Lists and Loops. browse Python docs and interactive help

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1 Lists and Loops 1 Help in Python browse Python docs and interactive help 2 Lists in Python defining lists lists as queues and stacks inserting and removing membership and ordering lists 3 Loops in Python for and while loops the composite trapezoidal rule MCS 507 Lecture 4 Mathematical, Statistical and Scientific Software Jan Verschelde, 4 September 2013 Scientific Software (MCS 507 L-4) Lists and Loops 4 September / 34

2 Lists and Loops 1 Help in Python browse Python docs and interactive help 2 Lists in Python defining lists lists as queues and stacks inserting and removing membership and ordering lists 3 Loops in Python for and while loops the composite trapezoidal rule Scientific Software (MCS 507 L-4) Lists and Loops 4 September / 34

3 getting help browse documentation Launch IDLE, go to the Help menu, and to Python docs (F1) which launches a browser for the documentation. the interactive command line help >>> help() this launches the interactive help system help> topics displays a list of available topics help> LISTS shows the available help on lists help> quit >>> Scientific Software (MCS 507 L-4) Lists and Loops 4 September / 34

4 getting help on a module To get help about a module, one must first import the module. For example: >>> import copy >>> help(copy) or with a relative import: >>> from copy import deepcopy >>> help(deepcopy) Printing the documentation string of a module: >>> print copy. doc Scientific Software (MCS 507 L-4) Lists and Loops 4 September / 34

5 Lists and Loops 1 Help in Python browse Python docs and interactive help 2 Lists in Python defining lists lists as queues and stacks inserting and removing membership and ordering lists 3 Loops in Python for and while loops the composite trapezoidal rule Scientific Software (MCS 507 L-4) Lists and Loops 4 September / 34

6 lists via range With range we make a list of consecutive integers: >>> L = range(5,11) >>> type(l) <type list > >>> L [5, 6, 7, 8, 9, 10] Note: range(a,b) terminates at b - 1. The default start of range is 0: >> range(5) [0, 1, 2, 3, 4] Instead of 1 as default step, use 3: >>> range(4,4+5*3,3) [4, 7, 10, 13, 16] Scientific Software (MCS 507 L-4) Lists and Loops 4 September / 34

7 selecting and assigning >>> L = range(5,10) >>> len(l) 5 >>> K = L; K [5, 6, 7, 8, 9] We can assign lists as any variable. Indexing starts at 0: >>> L[2] 7 >>> L[2] = 17 Assigning to L also changes K: >>> L [5, 6, 17, 8, 9] >>> K [5, 6, 17, 8, 9] Scientific Software (MCS 507 L-4) Lists and Loops 4 September / 34

8 taking slices of lists Generating 7 integers starting at 4 with step size 5: >>> L = range(4,4+7*5,5) >>> L [4, 9, 14, 19, 24, 29, 34] A slice: >>> L[2:5] [14, 19, 24] The first and last 3 numbers are selected as >>> L[:3] [4, 9, 14] >>> L[-3:] [24, 29, 34] Scientific Software (MCS 507 L-4) Lists and Loops 4 September / 34

9 Lists and Loops 1 Help in Python browse Python docs and interactive help 2 Lists in Python defining lists lists as queues and stacks inserting and removing membership and ordering lists 3 Loops in Python for and while loops the composite trapezoidal rule Scientific Software (MCS 507 L-4) Lists and Loops 4 September / 34

10 implementing a queue In a queue with first-come first-served protocol, we append items to the end and pop the front. >>> L = [] >>> L.append( a ) >>> L [ a ] >>> L.append( b ); L [ a, b ] >>> first = L.pop(0) >>> first a >>> L [ b ] Scientific Software (MCS 507 L-4) Lists and Loops 4 September / 34

11 implementing a stack In a stack (e.g.: a pile of books), we push to the front and also pop the front. >>> L = [] >>> L.insert(0, a ); L [ a ] >>> L.insert(0, b ); L [ b, a ] >>> top = L.pop(0) >>> top b >>> L [ a ] Scientific Software (MCS 507 L-4) Lists and Loops 4 September / 34

12 Lists and Loops 1 Help in Python browse Python docs and interactive help 2 Lists in Python defining lists lists as queues and stacks inserting and removing membership and ordering lists 3 Loops in Python for and while loops the composite trapezoidal rule Scientific Software (MCS 507 L-4) Lists and Loops 4 September / 34

13 the insert method The first argument of insert is the index of the new element in the list. After L.insert(k,i) we have: L[k] == i. >>> L = range(0,5); L [0, 1, 2, 3, 4] >>> L.insert(2, a ) >>> L [0, 1, a, 2, 3, 4] >>> L.insert(len(L), b ) >>> L [0, 1, a, 2, 3, 4, b ] >>> L.insert(1000, c ); L [0, 1, a, 2, 3, 4, b, c ] Scientific Software (MCS 507 L-4) Lists and Loops 4 September / 34

