callback, iterators, and generators

Transcription

1 callback, iterators, and generators 1 Adding a Callback Function a function for Newton s method a function of the user to process results 2 A Newton Iterator defining a counter class refactoring the Newton function into a class 3 Newton with a Generator defining a generator with yield putting yield in newton MCS 507 Lecture 10 Mathematical, Statistical and Scientific Software Jan Verschelde, 18 September 2013 Scientific Software (MCS 507 L-10) callback, iterators, generators 18 September / 34

2 callback, iterators, and generators 1 Adding a Callback Function a function for Newton s method a function of the user to process results 2 A Newton Iterator defining a counter class refactoring the Newton function into a class 3 Newton with a Generator defining a generator with yield putting yield in newton Scientific Software (MCS 507 L-10) callback, iterators, generators 18 September / 34

3 Newton s method from sympy import var, Subs, exp, sin def newton(fun, drv, init, mxit=5, eps=1.0e-8): Runs Newton s method on f(x) = 0. INPUT PARAMETERS : fun a function in one variable, drv derivative of the function f, init initial guess for the root, mxit maximum number of iterations, eps accurary requirement on f(x) and on the update to x, dx. ON RETURN : the tuple (x, dx, y, fail), with: x approximation for the root, dx magnitude of last correction, y residual f(x), fail is True if the accuracy is not reached. Scientific Software (MCS 507 L-10) callback, iterators, generators 18 September / 34

4 the body of the function def newton(fun, drv, init, mxit=5, eps=1.0e-8): "..." root = init fail = True for i in xrange(mxit): update = -fun(root)/drv(root) root = root + update residual = abs(fun(root)) if ((abs(update) <= eps) or (residual <= eps)): fail = False break return (root, abs(update), residual, fail) Observe the break: to exit the for loop when the accuracy requirement is reached. Scientific Software (MCS 507 L-10) callback, iterators, generators 18 September / 34

5 the main function def main(): Prompts the user for an expression in x, computes its derivative and calls newton. name = var( x ) expression = input( Give an expression in x : ) derivative = expression.diff(name) # using sympy fun = lambda v: Subs(expression,(name),(v)).doit() drv = lambda v: Subs(derivative,(name),(v)).doit() guess = input( Give an initial guess : ) guess = float(guess) # force float arithmetic print newton(fun, drv, guess) main() Scientific Software (MCS 507 L-10) callback, iterators, generators 18 September / 34

6 running newtonfun.py At the command prompt $, we type $ python newtonfun.py Give an expression in x : exp(-x**2)*sin(x) Give an initial guess : 0.3 ( e-9, , \ e-9, False) $ What if we want intermediate output? Scientific Software (MCS 507 L-10) callback, iterators, generators 18 September / 34

7 callback, iterators, and generators 1 Adding a Callback Function a function for Newton s method a function of the user to process results 2 A Newton Iterator defining a counter class refactoring the Newton function into a class 3 Newton with a Generator defining a generator with yield putting yield in newton Scientific Software (MCS 507 L-10) callback, iterators, generators 18 September / 34

8 a callback function def print_steps(data): Prints the content of the dictionary data and prompts the user to continue or not. result = "" for key in data.keys(): result = result + " " + key + = if ((key == dx ) or (key == res )): result = result + ( %.3e % data[key]) else: result = result + str(data[key]) print result answer = raw_input("continue? (y/n) ") return (answer == y ) Scientific Software (MCS 507 L-10) callback, iterators, generators 18 September / 34

9 using the callback def newton(fun, drv, init, mxit=5, eps=1.0e-8, back=none): Inside the for loop: if back!= None: data = { step : i+1, x : root, dx : update, \ res : residual} proceed = back(data) if proceed!= None: if not proceed: break In main: print Newton(f, df, x0, back=print_steps) Scientific Software (MCS 507 L-10) callback, iterators, generators 18 September / 34

10 running newtoncallback.py $ python newtoncallback.py Give an expression in x : exp(-x**2)*sin(x) Give an initial guess : 0.3 x = step = 1 \ dx = e-01 res = 7.924e-02 continue? (y/n) y x = step = 2 \ dx = 8.104e-02 res = 1.205e-03 continue? (y/n) y x = e-9 step = 3 \ dx = e-03 res = 4.085e-09 continue? (y/n) n ( e-9, , \ e-9, False) Scientific Software (MCS 507 L-10) callback, iterators, generators 18 September / 34

11 callback, iterators, and generators 1 Adding a Callback Function a function for Newton s method a function of the user to process results 2 A Newton Iterator defining a counter class refactoring the Newton function into a class 3 Newton with a Generator defining a generator with yield putting yield in newton Scientific Software (MCS 507 L-10) callback, iterators, generators 18 September / 34

