An introduction to C++

An introduction to C++#

Exercise 1.1. The sum of two integers#

a) Write a program that adds two integers a and b and writes out their sum.
b) Modify the program from a) so that it prompts the user for the values of the two integers.

Filename: sum_of_two.cpp

Exercise 1.2. Cooking an egg#

The following expression gives the time it takes (in seconds) for the center of the yolk to reach the temperature $T_{y}$ (in Celsius degrees):

t = \frac{M^{2 / 3} c p^{1 / 3}}{K π^{2} (4 π / 3)^{2 / 3}} \ln [0.76 \frac{T_{o} - T_{w}}{T_{y} - T_{w}}]

Here $M$ , $ρ$ , $c$ and $K$ are the mass, density, specific heat capacity and thermal conductivity of the egg respectively. Relevant values are $M = 47 g$ for a small egg and $M = 67 g$ for a large egg, $ρ = 1.038 g c m^{- 3}$ , $c = 3.7 J g^{- 1} K^{- 1}$ , and $K = 5.4 \cdot 10^{- 3} W c m^{- 1} K^{- 1}$ . Furthermore, $T_{w}$ is the temperature of the boiling water, and $T_{o}$ is the original temperature of the egg before being put in the water. For a hard boiled egg, the center of the yolk should reach $T_{y} = 70^{\circ} C$ . Make a program that calculates the time it takes to make a hard boiled egg when $T_{w} = 100^{\circ} C$ and the egg is coming directly from the fridge ( $T_{o} = 4^{\circ} C$ ).

Filename: egg.cpp

Exercise 1.3. Sum of first n integers#

a) Write a program that computes the sum of the integers from 1 up to (and including) $n$ . Compare with the value of $n (n + 1) / 2$ .
b) Modify the program so that n is asked for in the terminal.

Filename: integers.cpp

Exercise 1.4. Generate an approximate Fahrenheit-Celsius conversion table#

The formula for converting $F$ degrees Fahrenheit to $C$ degrees Celsius is $C = 5 / 9 (F - 32)$ . This can be approximated by $C \approx \hat{C} = (F - 30) / 2$ . Write a program that prints a nicely formatted table with three columns: $F$ , $C$ and the approximate value $\hat{C}$ . Also write the table to a file.

Filename: f2c_approx.cpp

Exercise 1.5. The sum of n integers#

Write a program that asks the user how many numbers he wants to add and asks for the value of these numbers. Then the program should print the sum of the numbers.

Filename: sum_many.cpp 1

Exercise 1.6. The sum of n integers from command line#

Write a program that prints the sum of the command line arguments. Hint. For converting from char to int you can use the function atoi.

Filename: sum_command.cpp

Exercise 1.7. Making a function in C++#

Make a function called half that takes an integer argument. The function must print the number it received to the screen, then the program should divide that number by two to make a new number. If the new number is not zero the function then calls the function half passing it the new number as its argument. If the number is zero then the function exits. Call the function half with an argument of 100, the screen output should be

100
50
25
...
...
1

Filename: half.cpp

Exercise 1.8. Making an array#

Make an array with $N$ uniformly spaced values between $a$ and $b$ . Begin with declaring the array, then ﬁll it with a for loop. Print out the elements of the array to check that the result is as wanted.

Filename: array.cpp

Exercise 1.9. Cooking more eggs#

a) Modify your program from Exercise 1.2 so that you get a function returning the time it takes for the center of the yolk to reach a temperature $T_{y}$ when the egg had a temperature $T_{o}$ before cooking. Check that you get the same result as in Exercise 1.2 for $T_{o} = 4^{\circ} C$ and $T_{y} = 70^{\circ} C$ .
b) Make an array of $T_{y}$ values, $T_{y} = {60, 62, 64, 66, 68, 70, 72}$ . Then declare an array t of the same length for time values. Make a for loop and fill in the t array.
c) Use a for loop to print out a nicely formatted table of $T_{y}$ values and the corresponding t values.

Filename: eggs.cpp

Exercise 1.10. Stirling’s approximation#

Stirling’s approximation is

\ln x! \approx x \ln x - x .

a) Write a function taking an integer value $x$ as argument that returns Stirling’s approximation to $\ln x!$ .
b) Print a nicely formatted table with three columns: $x$ , $\ln x!$ and Stirling’s approximation to $\ln x!$ for $x = 2, 5, 10, 50, 100, 1000$ . Also write this table to a file named stirling.txt. Hint. To compute $\ln x!$ you can use lgamma from <cmath> Then $\ln x! = ‘ l g a m m a ‘ (x + 1)$ .
c) Make sure you have stored the $x$ values in an array x. Then declare two arrays exact and approx of the same length as x. Use a for loop to fill these arrays with the exact value and the approximated value of $\ln x!$ for each value in the array x.

