Operator Overloading. a flop = a floating-point operation overloading arithmetical operators counting number of flops in a sum
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1 Operator Overloading 1 OOP to count Flops a flop = a floating-point operation overloading arithmetical operators counting number of flops in a sum 2 Quaternions hypercomplex numbers application in computer graphics 3 Operator Overloading the classquaternion defining functional attributes 4 Summary + Assignments MCS 260 Lecture 25 Introduction to Computer Science Jan Verschelde, 9 March 2016 Intro to Computer Science (MCS 260) operator overloading L-25 9 March / 41
2 Object-Oriented Programming A definition from Grady Booch et al.: Object-oriented programming is a method of implementation in which 1 programs are organized as cooperative collections of objects, 2 each of which represents an instance of some class, 3 and whose classes are all members of a hierarchy of classes united via inheritance relationships. Objects not algorithms are the building blocks. Algorithms are central in procedure-oriented programming. Intro to Computer Science (MCS 260) operator overloading L-25 9 March / 41
3 Operator Overloading 1 OOP to count Flops a flop = a floating-point operation overloading arithmetical operators counting number of flops in a sum 2 Quaternions hypercomplex numbers application in computer graphics 3 Operator Overloading the classquaternion defining functional attributes 4 Summary + Assignments Intro to Computer Science (MCS 260) operator overloading L-25 9 March / 41
4 Counting Flops floating-point operations A flop is short for floating-point operation. In scientific computation, the cost analysis is often measured in flops. Note: before version 6, MATLAB had aflops command. Using Object Oriented Programming: 1 we define a classflopfloat, 2 every object stores its #flops: these are the flops used to compute the number, 3 the overloaded arithmetical operators count also the flops for each result. Intro to Computer Science (MCS 260) operator overloading L-25 9 March / 41
5 the classflopfloat class FlopFloat(object): An object of the class FlopFloat records the number of floating-point operations executed to compute the float. def init (self, f=0.0, n=0): Construct a FlopFloat from a number f and an operational cost n, by default 0. self.float = float(f) self.flops = n Intro to Computer Science (MCS 260) operator overloading L-25 9 March / 41
6 interactive usage If we save the class definition inflopfloats.py, then we can import the definition as follows: >>> from flopfloats import FlopFloat >>> x = FlopFloat( ) >>> x e+00 >>> str(x) e+00 >>> print(x) e+00 >>> x. str ( %.8f ) The representation of aflopfloat is defined by repr (). The string representation is defined by str (). Intro to Computer Science (MCS 260) operator overloading L-25 9 March / 41
7 representations offlopfloat def str (self, form= %.4e ): Uses the formatting string in the % to return a string representation. return form % self.float def repr (self): Returns the default string representation. return str(self) Because of the definition of repr (), we have: >>> x e+00 Intro to Computer Science (MCS 260) operator overloading L-25 9 March / 41
8 Operator Overloading 1 OOP to count Flops a flop = a floating-point operation overloading arithmetical operators counting number of flops in a sum 2 Quaternions hypercomplex numbers application in computer graphics 3 Operator Overloading the classquaternion defining functional attributes 4 Summary + Assignments Intro to Computer Science (MCS 260) operator overloading L-25 9 March / 41
9 overloading operators Recall the addition of strings: >>> "ab" + "bc" abbc and the addition of lists: >>> [1,2] + [2,3] [1, 2, 2, 3] The+operator is defined via the add method: >>> L = [1,2] >>> L. add ([3,4]) [1, 2, 3, 4] Intro to Computer Science (MCS 260) operator overloading L-25 9 March / 41
10 the classlist Typinghelp(list) in a Python session shows: Help on class list in module builtin : class list(object) list() -> new list list(sequence) -> new list initialized from sequence s items Methods defined here: add (...) x. add (y) <==> x+y... Intro to Computer Science (MCS 260) operator overloading L-25 9 March / 41
11 comparison operators and methods comparison operation operator method equality == eq inequality!= ne less than < lt greater than > gt less or equal <= le greater or equal >= ge Intro to Computer Science (MCS 260) operator overloading L-25 9 March / 41