12 object oriented programming Python is an object oriented programming language. From Object-Oriented Analysis and Design with Applications (Addison-Wesley 2007) by Grady Booch et al.: Object-oriented programming is a method of implementation in which programs are organized as cooperative collections of objects, each of which represents an instance of some class, and whose classes are all members of a hierarchy of classes united via inheritance relationships. Each object has data and functional attributes. A class definition in Python usually begins with specifying the data attributes in the init method; override the str method to display an object. Scientific Software (MCS 507 L-10) callback, iterators, generators 18 September / 34

13 a Counter class class Counter(object): Our first class stores a counter. def init (self, val=0): Sets the counter to the value val. self.count = val def str (self): Returns the string representation. return "counter is " + str(self.count) Scientific Software (MCS 507 L-10) callback, iterators, generators 18 September / 34

14 use in Python session If the class definition is saved in classcounter.py: >>> from classcounter import Counter >>> c = Counter(3) >>> print c counter is 3 >>> str(c) counter is 3 Although we are not supposed to, we can manipulate the data attribute of an object of the class Counter directly: >>> c.count 3 >>> c.count = -4 >>> c counter is -4 Scientific Software (MCS 507 L-10) callback, iterators, generators 18 September / 34

15 class definition continued If c is a Counter object, then typing c at the interactive prompt is the same as print c. repr (). def repr (self): Defines the representation. return self. str () def next(self): Adds one to the counter. self.count = self.count + 1 The next() method models an iterator. Scientific Software (MCS 507 L-10) callback, iterators, generators 18 September / 34

16 a test function def test(): Performs a simple test. cnt = Counter(3) print cnt cnt.next() print cnt if name == " main ": test() Because of the last line, we run as $ python classcounter.py counter is 3 counter is 4 Scientific Software (MCS 507 L-10) callback, iterators, generators 18 September / 34

17 callback, iterators, and generators 1 Adding a Callback Function a function for Newton s method a function of the user to process results 2 A Newton Iterator defining a counter class refactoring the Newton function into a class 3 Newton with a Generator defining a generator with yield putting yield in newton Scientific Software (MCS 507 L-10) callback, iterators, generators 18 September / 34

18 a Newton iterator Problem: Adding a callback function to a Newton function we can see intermediate results and even stop the iteration, but the callback function remains subordinate to the main function that performs the Newton iterations. The class NewtonIterator defines an object that stores the state of all variables in a Newton iteration. Applying the next() method to an object of the class NewtonIterator executes one Newton step. Control of the execution is reverted to the consumer. Scientific Software (MCS 507 L-10) callback, iterators, generators 18 September / 34

19 using newtoniterator >>> from newtoniterator import * >>> f = lambda x: x**2-2.0 >>> df = lambda x: 2.0*x >>> nf = NewtonIterator(f,df,2.0) >>> print nf step 0 : x = e+00, \ dx = 1.000e+00, f(x) = 2.000e+00 >>> nf.next() >>> print nf step 1 : x = e+00, \ dx = 5.000e-01, f(x) = 2.500e-01 Scientific Software (MCS 507 L-10) callback, iterators, generators 18 September / 34

20 storing the state class NewtonIterator(object): A Newton iterator applies the Newton operator to find a solution to the equation f(x) = 0. def init (self, fun, drv, x0, n=5, eps=1.0e-8): Initializes the state of the iterator. self.step = 0 self.fun = fun self.drv = drv self.root = float(x0) self.max = n self.eps = eps self.update = 1.0 self.res = abs(fun(x0)) Scientific Software (MCS 507 L-10) callback, iterators, generators 18 September / 34

21 printing the state def str (self): Returns the string representation of the data attributes. result = "step %d : " % self.step result = result + ("x = %.16e" % self.root) result = result + (", dx = %.3e" % self.update) result = result + (", f(x) = %.3e" % self.res) return result def repr (self): Defines the representation. return self. str () Scientific Software (MCS 507 L-10) callback, iterators, generators 18 September / 34

22 a Newton step def next(self): Does one Newton step. self.step = self.step + 1 if (self.step > self.max): raise StopIteration\ ("only %d steps allowed" % self.max) else: work = self.root self.update = -self.fun(work)/self.drv(work) self.root = work + self.update self.update = abs(self.update) self.res = abs(self.fun(self.root)) Scientific Software (MCS 507 L-10) callback, iterators, generators 18 September / 34

23 raising an exception If we have reached the maximum number of steps: >>> nf.next() Traceback (most recent call last): File "<stdin>", line 1, in <module> File "newtoniterator.py", line 63, in next raise StopIteration("only %d steps allowed" % self.max) StopIteration: only 5 steps allowed Scientific Software (MCS 507 L-10) callback, iterators, generators 18 September / 34