Filename: stirling.cpp

Exercise 1.11. Primality checker#

Recall that a prime number is a number greater than 1 that has exactly 2 divisors. Said differently, a number greater than one is a prime if it is divisible by only itself and one. Every number $n$ can be written as a unique product of primes (e.g. $12 = 2 \cdot 2 \cdot 3$ ), this is called the prime factorization of $n$ .

Make a function that takes a number $n$ , and returns true if it’s prime, and false if it’s not. Use the program to ﬁnd all prime numbers up to 100.

Hint: You will only need to check divisibility for numbers up to and including $\sqrt{n}$ , because any greater divisor will imply that there is a divisor less than this.

Filename: prime.cpp

Exercise 1.12. Euler’s totient function#

Two numbers $n$ and $m$ are called relatively prime if they have no common divisors except for 1. That is, no number greater than one should divide both numbers with no residue.

a) Make a function that takes two numbers and returns true if they’re relatively prime and false if they’re not.
b) Euler’s totient function is defined as

$φ (d) = # {Numbers less than d which are relatively prime to d} .$

Implement Euler’s totient function and print $φ (d)$ for $d = 10, 50, 100, 200$ .

Filename: euler.cpp

Exercise 1.13. Converting from base n to decimal#

Make a function long convert_n(long number, int n) that converts a number from base $n$ to decimal for $n = 2, 3, 4, 5, 6, 7, 8, 9, 10$ .

Hint: Remember that a number $d_{0} d_{1} \dots d_{k}$ in base $n$ is the following in decimal: $d_{0} \cdot n^{k} + d_{1} \cdot n^{k - 1} + \dots + d_{k} \cdot n^{0}$ . Some useful operations: / and %.

Filename: convert_n.cpp

Exercise 1.14. Converting from hexadecimal to decimal#

Make a function long convert_hex(string number) that converts a number of the type string from hexadecimal to decimal. The answer should be returned as type long.

Hint: When converting from string to int or long, subtract the zero string '0' before converting or use the function atoi.

Filename: convert_hex.cpp

Exercise 1.15. Add two binary numbers#

a) Make a function long add_binary(long a, long b) that adds two binary numbers $a$ and $b$ .

Hint: Add like you do by hand: start with the last digits. You can use a while loop containing

sum += long(a%10 + b%10 + r)%2)*pow(10, i++);
r = int((a % 10 + b % 10 + r) / 2);
a /= 10;
b /= 10;

Think about what r is. What should you do if r is non-zero after you have gone through all the digits?

b) Modify your program from a) so that you can add any two numbers of base $n$ for $n = 2, 3, 4, 5, 6, 7, 8, 9, 10$ .

Filename: add_n.cpp

Exercise 1.16. Adding fractions#

Write a program to add two fractions and display the resulting fraction. Your program will prompt the user to input the two fractions. The numerator and denominator of each fraction are input separately by space, as illustrated below.

Enter fraction 1 (numerator denominator): 1 3
Enter fraction 2 (numerator denominator): 2 5
Result: 11/15

Hint: You will need to use a struct to define a fraction. The struct has two members: numerator and denominator.

Filename: fraction.cpp

Exercise 1.17. Adding and simplifying fractions#

Modify your program from Exercise 1.16 so that the result is the fraction in it’s simplest form. You should make a function simplify that takes in a fraction and simplifies it. Let the function be of the type void. An example from the terminal:

Enter fraction 1 (numerator denominator): 1 3
Enter fraction 2 (numerator denominator): 2 6
Result: 2/3

Hint: For the function to be able to change the value of the fraction, the argument of the function should be a reference variable: simplify(Fraction& fract) where Fraction is the name of the struct for fractions.

Filename: fraction2.cpp

Exercise 1.18. Make a class for rectangles#

Make a class Rectangle that has two private variables and one member function which will return the area of the rectangle.

Filename: Rectangle.cpp

Exercise 1.19. Make a class for cooking eggs#

a) Make a class Cook_egg that has a public function taking no arguments which returns the time it takes for the egg to be cooked.
b) Add two different methods for changing the mass of the egg. The first method takes the mass of the egg in grams as argument and reassigns the mass of the egg to this mass. The other method takes 'S', 'M' or 'L' as arguments and changes the mass to 47g, 57g or 67g respectively.
c) Add two new methods: one for changing the initial temperature of the egg and one for changing the desired final temperature of the yolk.