12 motivation for an equality test Without having defined the equality inflopfloat: >>> x = FlopFloat(3) >>> x == 3 False But we have: >>> x.float == 3 True And consider: >>> y = FlopFloat(3) >>> x == y False We getfalse becausexandyare different objects. Intro to Computer Science (MCS 260) operator overloading L-25 9 March / 41
13 after extendingflopfloat with eq >>> from flopfloats import FlopFloat >>> x = FlopFloat(3) >>> x == 3 True >>> y = FlopFloat(3) >>> x == y True >>> z = x + 0 >>> z e+00 >>> z == x False >>> z.flops 1 >>> x.flops 0 Intro to Computer Science (MCS 260) operator overloading L-25 9 March / 41
14 defining the equality operator def eq (self, other): Two flopfloats are equal if both their float and flops match. if isinstance(other, FlopFloat): return (other.float == self.float) \ and (other.flops == self.flops) else: return self.float == other Intro to Computer Science (MCS 260) operator overloading L-25 9 March / 41
15 arithmetical operators and methods arithmetical operation operator method negation - neg addition + add inplace addition += iadd reflected addition + radd subtraction - sub inplace subtraction -= isub reflected subtraction -= rsub multiplication * mul inplace multiplication *= imul reflected multiplication * rmul division / div inplace division /= idiv reflected division /= rdiv invert invert power ** pow Intro to Computer Science (MCS 260) operator overloading L-25 9 March / 41
16 Overloading Operators, e.g.: addition Theother may be an ordinaryfloat: def add (self, other): Returns the result of the addition, the other may be an ordinary number. if isinstance(other, FlopFloat): result = FlopFloat(self.float + other.float) result.flops = self.flops + other.flops + 1 else: # treat other just as ordinary number result = FlopFloat(self.float + other) result.flops = self.flops + 1 return result Other arithmetical operations are defined similarly. We store the class definition in the fileflopfloats.py and import it as a module. Intro to Computer Science (MCS 260) operator overloading L-25 9 March / 41
17 reflected addition The reflected (or swapped) addition happens when the first operand in + is not aflopfloat but an ordinary number. def radd (self, other): Addition when operand is not a FlopFloat, as in x or 3 + x (reflected operand). result = self + other return result Whenxis aflopfloat, thenx+y is executed asx. add (y), wherexisself andyis other. Forx + y whenxis not aflopfloat, butyis aflopfloat, theny. radd (x) is executed. Intro to Computer Science (MCS 260) operator overloading L-25 9 March / 41
18 inplace addition and negation The inplace operator allows for shorter notation. def iadd (self, other): Defines the inplace addition, x += y. result = self + other return result Flipping sign is a unary operator (noother): def neg (self): Returns -x. result = self result.float = -result.float # flipping sign does not count as a flop return result Intro to Computer Science (MCS 260) operator overloading L-25 9 March / 41
19 Operator Overloading 1 OOP to count Flops a flop = a floating-point operation overloading arithmetical operators counting number of flops in a sum 2 Quaternions hypercomplex numbers application in computer graphics 3 Operator Overloading the classquaternion defining functional attributes 4 Summary + Assignments Intro to Computer Science (MCS 260) operator overloading L-25 9 March / 41
20 A Simple Test of Use summing numbers inflopsum.py We use FlopFloats to count the number of operations when summing n floats. from flopfloats import FlopFloat from random import gauss print( counting flops in a sum + \ of n floating-point numbers ) N = int(input( Give n : )) RANDSUM = FlopFloat() for i in range(0, N): RANDSUM = RANDSUM + gauss(0, 1) print( sum = + str(randsum) + \ #flops is %d % RANDSUM.flops) Intro to Computer Science (MCS 260) operator overloading L-25 9 March / 41
21 runningflopsum At the command prompt$: $ python flopsum.py counting flops in a sum of n floats Give n : 100 sum = e+00 #flops is 100 It works! A less trivial application: UseFlopFloats to count #operations to evaluate p(x) = 6x x 3 x x Intro to Computer Science (MCS 260) operator overloading L-25 9 March / 41