14 popping and removing Observe the difference between pop and remove: >>> L = range(0,5); L [0, 1, 2, 3, 4] >>> L.remove(0) >>> L [1, 2, 3, 4] >>> L.pop(0) 1 Also their arguments have different type: >>> L.pop(2) 4 >>> L [2, 3] >>> L.remove(3); L [2] Scientific Software (MCS 507 L-4) Lists and Loops 4 September / 34

15 deleting elements >>> x = a ; y = b ; L = [x,y] >>> L [ a, b ] >>> del L[0] >>> L [ b ] >>> x a Although the first element a of the list is deleted, we can still refer to that a via the name x Difference with remove? The del applies to any variable. Scientific Software (MCS 507 L-4) Lists and Loops 4 September / 34

16 memory locations addresses and values >>> x = 3 >>> id(x) The machine view: address of x 3 value of x In Python, the name x refers to the object 3. x name reference 3 object Scientific Software (MCS 507 L-4) Lists and Loops 4 September / 34

17 visualizing lists from scitools.lumpy import Lumpy lumpy = Lumpy() lumpy.make_reference() L0 = [1, 4, 3] L1 = L0 L2 = L1[:-1] L1[0] = 100 lumpy.object_diagram() Scientific Software (MCS 507 L-4) Lists and Loops 4 September / 34

18 Lists and Loops 1 Help in Python browse Python docs and interactive help 2 Lists in Python defining lists lists as queues and stacks inserting and removing membership and ordering lists 3 Loops in Python for and while loops the composite trapezoidal rule Scientific Software (MCS 507 L-4) Lists and Loops 4 September / 34

19 testing membership The in operator returns True or False: >>> L = range(5,10); L [5, 6, 7, 8, 9] >>> 6 in L True >>> 3 in L False If e in L, then L[L.index(e)] == e: >>> L.index(6) 1 >>> L[1] 6 Scientific Software (MCS 507 L-4) Lists and Loops 4 September / 34

20 ordering lists >>> L = range(1,11) >>> import random >>> random.shuffle(l) >>> L [3, 6, 9, 10, 7, 8, 1, 4, 2, 5] >>> max(l); min(l) 10 1 >>> L.reverse() >>> L [5, 2, 4, 1, 8, 7, 10, 9, 6, 3] >>> L.sort() >>> L [1, 2, 3, 4, 5, 6, 7, 8, 9, 10] Scientific Software (MCS 507 L-4) Lists and Loops 4 September / 34

21 Lists and Loops 1 Help in Python browse Python docs and interactive help 2 Lists in Python defining lists lists as queues and stacks inserting and removing membership and ordering lists 3 Loops in Python for and while loops the composite trapezoidal rule Scientific Software (MCS 507 L-4) Lists and Loops 4 September / 34

22 for loops Computing the sum of all elements in a list L: e L e. s = 0 for e in L: s = s + e print s The indentation is important! To see how s evolves: s = 0 for e in L: s = s + e print s The second code fragment prints as many times as len(l), while the first code prints only once. Scientific Software (MCS 507 L-4) Lists and Loops 4 September / 34

23 while loops The for loop is appropriate when the number of steps is fixed and known in advance. A while loop is more flexible. s = 0 for e in L: s = s + e is equivalent to s = 0; i = 0 while (i < len(l)): s = s + L[i] i = i + 1 The while loop continues as long as the condition is True. Scientific Software (MCS 507 L-4) Lists and Loops 4 September / 34

24 Lists and Loops 1 Help in Python browse Python docs and interactive help 2 Lists in Python defining lists lists as queues and stacks inserting and removing membership and ordering lists 3 Loops in Python for and while loops the composite trapezoidal rule Scientific Software (MCS 507 L-4) Lists and Loops 4 September / 34

25 the composite trapezoidal rule To approximate the integral of a function f(x) over [a, b], the trapezoidal rule is b a f(x)dx 1 (f(a) + f(b))(b a). 2 Geometrically, we approximate the area under f(x) for x [a, b] by the area of a trapezium, with base [a, b] and heights f(a) and f(b). Dividing [a, b] into n intervals of length h = (b a)/n, denote x k = a + kh, x 0 = a, x n = b, the composite trapezoidal rule is b a f(x)dx h n 1 2 (f(a) + f(b)) + h f(a + kh). k=1 Scientific Software (MCS 507 L-4) Lists and Loops 4 September / 34

26 evaluating functions At the command prompt $: $ python testevalfun.py give a function in x : exp(x)*sin(x) give a value for x : 1 formula exp(x)*sin(x) at 1 is The script testevalfun.py uses StringFunction of the module scitools. Alternative to convert a formula to a function: lambdify of the package sympy. Scientific Software (MCS 507 L-4) Lists and Loops 4 September / 34

