”? Think about these questions carefully before coding.

Regarding the floating-point errors: we will not test this on purpose, but such problems might show up accidentally and cause unexpected results. To avoid such risks, it is suggested to always use isZero(x) and isZero(a - b), instead of directly comparing floating-point numbers.

Compile-time requirements test#

We have provided a compile-time test compile_test.cpp for you.

Attachments#

👉 Go to GitHub

Expression#

In recitations, we showed an example of designing a class hierarchy for different kinds of nodes in an abstract syntax tree (AST). In this problem, we will support more functionalities:

A new kind of node called Variable, which represents the varaible x. With this supported, our AST can be used to represent unary functions, instead of only arithmetic expressions with constants.
Evaluation of the function at a certain point $x=x_0$ .
Evaluation of the derivative of the function at a certain point $x=x_0$ .

The operations that need to be supported are declared in this base class:

class NodeBase {
 public:
  // Make any of these functions virtual, or pure virtual, if necessary.
  NodeBase() = default;
  double eval(double x) const; // Evaluate f(x)
  double derivative(double x) const; // Evaluate df(x)/dx
  std::string rep() const; // Returns the parenthesized representation of the function.
  ~NodeBase() = default;
};

In recitations, we used only one class BinaryOperator to represent four different kinds of binary operators. Such design may not be convenient for this problem, because the ways of evaluating the function and its derivative differ to a greater extent between different operators. Therefore, we separate them into four classes:

class BinaryOperator : public NodeBase {
 protected:
  Expr m_left;
  Expr m_right;
 public:
  BinaryOperator(const Expr &left, const Expr &right)
      : m_left{left}, m_right{right} {}
};

class PlusOp : public BinaryOperator {
  using BinaryOperator::BinaryOperator;
  // Add functions if necessary.
};

class MinusOp : public BinaryOperator {
  using BinaryOperator::BinaryOperator;
  // Add functions if necessary.
};

class MultiplyOp : public BinaryOperator {
  using BinaryOperator::BinaryOperator;
  // Add functions if necessary.
};

class DivideOp : public BinaryOperator {
  using BinaryOperator::BinaryOperator;
  // Add functions if necessary.
};

By declaring using BinaryOperator::BinaryOperator;, we are inheriting the constructor from BinaryOperator. For example, the class PlusOp now has a constructor like this:

class PlusOp : public BinaryOperator {
 public:
  PlusOp(const Expr &left, const Expr &right) : BinaryOperator(left, right) {}
};

and the same for MinusOp, MultiplyOp and DivideOp.

To support the variable x, we define a new class Variable:

class Variable : public NodeBase {
  // Add functions if necessary.
};

Then we define a static member of Expr as follows.

class Expr {
  std::shared_ptr<NodeBase> m_ptr;
  Expr(std::shared_ptr<NodeBase> ptr) : m_ptr{ptr} {}

 public:
  Expr(double value);

  static const Expr x;
  // Other members ...
};

// After the definition of `Variable`:
const Expr Expr::x{std::make_shared<Variable>()};

Note that Expr also has a non-explicit constructor that receives a double and creates a Constant node:

Expr::Expr(double value) : m_ptr{std::make_shared<Constant>(value)} {}

With all of these, and the overloaded arithmetic operators defined, we can create functions in a very convenient manner:

auto &x = Expr::x;
auto f = x * x + 2 * x + 1; // x^2 + 2x + 1
std::cout << f.rep() << std::endl; // (((x) * (x)) + ((2.000000) * (x))) + (1.000000)
std::cout << f.eval(3) << std::endl; // 16
std::cout << f.derivative(3) << std::endl; // 8
auto g = f / (-x * x + x - 1); // (x^2 + 2x + 1)/(-x^2 + x - 1)
std::cout << g.eval(3) << std::endl; // -2.28571
std::cout << g.derivative(3) << std::endl; // 0.489796

Requirements for `rep()`#

Note: This is a very simple and naive way to convert an expression to a string, because we don’t want to set barriers on this part. You can search for some algorithms to make the expression as simplified as possible, but it may not pass the OJ tests.

Any operand of the unary operators ( +, - ) or the binary operators ( +, -, *, / ) should be surrounded by a pair of parentheses. Correct examples:

$2+3$ : (2.000000) + (3.000000)
$2x+3$ : ((2.000000) * (x)) + (3.000000)
$2\cdot(-x)+3$ : ((2.000000) * (-(x))) + (3.000000)

There should be a space before and after each binary operator. For example, (expr1) + (expr2) instead of (expr1)+(expr2).

To convert a floating-point number to std::string, just use std::to_string and you will be free of troubles caused by precisions and floating-point errors.

Note: If the floating-point value of a Constant node is negative, it is not treated as a unary minus sign applied to its absolute value. For example,

Expr e(-2);
std::cout << e.rep() << std::endl;

The output should be -2.000000 instead of -(2.000000).

example.cpp contains these simple tests.

Requirements for `Expr`#

From the user’s perspective, only Expr and its arithmetic operators are interfaces. Anything else in your code is implementation details, which user code will not rely on. In other words, you are free to implement these things in any way, as long as the interfaces of Expr meet our requirements.

Expr is copy-constructible, copy-assignable, move-constructible, move-assignable and destructible. These operations should just perform the corresponding operations on the member m_ptr, and let the corresponding function of std::shared_ptr handle everything. The move operations should be noexcept.
Expr is constructible from a double value, and this constructor is non-explicit.
Let e be const Expr and x0 be double, then the subtree rooted at e represents a function. e.eval(x0) returns the value of this function at $x=x_0$ . e.derivative(x0) returns the value of the derivative of this function at $x=x_0$ . e.rep() returns a std::string representing this function.
Let e1 and e2 be two objects of type const Expr. The following expressions return an object of type Expr, which creates a new node corresponding to the operators.
- -e1
- +e1
- e1 + e2
- e1 - e2
- e1 * e2
- e1 / e2
Expr::x is an object of type const Expr, which represents the variable x.

Use compile_test.cpp to check whether your code compiles.

Submission#

Submit expr.hpp or its contents to OJ.

Thinking#

Why do we use std::shared_ptr instead of std::unique_ptr?

Can you add support for more functionalities, e.g. the elementary functions $\sin(e)$ , $\cos(e)$ , exponential expressions $e_1^{e_2}$ , …? More variables? Print expressions to $\LaTeX$ ?

Attachments#

👉 Go to GitHub

CS100 Homework 6 (Spring 2023)#

Unit Test: Inheritance and Operator Overloading#

Dynamic Array 3.0#

Subscript operator#

Relational operators#

Output operator#

OJ tests#

Polynomial#

Constructors and copy control#

Basic operations#

Arithmetic operations#

Derivatives and integrals#

Notes#

Compile-time requirements test#

Attachments#

Expression#

Requirements for rep()#

Requirements for Expr#

Submission#

Thinking#

Attachments#

Requirements for `rep()`#

Requirements for `Expr`#