22 Operator Overloading 1 OOP to count Flops a flop = a floating-point operation overloading arithmetical operators counting number of flops in a sum 2 Quaternions hypercomplex numbers application in computer graphics 3 Operator Overloading the classquaternion defining functional attributes 4 Summary + Assignments Intro to Computer Science (MCS 260) operator overloading L-25 9 March / 41
23 Complex Numbers Typinghelp(complex) in a Python session shows Help on class complex in module builtin : class complex(object) complex(real[, imag]) -> complex number Create a complex number from a real part and an optional imaginary part. This is equivalent to (real + imag*1j) where imag defaults to 0. Methods defined here: abs (...) x. abs () <==> abs(x)... Intro to Computer Science (MCS 260) operator overloading L-25 9 March / 41
24 Quaternion hypercomplex numbers A quaternion q (or a hypercomplex number) is q = a 0 + a 1 i + a 2 j + a 3 k, where the tuple (a 0, a 1, a 2, a 3 ) is the coefficient vector of q and the symbols i, j, and k satisfy i 2 = 1 j 2 = 1, k 2 = 1, ij = k, jk = i, ki = j, ji = k, kj = i, ik = j, defining the multiplication of two quaternions. The set of all quaternions is often denoted by H, in honor of Sir William Rowan Hamilton who introduced them in 1843 before vector algebra was known. Intro to Computer Science (MCS 260) operator overloading L-25 9 March / 41
25 Operator Overloading 1 OOP to count Flops a flop = a floating-point operation overloading arithmetical operators counting number of flops in a sum 2 Quaternions hypercomplex numbers application in computer graphics 3 Operator Overloading the classquaternion defining functional attributes 4 Summary + Assignments Intro to Computer Science (MCS 260) operator overloading L-25 9 March / 41
26 Applications of Quaternions computer graphics Quaternions represent coordinate transformations in 3-space more compactly than matrices: q = (a 0, a), a = (a 1, a 2, a 3 ). Also composition of coordinate transformations goes faster with quaternions. Quaternion multiplication with scalar, dot ( ), and cross product ( ): (a 0, a) (b 0, b) = a 0 b 0 a b + a 0 a + b 0 b + a b. Intro to Computer Science (MCS 260) operator overloading L-25 9 March / 41
27 Rotations in Space Rotation about a unit vector u by angle θ: q = (s, v ) where { s = cos(θ/2), v = sin(θ/2) u. Applying the rotation to a point p = (x, y, z): 1 represent p by the quaternion P = (0, p), 2 compute q P q 1. For q = (q 0, q 1, q 2, q 3 ), its inverse is q 1 = q q 2, where the conjugate of q is q = (q 0, q 1, q 2, q 3 ), and the magnitude of q satisfies q 2 = q q2 1 + q2 2 + q2 3. Intro to Computer Science (MCS 260) operator overloading L-25 9 March / 41
28 Operator Overloading 1 OOP to count Flops a flop = a floating-point operation overloading arithmetical operators counting number of flops in a sum 2 Quaternions hypercomplex numbers application in computer graphics 3 Operator Overloading the classquaternion defining functional attributes 4 Summary + Assignments Intro to Computer Science (MCS 260) operator overloading L-25 9 March / 41
29 the classquaternion in the filequaternion.py Another illustration of operator overloading. class Quaternion(object): Quaternions are hypercomplex numbers. With the class definition in a file: >>> import quaternion >>> quaternion. doc \nanother illustration of operator overloading.\n >>> from quaternion import Quaternion >>> Quaternion. doc \n Quaternions are hypercomplex numbers.\n The doc gives the documentation strings of the modulequaternion and the classquaternion. Intro to Computer Science (MCS 260) operator overloading L-25 9 March / 41
30 data attributes for classes and objects class Quaternion(object): # documentation string omitted count = 0 # class data attributes who = [] def init (self, a=0, b=0, c=0, d=0): Constructs the Quaternion with coefficients a,b,c,d. self.cre = a # real part self.cim = b # imaginary part self.cjj = c # j-part self.ckk = d # k-part Quaternion.count += 1 Quaternion.who.append(self) count andwho are class data attributes a,b,c, anddare object data attributes Intro to Computer Science (MCS 260) operator overloading L-25 9 March / 41
31 instances of data attributes for classes and objects Consider >>> from quaternion import Quaternion >>> x = Quaternion(2,3,0,1) >>> y = Quaternion(0,1,0,0) >>> Quaternion.count 2 The numbers 2,3,0,1 are attributes for the objectx, just as 0,1,0,0 are fory. The value 2 counts the number of quaternions, stored by the class data attributecount. The other class data attribute,quaternion.who keeps the list of all quaternions. Intro to Computer Science (MCS 260) operator overloading L-25 9 March / 41