27 the script testevalfun.py # L-4 MCS 507 Wed 4 Sep 2013 : testevalfun.py """ Using the StringFunction of scitools to make a script which prompts the user for a function. """ from scitools.stringfunction import StringFunction FORMULA = raw_input( give a function in x : ) FUN = StringFunction(FORMULA) VAL = input( give a value for x : ) RES = FUN(VAL) print formula, FORMULA, at, VAL, is, RES Scientific Software (MCS 507 L-4) Lists and Loops 4 September / 34

28 using lambdify of sympy # L-4 MCS 507 Wed 4 Sep 2013 : testevalfun2.py """ Using the lambdify of sympy to make a script which prompts the user for a function. """ from sympy import lambdify FORMULA = raw_input( give a function in x : ) FUN = lambdify( x, FORMULA) VAL = input( give a value for x : ) RES = FUN(VAL) print formula, FORMULA, at, VAL, is, RES Scientific Software (MCS 507 L-4) Lists and Loops 4 September / 34

29 running comptraprule.py $ python comptraprule.py give a function in x : exp(x)*sin(x) what is left end of [a,b]? 0 what is right end of [a,b]? 2 give number of evaluations : 100 approximate integral : asterix:lec04 jan$ python comptraprule.py give a function in x : exp(x)*sin(x) what is left end of [a,b]? 0 what is right end of [a,b]? 2 give number of evaluations : 1000 approximate integral : asterix:lec04 jan$ python comptraprule.py give a function in x : exp(x)*sin(x) what is left end of [a,b]? 0 what is right end of [a,b]? 2 give number of evaluations : approximate integral : Scientific Software (MCS 507 L-4) Lists and Loops 4 September / 34

30 the script comptraprule.py from scitools.stringfunction import StringFunction FORMULA = raw_input( give a function in x : ) INTEGRAND = StringFunction(FORMULA) LEFT = input( what is left end of [a,b]? ) RIGHT = input( what is right end of [a,b]? ) NBEVALS = input( give number of evaluations : ) STEP = (RIGHT - LEFT)/float(NBEVALS) SUM = (INTEGRAND(LEFT) + INTEGRAND(RIGHT))/2 for i in range(1, NBEVALS): SUM = SUM + INTEGRAND(LEFT+i*STEP) SUM = SUM*STEP print approximate integral :, SUM Scientific Software (MCS 507 L-4) Lists and Loops 4 September / 34

31 comptraprule2.py uses sympy.lambdify from sympy import lambdify FORMULA = raw_input( give a function in x : ) INTEGRAND = lambdify( x, FORMULA) LEFT = input( what is left end of [a,b]? ) RIGHT = input( what is right end of [a,b]? ) NBEVALS = input( give number of evaluations : ) STEP = (RIGHT - LEFT)/float(NBEVALS) SUM = (INTEGRAND(LEFT) + INTEGRAND(RIGHT))/2 for i in range(1, NBEVALS): SUM = SUM + INTEGRAND(LEFT+i*STEP) SUM = SUM*STEP print approximate integral :, SUM Scientific Software (MCS 507 L-4) Lists and Loops 4 September / 34

32 using scipy Romberg integration extrapolates on the composite trapezoidal rule, available in the module integrate of the package scipy. >>> from scipy.integrate import romberg >>> from math import exp, sin >>> romberg(lambda x: exp(x)*sin(x),0,2) With lambda we quickly define a function. Calling romberg with show=true shows the extrapolation table; and tells the number of function evaluations. Scientific Software (MCS 507 L-4) Lists and Loops 4 September / 34

33 Summary + Exercises We started chapter 2 of the text book. Exercises: 1 Type L = [3, 9]; K = L; K[1] = 6 in an interactive Python session. Sketch (or use Lumpy) the relations between K and L. 2 Write a script that prompts the user for a string definition of a function f (in x), the end points of an interval [a, b] and the number n of equidistant samples in [a, b]. The script prints the maximum and minimum value of f at the n equidistant samples in [a, b]. 3 Replace the for loop in comptraprule.py by an equivalent while loop. Scientific Software (MCS 507 L-4) Lists and Loops 4 September / 34

34 more exercises 4 Write a script that asks the user ( for ) two integers n and k. n n! The script prints the value of = k k!(n k)!. 5 Modify comptraprule.py so that instead of n the script prompts the user for a tolerance ǫ (e.g.: 0.001). Starting with n = 2, the number of evaluations n doubles in every step till the difference between two consecutive approximations for the integral < ǫ. The first homework collection is on Monday 9 September, at 9AM. Bring to class your answers to exercise 3 of Lecture 1; exercises 1, 2 of Lecture 2; and exercises 1, 3 of Lecture 3. Along with a paper version, also me your scripts. Scientific Software (MCS 507 L-4) Lists and Loops 4 September / 34

Lists and Loops. defining lists lists as queues and stacks inserting and removing membership and ordering lists

Lists and Loops. defining lists lists as queues and stacks inserting and removing membership and ordering lists Lists and Loops 1 Lists in Python defining lists lists as queues and stacks inserting and removing membership and ordering lists 2 Loops in Python for and while loops the composite trapezoidal rule MCS