CS 135 Lab Assignments Week 13

Transcription

1 CS 135 Lab Assignments Week 13 Professor: Matt B Pedersen This handout is the assignment that you must finish for the lab portion of the course in week 13 You must finish the assignments yourself; if you need any help ask the TA or the professor You may not work together with other students in solving the assignments The weekly assignments must be handed in on a piece of paper, typed neatly (spell checked etc) and look nice The assignments are due in the the professor s mailbox labeled Matt Pedersen in TBE-A11 the following Wednesday before the computer science office closes at 5PM All lab assignments must be handed in if you wish to attend the final exam Late assignments are not accepted for any reason except sickness or family emergencies, and the former requires a doctor s notice 1 Introduction This is the last lab of the semester, and it deals with structs We are going to implement a library (or at least a set of functions) to work on complex numbers Complex Numbers A complex number can most easily be thought of as a point in the plane, like (a, b); so a along the x-axis and b along the y axis; such a number is written as a + bi where i is 1 Both a and b are real numbers, and i is an imaginary number For a complex number a + bi, a is called the real part of the number, and b is called the imaginary part The following pictures illustrates the interpretation of a + bi as a point in the plane 1

2 3 Representation in C++ Representing a complex number in C++ is easy; we simply use a struct with two fields: one for the real part and one for the imaginary part: struct Complex { double real; double imag; } ; Recall, if c is a variable of type Complex, we can access the fields as creal and cimag 4 Operations on Complex Numbers Like a whole bunch of operations like addition, multiplication, inverse etc exist for real numbers, so do they for complex numbers Let us start with addition If a + bi and c + di are complex numbers, then (a + bi) + (c + di) = (a + c) + (b + d)i We can now easily implement this function in C++ in the following way: Complex add(complex x, Complex y) { Complex c; // declare a new Complex creal = xreal + yreal; // Compute the real part cimag = ximag + yimag; // Compute the imag part return c; // return the resulting sum }

3 I have written the following test code in a main (Remember you can write literals of structs by using a set of { } like in the first two lines of main): int main() { Complex a = {3,8}; Complex b = {-4,}; cout << "a = " << tostring(a) << endl; cout << "b = " << tostring(b) << endl; cout << "Real(a) = " << realpart(a) << endl; cout << "Img(a) = " << imagpart(a) << endl; cout << "a + b = " << tostring(add(a,b)) << endl; cout << "a - b = " << tostring(subtract(a,b)) << endl; cout << "a * b = " << tostring(multiply(a,b)) << endl; cout << "a / b = " << tostring(divide(a,b)) << endl; cout << " a = " << length(a) << endl; cout << " b = " << length(b) << endl; cout << "5 * a = " << tostring(multiplyreal(a,5)) << endl; cout << "con(a) = " << tostring(conjugate(a)) << endl; cout << "1/a = " << tostring(reciprocal(a)) << endl; cout << "a*1/a = " << tostring(multiply(a,reciprocal(a))) << endl; cout << "a*con(a) = " << tostring(multiply(a, conjugate(a))) << endl; cout << "sqrt(a) = " << tostring(squareroot(a)) << endl; cout << "polar(a) = " << topolarstring(a) << endl; Complex c = {0, 1}; cout << "c = " << tostring(c) << endl; cout << "c*c = " << tostring(multiply(c,c)) << endl; } which you must use as is This means you have to implement a whole lot of functions, which we will define now You can grab the main above from the course website if you don t want to type it in! Let us start with a function called tostring() which takes in a complex number and returns a string with the number in For example if Complex c = 4,5 then passing c to tostring(c) should return the string "4 + 5i" Consider this little snippet of code (which will help you implement tostring(): #include <sstream> 3

4 a = 10; stringstream ss; ss << "a has the value " << a; string s = ssstr(); At the end, s contains the string "a has the value 10" Implement tostring() to take in a Complex number and return a string with that number in; be careful if the imaginary part is negative, then we should not get i but 4-5i 5 Simple Binary Operations Let us define how to subtract, multiply and divide complex numbers: Subtraction: Multiplication: (a + bi) (c + di) = (a c) + (b d)i Division: (a + bi) (c + di) = (a c b d) + (a d + b c)i ( ) a c + b d (a + bi)/(c + di) = c + d + Multiply by real number c: Equality: c (a + bi) = (c a) + (c b)i ( b c a d c + d (a + bi) = (c + di) (a = c) (b = d) Implement subtract(), multiply(), and divide() which all take in two complex numbers and return a new one Implement multiplyreal() which takes in a complex number and a double value and returns a new complex number Implement equal() which takes in two complex numbers and returns a Boolean value ) i 4

5 6 Unary Operations Two have two noteworthy unary operations on complex numbers: conjugation and reciprocal Let us start with the conjugate You can think of the conjugate of a complex number as the result of simply flipping it around the x-axis; the conjugate of a complex number a is denoted a so: Conjugate: a + bi = a bi The reciprocal of a complex number z is just 1/z: Reciprocal: 1 a + bi = a + bi (a + bi) (a + bi) = a bi a + b Implement conjugate() and reciprocal() which both take in a complex number and returns a new complex number A complex number, interpreted as a point in the plane, and thus as a vector, has a length (which we denote ): Length: a + bi = a + b Implement the length() function, which takes in a complex number and returns a double In order to compute square roots of complex numbers, we need a helper function called sgn (short for signum), and it is defined as follows: 1 if d < 0 sgn(d) = 0 if d = 0 1 if d > 0 sgn() takes in a double number and returns an integer The square root function is defined as follows for the positive root (we will skip the negative root for now): 5

6 a + bi = a + a + b + sgn(b) a + a + b i Implement sgn() which takes in a double and returns am integer, and also implement squareroot() which takes in and returns a complex number Implement a function called realpart() that takes in a complex number and returns its real part Also implement a function called ImagPart() which takes in a complex number and returns its imaginary part Both functions return values of type double 7 Polar Coordinates As you probably know, any point in the plane is also a representative for a vector starting at (0,0) and ending in the point in question We could express such a vector as a length and an angle from the x-axis to the vector Therefore, a complex number can be converted to polar coordinates as well We shall write a function that takes in a complex number and returns a string For example, the complex number 3 + 8i in polar coordinates is 8544[cos(1103) + i sin(1103)] As you can seem the 8544 is the length ( = 8544) but the cos and sin are not obvious The rule is as follows: if x + yi is a complex number with length r, then its polar form is: r[cos(ϕ) + i sin(ϕ)], where ϕ is computed in the following way: ϕ = arctan( y x ) if x > 0 arctan( y x ) + π if x < 0 y 0 arctan( y x ) π if x < 0 y < 0 π if x = 0 y > 0 π if x = 0 y < 0? if x = 0 y = 0 Implement topolarstring() which takes in a complex number and returns a string of the form r[cos(ϕ) + i sin(ϕ)] with the appropriate values for r and ϕ as described above You might want to add the following global constant to your code: const double PI = ; 6

7 8 Testing Here is the result of running the main() I stated earlier in the text: a = 3 + 8i b = -4 + i Real(a) = 3 Img(a) = 8 a + b = i a - b = 7 + 6i a * b = -8-6i a / b = 0-19i a = 8544 b = * a = i con(a) = 3-8i 1/a = i a*1/a = 1 a*con(a) = 73 sqrt(a) = i polar(a) = 8544[cos(1103) + i*sin(1103)] c = 0 + 1i c*c = -1 7