Filename: Cook_egg.cpp

Exercise 1.20. Make a class for quadratic functions#

Consider a quadratic function $f (x; a, b, c) = a x + b x + c$ . Make a class Quadratic for representing $f$ , where a, b and c are initial arguments. The class should have three methods: value, table and roots. The value method should compute the value of $f$ at a point $x$ . The table method should write out a table of $x$ and $f$ values for $n$ uniformly spaced $x$ values in the interval $[L, R]$ . The roots method should compute the two roots of the quadratic function. It should accept complex roots. Hint. For the roots method to be able to return two values consider making a structure Two_vals containing two values.

Filename: Quadratic.cpp

Exercise 1.21. Points in different coordinate systems#

Make a class Point to represent a point $(x, y, z)$ in space. This class should have methods for getting the value of $x$ , $y$ and $z$ . Make a subclass SphericalPoint that inherits from Point. The subclass should take the spherical representation of a point $(r, φ, θ)$ as arguments. Call the superclass constructor with the corresponding $x$ , $y$ and $z$ values (recall that $x = r \cos φ \sin θ$ , $y = r \sin φ \sin θ$ and $z = r \cos θ$ ).

Verify the implementation by initializing three points (e.g. the three Cartesian unit vectors) as spherical points and print the corresponding Cartesian coordinates by calling the methods for getting the value of $x$ , $y$ and $z$ .

Filename: Point.cpp

Exercise 1.22. Numerical approximations for the derivative#

Let f(x) be a function and f'(x) its derivative. There are many ways to approximate the derivative, some of which are:

f^{'} (x) \approx \frac{f (x + h) - f (x)}{h},

f^{'} (x) \approx \frac{f (x + h) - f (x - h)}{2 h},

f^{'} (x) \approx \frac{- f (x + 2 h) + 8 f (x + h) - 8 f (x - h) + f (x - 2 h)}{12 h} .

Make a class Diff with a function $f$ as initial argument and implement three methods diff1, diff2, and diff3 for approximating the derivative using the above formulas. The class should also have a method set_h for changing the value of $h$ . ′ Let $f (x) = e^{x}$ and compute $f^{'} (1)$ with the three different methods for $h \in {1, 0.5, 0.2, 0.1, 0.01, 0.001}$ .

You should let $h$ be an array with the different values and loop over it.

Hint: To send a function as an argument to a class you should use pointers, i.e. let the beginning of the class be

class Diff
{
  private:
    double (*f)(double x);
    double h;

  public:
    Diff(double function(double x), double _h = 0.001)
    {
        f = function;
        h = _h;
    }

Filename: Diff.cpp

Exercise 1.23. Numerical approximations for integration#

Let $f (x)$ be a function we want to integrate. The integral $\int_{a}^{b} f (x) d x$ can be approximated in many different ways, some of which are: the midpoint rule, the trapezoidal rule and Simpson’s rule. The midpoint rule gives the following approximation to the integral:

\int_{a}^{b} f (x) d x \approx \sum_{i = 0}^{N - 1} f (a + Δ x (i + 1 / 2)) \cdot Δ x,

where $Δ x = (b - a) / (N)$ and $N$ is the number of intervals the integral is divided into. The trapezoidal rule gives the following approximation to the integral:

\int_{a}^{b} f (x) d x \approx \frac{1}{2} (f (a) + 2 \sum_{i = 1}^{N - 1} f (a + Δ x \cdot i) + f (b)) Δ x

Simpson’s rule gives the following approximation to the integral:

\int_{a}^{b} f (x) d x \approx \frac{1}{3} (f (a) + 4 f (x_{1}) + 2 f (x_{2}) + 4 f (x_{3}) + 2 f (x_{4}) + \dots + 4 f (x_{N - 1})) + f (b)) Δ x,

where $x = a + Δ x \cdot i$ .

Make a class Integration where you implement the three different rules. Include also a method for changing the number of intervals $N$ .

Filename: Integration.cpp

Exercise 1.24. Newton’s method#

Make a class Function which is a subclass of Diff from Exercise 1.22. It should take a function $f (x)$ as an initial variable. Make a method call that takes $x$ as an argument and returns the value of the function for that $x$ . We would like the class to give estimated values for roots of $f$ . That is, points such that $f (x) = 0$ . To do this we implement Newton’s formula. It is given recursively as

x_{n + 1} = x_{n} - \frac{f (x_{n})}{f^{'} (x_{n})}

where we give a starting point $x_{0}$ . In some cases (not all) x_n will approach a root of $f$ . Implement this in a method approx_root that takes a starting point and a bound $ϵ < 1$ as arguments and approximates $x_{n}$ such that $f (x_{n}) < ϵ$ .

Hint: Implement a simple convergence test. Check that $f (x_{n}) < 1$ after 100 iterations. If not terminate the loop and inform the user that there is no convergence for that starting point. It is still a possibility for convergence, but unlikely.

Filename: Function.cpp