Defining Functions. turning expressions into functions. writing a function definition defining and using modules

Transcription

1 Defining Functions 1 Lambda Functions turning expressions into functions 2 Functions and Modules writing a function definition defining and using modules 3 Computing Series Developments exploring an example with a script series for expressions in two variables MCS 507 Lecture 7 Mathematical, Statistical and Scientific Software Jan Verschelde, 12 September 2012 Scientific Software (MCS 507) Defining Functions 12 Sep / 26

2 Defining Functions 1 Lambda Functions turning expressions into functions 2 Functions and Modules writing a function definition defining and using modules 3 Computing Series Developments exploring an example with a script series for expressions in two variables Scientific Software (MCS 507) Defining Functions 12 Sep / 26

3 evaluating expressions Suppose we want to evaluate x 2 cos(y) + 4e z sin(x). >>> from math import cos, sin, exp >>> f = lambda x,y,z: x**2*cos(y) \ *exp(z)*sin(x) >>> f(1,2,3) >>> f(z=3,y=2,x=1) In f(z=3,y=2,x=1), we call f by its keyword arguments, linking the formal parameters x, y, z to the values 1, 2, 3. Scientific Software (MCS 507) Defining Functions 12 Sep / 26

4 evaluating strings Restarting the interactive Python session, making a function for x 2 cos(y) + 4e z sin(x), stored in a string: >>> e = x**2*cos(y) + 4*exp(z)*sin(x) >>> from math import cos, sin, exp >>> f = lambda x,y,z: eval(e) >>> f(1,2,3) >>> f(z=3,x=1,y=2) Scientific Software (MCS 507) Defining Functions 12 Sep / 26

5 symbolic substitution With the substitution of sympy, we can evaluate an expression in symbolically, e.g., to permute the variables (x, y, z) (z, y, x): >>> from sympy import * >>> x,y,z = var( x,y,z ) >>> e = x**2*cos(y) + 4*exp(z)*sin(x) >>> Subs(e,(x,y,z),(z,y,x)) Subs(_x**2*cos(_y) + 4*exp(_z)*sin(_x), \ (_x, _y, _z), (z, y, x)) >>> _.doit() z**2*cos(y) + 4*exp(x)*sin(z) Scientific Software (MCS 507) Defining Functions 12 Sep / 26

6 approximating functions With series we do symbolic-numeric computation: >>> from sympy import * >>> x = var( x ) >>> sin(x).series(x,x0=0,n=7) x - x**3/6 + x**5/120 + O(x**7) Using an iterator: >>> s = sin(x).series(x,n=none) >>> L = [s.next() for i in range(3)]; L [x, -x**3/6, x**5/120] We experience the O(x**7) when evaluating: >>> S = sum(l) >>> Subs(S,(x),(0.01)).doit() >>> Subs(sin(x),(x),(0.01)).doit() Scientific Software (MCS 507) Defining Functions 12 Sep / 26

7 Defining Functions 1 Lambda Functions turning expressions into functions 2 Functions and Modules writing a function definition defining and using modules 3 Computing Series Developments exploring an example with a script series for expressions in two variables Scientific Software (MCS 507) Defining Functions 12 Sep / 26

8 a function definition def f(x,y,z): """ Returns the value of the expression x**2*cos(y) + 4*exp(z)*sin(x) for numerical values of x, y, and z. """ from math import exp, cos, sin v = x**2*cos(y) + 4*exp(z)*sin(x) return v Scientific Software (MCS 507) Defining Functions 12 Sep / 26

9 function definitions A function header (e.g., def f(x,y,z):) consists of 1 The name of the function follows def. 2 Arguments of the function are between round brackets ( and ); separated by commas. Round brackets are needed even if no arguments. 3 The colon : follows ). The documentation string (between triple quotes) is optional, but is strongly recommended. The function body may have local variables, e.g., v. Values (e.g., v) are returned with return v. Scientific Software (MCS 507) Defining Functions 12 Sep / 26

10 testing a function If we store the function definition for f in the file ourfirstmodule.py, we can do >>> from ourfirstmodule import f >>> f(1,2,3) We import the function f into an interactive Python session. Scientific Software (MCS 507) Defining Functions 12 Sep / 26

11 Defining Functions 1 Lambda Functions turning expressions into functions 2 Functions and Modules writing a function definition defining and using modules 3 Computing Series Developments exploring an example with a script series for expressions in two variables Scientific Software (MCS 507) Defining Functions 12 Sep / 26

12 help(ourfirstmodule) Help on module ourfirstmodule: NAME ourfirstmodule - # L-7 MCS 507 Wed 12 Sep 2012 : ourfir FILE /Users/jan/Courses/MCS507/Lec07/ourfirstmodule.py FUNCTIONS f(x, y, z) Returns the value of the expression x**2*cos(y) + 4*exp(z)*sin(x) for numerical values of x, y, and z. main() Prompts the user for three values for the variables x, y, z and prints x**2*cos(y) + 4*exp(z)*sin(x). Scientific Software (MCS 507) Defining Functions 12 Sep / 26

13 the main function def main(): """ Prompts the user for three values for the variables x, y, z and prints x**2*cos(y) + 4*exp(z)*sin(x). """ print v = x**2*cos(y) + 4*exp(z)*sin(x) x = input( give x : ) y = input( give y : ) z = input( give z : ) v = f(x,y,z) print v =, v Scientific Software (MCS 507) Defining Functions 12 Sep / 26