32 Operator Overloading 1 OOP to count Flops a flop = a floating-point operation overloading arithmetical operators counting number of flops in a sum 2 Quaternions hypercomplex numbers application in computer graphics 3 Operator Overloading the classquaternion defining functional attributes 4 Summary + Assignments Intro to Computer Science (MCS 260) operator overloading L-25 9 March / 41
33 constructing quaternions To define a general quaternion, we supply 4 arguments: >>> from quaternion import Quaternion >>> q = Quaternion(1,2,3,4) # i + 3 j + 4 k but sometimes we want to supply less: >>> c = Quaternion(5) # defines a constant >>> k = Quaternion(d = 1) # defines k The constructor init has 4 default values: def init (self, a=0, b=0, c=0, d=0): Constructs the Quaternion with coefficients a,b,c,d. Intro to Computer Science (MCS 260) operator overloading L-25 9 March / 41
34 representing quaternions def str (self): The string representation of a Quaternion uses i, j, and k. Note the + in the format. return %.2f %+.2f i %+.2f j %+.2f k \ % (self.cre, self.cim, self.cjj, self.ckk) >>> I = Quaternion(b=1) >>> str(i) i j k >>> I <quaternion.quaternion instance at 0x402d5dcc> To represent with a string by default, overload repr (): def repr (self): "Quaternion Representation is a String" return str(self) TypingIgives now i j k. Intro to Computer Science (MCS 260) operator overloading L-25 9 March / 41
35 overloading equality With the builtin== operator: >>> x = Quaternion(1) >>> y = Quaternion(1,0) >>> x i j k >>> y i j k >>> x == y False But we wanted (and expected)true as answer! It takes two to compare, besidesself, we need theother. Intro to Computer Science (MCS 260) operator overloading L-25 9 March / 41
36 overloading equality: defining eq Theother is likeself, used in binary operators: def eq (self, other): Defines Quaternion Equality. return self.cre == other.cre and \ self.cim == other.cim and \ self.cjj == other.cjj and \ self.ckk == other.ckk Now, we will havetrue: >>> x = Quaternion(1) >>> y = Quaternion(1,0) >>> x == y True Intro to Computer Science (MCS 260) operator overloading L-25 9 March / 41
37 overloading arithmetical operators We want to writez = x + y and evenz += x when adding two quaternions. Consider first the negation, i.e.: z = -x def neg (self): The negation of a Quaternion. return Quaternion(-self.cre, \ -self.cim, \ -self.cjj, \ -self.ckk) Defining neg we overloaded the negation-. Intro to Computer Science (MCS 260) operator overloading L-25 9 March / 41
38 overloading addition the other and inplace addition It takes two to add, besidesself, we need theother. def add (self, other): Adding two Quaternions" return Quaternion(self.cre + other.cre, \ self.cim + other.cim, \ self.cjj + other.cjj, \ self.ckk + other.ckk) If we also want to update quaternions asz += x, then we overload the inplace addition operator iadd. Intro to Computer Science (MCS 260) operator overloading L-25 9 March / 41
39 magnitude and conjugate def abs (self): Returns the magnitude of a Quaternion. from math import sqrt return sqrt(self.cre**2+self.cim**2 \ +self.cjj**2+self.ckk**2) def conjugate(self): Returns the conjugate of a Quaternion. return Quaternion(self.cre, -self.cim, \ -self.cjj, -self.ckk) Intro to Computer Science (MCS 260) operator overloading L-25 9 March / 41
40 the inverse of a quaternion def scamul(self, scx): Product of Quaternion with scalar scx. return Quaternion(self.cre*scx, self.cim*scx, \ self.cjj*scx, self.ckk*scx) def invert (self): The Inverse of the Quaternion. return self.conjugate().scamul(1/abs(self)) Intro to Computer Science (MCS 260) operator overloading L-25 9 March / 41
41 Summary + Assignments Assignments: 1 Overload the subtraction operator for quaternions. Also do the inplace version. 2 Provide a methodcoefficients that returns a tuple with the coefficients of the quaternion. 3 Extendscamul so that you can compute the multiplication of any quaternion with any scalar multiple of i, j, and k. 4 Write Python code for and verify whether q q 1 = 1, for a random quaternion q. 5 Use operator overloading in the Python code for the class Rational to work with rational numbers. Intro to Computer Science (MCS 260) operator overloading L-25 9 March / 41
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