24 the stop test def accurate(self): Returns True if the accuracy requirements are satisfied. Returns False otherwise. return ((self.update <= self.eps) or (self.res <= self.eps)) Scientific Software (MCS 507 L-10) callback, iterators, generators 18 September / 34

25 the test function def test(): Approximates the square root of 2. fun = lambda x: x**2-2.0 drv = lambda x: 2.0*x nft = NewtonIterator(fun, drv, 2.0) print nft for i in xrange(5): nft.next() print nft if nft.accurate(): break if name == " main ": test() Scientific Software (MCS 507 L-10) callback, iterators, generators 18 September / 34

26 running newtoniterator.py $ python newtoniterator.py step 0 : x = e+00, \ dx = 1.000e+00, f(x) = 2.000e+00 step 1 : x = e+00, \ dx = 5.000e-01, f(x) = 2.500e-01 step 2 : x = e+00, \ dx = 8.333e-02, f(x) = 6.944e-03 step 3 : x = e+00, \ dx = 2.451e-03, f(x) = 6.007e-06 step 4 : x = e+00, \ dx = 2.124e-06, f(x) = 4.511e-12 Scientific Software (MCS 507 L-10) callback, iterators, generators 18 September / 34

27 callback, iterators, and generators 1 Adding a Callback Function a function for Newton s method a function of the user to process results 2 A Newton Iterator defining a counter class refactoring the Newton function into a class 3 Newton with a Generator defining a generator with yield putting yield in newton Scientific Software (MCS 507 L-10) callback, iterators, generators 18 September / 34

28 defining a generator with yield def counter(value=0): Maintains a counter. count = value while True: count = count + 1 yield count If saved in show_yield.py, then we can do >>> from show_yield import counter >>> mycounter = counter(3) >>> mycounter.next() 4 >>> mycounter.next() 5 >>> Scientific Software (MCS 507 L-10) callback, iterators, generators 18 September / 34

29 using a generator def main(): Example of the use of a generator. print initializing counter... mycounter = counter(3) print incrementing counter... print mycounter.next() print incrementing counter... print mycounter.next() if name == " main ": main() Scientific Software (MCS 507 L-10) callback, iterators, generators 18 September / 34

30 callback, iterators, and generators 1 Adding a Callback Function a function for Newton s method a function of the user to process results 2 A Newton Iterator defining a counter class refactoring the Newton function into a class 3 Newton with a Generator defining a generator with yield putting yield in newton Scientific Software (MCS 507 L-10) callback, iterators, generators 18 September / 34

31 putting yield in newton def newton(fun, drv, init, mxit=5, eps=1.0e-8): "..." root = init fail = True for i in xrange(mxit): update = -fun(root)/drv(root) root = root + update residual = abs(fun(root)) fail = ((abs(update) > eps) and (residual > eps)) yield { step : i+1, x : root, dx : update, \ res : residual} if not fail: break Scientific Software (MCS 507 L-10) callback, iterators, generators 18 September / 34

32 the main function def main(): Prompts the user for an expression in x, computes its derivative and calls newton. name = var( x ) expression = input( Give an expression in x : ) derivative = expression.diff(name) fun = lambda v: Subs(expression, (name),(v)).doit() drv = lambda v: Subs(derivative, (name),(v)).doit() guess = input( Give an initial guess : ) guess = float(guess) ntg = newton(fun, drv, guess) nxt = ntg.next() while nxt[ res ] > 1.0e-7: nxt = ntg.next() print nxt Scientific Software (MCS 507 L-10) callback, iterators, generators 18 September / 34

33 Summary + Exercises Refactoring code defined by a loop as an iterator via a class gives the user more flexibility. The yield in Python is very useful. Exercises: 1 Replace the break and the enclosing for loop in the function Newton of the script newtonfun.py by an equivalent while loop. 2 Write a script that prompts the user for n, the number of iterations in a for loop that calls scipy.integrate.simps in its body. With each new call the number of function evaluations in scipy.integrate.simps doubles. Put the loop in a function which takes n as an input parameter. Add a callback function to this function. Scientific Software (MCS 507 L-10) callback, iterators, generators 18 September / 34

34 more exercises 3 Make an iterator for the composite Simpson rule, using scipy.integrate.simps. In each step of the iterator, the total number of function evaluations doubles. 4 Write code for your own composite Simpson rule to be used in an iterator fashion. Organize the computations so that all previous function evaluations are recycled. The second homework is due on Friday 20 September, at 9AM: solve exercises 1 and 3 of Lecture 4; exercises 4 and 5 of Lecture 5; exercises 1, 2 and 5 of Lecture 6; exercises 1, 2, and 4 of Lecture 7. Scientific Software (MCS 507 L-10) callback, iterators, generators 18 September / 34