14 running main() We can also import the function main(). In order to run main() as a program at the command prompt $ $ python ourfirstmodule.py the last line in ourfirstmodule.py is if name == " main ": main() Scientific Software (MCS 507) Defining Functions 12 Sep / 26

15 Defining Functions 1 Lambda Functions turning expressions into functions 2 Functions and Modules writing a function definition defining and using modules 3 Computing Series Developments exploring an example with a script series for expressions in two variables Scientific Software (MCS 507) Defining Functions 12 Sep / 26

16 example_series.py from sympy import sin, cos, exp from sympy.abc import x, y, z e = x**2*cos(y) + 4*exp(z)*sin(x) # developing e about x = 0, 4th order print e.series(x,x0=0,n=4) # using an iterator of the series tx = e.series(x,x0=0,n=none) Lx = [tx.next() for i in range(3)] print Lx =, Lx e3x = sum(lx) # observe there is no O() in e3x print sum(lx) =, e3x produces 4*x*exp(z) + x**2*cos(y) - 2*x**3*exp(z)/3 + O(x**4) Lx = [4*x*exp(z), x**2*cos(y), -2*x**3*exp(z)/3] sum(lx) = -2*x**3*exp(z)/3 + x**2*cos(y) + 4*x*exp(z) Scientific Software (MCS 507) Defining Functions 12 Sep / 26

17 the script continued Only the middle term of Lx contains a function in y: prints # developing e3x about y = 1 ty = Lx[1].series(y,n=None) Ly = [ty.next() for i in range(2)] print Ly =, Ly e2y = sum(ly) print e2y =, e2y Ly = [x**2, -x**2*y**2/2] e2y = -x**2*y**2/2 + x**2 Scientific Software (MCS 507) Defining Functions 12 Sep / 26

18 developing in z gives print e2y =, e2y # developing z about z = 0, 3rd order tz0 = Lx[0].series(z,n=None) tz2 = Lx[2].series(z,n=None) Lz0 = [tz0.next() for i in range(2)] print Lz0 =, Lz0 Lz2 = [tz2.next() for i in range(2)] print Lz2 =, Lz2 s = sum(lz0) + sum(ly) + sum(lz2) print s =, s Lz0 = [4*x, 4*x*z] Lz2 = [-2*x**3/3, -2*x**3*z/3] s = -2*x**3*z/3-2*x**3/3 - x**2*y**2/2 \ + x**2 + 4*x*z + 4*x Scientific Software (MCS 507) Defining Functions 12 Sep / 26

19 checking the series from sympy import Subs v = (0.01,1.01,0.01) ev = Subs(e,(x,y,z),v).doit() sv = Subs(s,(x,y,z),v).doit() print expression value =, ev print series value =, sv print difference =, abs(ev - sv) shows expression value = series value = difference = e-6 Scientific Software (MCS 507) Defining Functions 12 Sep / 26

20 Defining Functions 1 Lambda Functions turning expressions into functions 2 Functions and Modules writing a function definition defining and using modules 3 Computing Series Developments exploring an example with a script series for expressions in two variables Scientific Software (MCS 507) Defining Functions 12 Sep / 26

21 running define_series.py The series() of sympy does not seem to apply for functions of several variables... $ python define_series.py give an expression : sin(x)*cos(y) give pair of variables : x,y give pair of orders : 2,2 x**3*y**2/12 - x**3/6 - x*y**2/2 + x Scientific Software (MCS 507) Defining Functions 12 Sep / 26

22 the function main() def main(): """ Prompts user for an expression, a pair of variables and orders. """ e = raw_input( give an expression : ) v = raw_input( give pair of variables : ) o = input( give pair of orders : ) w = var(v) print bivariate_series(eval(e),w,o) if name ==" main ": main() Scientific Software (MCS 507) Defining Functions 12 Sep / 26

23 list comprehensions def bivariate_series(e,v,o): """ Returns a series of the expression e in the pair of variables in v of respective orders in the pair o. """ t1 = e.series(v[0],n=none) L1 = safe_expand(t1,o[0]) t2 = [a.series(v[1],n=none) for a in L1] L2 = [safe_expand(t,o[1]) for t in t2] return sum([sum(l) for L in L2]) Scientific Software (MCS 507) Defining Functions 12 Sep / 26

24 exception handling def safe_expand(t,o): """ Given an iterator, returns the list of terms up to the order o. Uses an exception handler to catch cases when there is no next term. """ L = [] for i in xrange(o): try: L.append(t.next()) except StopIteration: return L return L Scientific Software (MCS 507) Defining Functions 12 Sep / 26

25 Summary + Exercises We started chapter 3 of the text book. Exercises: 1 The function numpy.linspace(a,b,n) returns a list of n equally spaced points in [a,b]. Write your own function definition for linspace. You may assume that n is at least 2 and a < b. 2 Compute the length of a path in the plane given by a list of coordinates (as tuples), see Exercise Extend the script define_series.py so it works for expressions in three variables. Test your solution on the expression x 2 cos(y) + 4e x sin(x). Scientific Software (MCS 507) Defining Functions 12 Sep / 26

26 more exercises 4 The formula T(t) = T + (T(0) T )e kt models the temperature T in function of time t. For k > 0, T(t) declines and models how an object cools off. Define a Python function that takes as arguments T(0), T, k, t and that returns T(t). 5 Use Sage or sympy to make a series approximation for T(t) from the previous exercise. Make a Python function from the series. Take various orders and compare the values at t = 0.1. The second homework is due on Friday 21 September, at 10AM: 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) Defining Functions 12 Sep / 26