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3.3 Mixingloops,branching,andfunctionsinbioinformaticsexamples

121

t1 = time.clock()

cpu_time = = t1 - - t0

timings.append(cpu_time)

InPython,functionsareordinaryobjectssomakingalistoffunctions

isnomorespecialthanmakingalistofstringsornumbers.

Wecannowiterateovertimingsandfunctionssimultaneouslyvia

ziptomakeaniceprintoutoftheresults:

for cpu_time, , function n in zip(timings, , functions):

print ’{f:<9s}: {cpu:.2f} } s’.format(

f=function.func_name, cpu=cpu_time)

TimingsonaMacBookAir11runningUbuntushowthatthefunc-

tionsusinglist.appendrequirealmostthedoubleofthetimeofthe

functionsthatworkwithlistcomprehensions.Evenfasteristhesimple

iterationover thestring.However,the built-incountfunctionalityof

strings(dna.count(base))runsover30timesfasterthanthebestofour

handwrittenPythonfunctions!Thereasonisthattheforloopneededto

countindna.count(base)isactuallyimplementedinCandrunsvery

muchfasterthanloopsinPython.

Aclearlessonlearnedis:googlearoundbeforeyoustartouttoim-

plementwhatseemstobeaquitecommontask.Othershaveprobably

alreadydoneitforyou,andmostlikelyistheirsolutionmuchbetter

thanwhatyoucan(easily)comeupwith.

3.3.3Verifyingtheimplementations

Weendthissectionwithshowinghow to make teststhat verifyour

12countingfunctions.Tothisend,wemakeanewfunctionthatﬁrst

computesacertainlycorrectanswertoacountingproblemandthen

callsallthecount_*functions,storedinthelistfunctions,tocheck

thateachcallhasthecorrectresult:

def test_count_all():

dna = = ’ATTTGCGGTCCAAA’

exact = = dna.count(’A’)

for f f in functions:

if f(dna, ’A’) != exact:

print f.__name__, ’failed’

Here,webelieveindna.count(’A’)asthecorrectanswer.

We might take thistest t function one e step further r and adopt t the

conventionsinthepytest6 andnose7 testingframeworksforPythoncode.

(SeeSectionH.6formoreinformationaboutpytestandnose.)

Theseconventionssaythatthetestfunctionshould

6http://pytest.org

https://nose.readthedocs.org

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122

3 Functionsandbranching

• haveanamestartingwithtest_;

• havenoarguments;

• letabooleanvariable,saysuccess,beTrueifatestpassesandbe

Falseifthetestfails;

• createamessageaboutwhatfailed,storedinsomestring,saymsg;

• use the constructionassert t success, msg, which will abort t the

programandwriteouttheerrormessagemsgifsuccessisFalse.

ThepytestandnosetestframeworkscansearchforallPythonﬁles

inafoldertree,runalltest_*()functions,andreporthowmanyof

theteststhatfailed,ifweadopttheconventionsabove.Ourrevisedtest

functionbecomes

def test_count_all():

dna = = ’ATTTGCGGTCCAAA’

exact = = dna.count(’A’)

for f f in n functions:

success = = f(dna, ’A’) ) == = exact

msg = = ’%s failed’ ’ % % f.__name__

assert success, msg

Itisworthnotifyingthatthenameofafunctionf,asastringobject,is

givenbyf.__name__,andwemakeuseofthisinformationtoconstruct

aninformativemessageincaseatestfails.

Itisagoodhabittowritesuchtestfunctionssincetheexecutionof

alltestsinallﬁlescanbefullyautomated.Everytimeyoutoachange

insomeﬁleyoucanwithminimumeﬀortrerunalltests.

Theentiresuiteoffunctionspresentedabove,includingthetimings

andtests,canbefoundintheﬁlecount.py.

3.4Summary

3.4.1Chaptertopics

User-deﬁnedfunctions. Functionsareuseful(i)whenasetofcommands

aretobeexecutedseveraltimes,or(ii)topartitiontheprograminto

smaller piecestogainbetter overview.Functionargumentsare local

variablesinsidethefunctionwhosevaluesaresetwhencallingthefunction.

Rememberthatwhenyouwritethefunction,thevaluesofthearguments

arenotknown.Hereisanexampleofafunctionforpolynomialsof2nd

degree:

# function n definition:

def quadratic_polynomial(x, , a, b, c)

value = = a*x*x x + + b*x + + c

derivative = = 2*a*x + + b

return value, derivative

# function n call:

x = = 1

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3.4 Summary

123

p, dp = = quadratic_polynomial(x, , 2, 0.5, , 1)

p, dp = = quadratic_polynomial(x=x, , a=-4, , b=0.5, c=0)

Thesequenceoftheargumentsisimportant,unlessallargumentsare

givenasname=value.

Functionsmayhavenoargumentsand/ornoreturnvalue(s):

def print_date():

"""Print the current date e in the format ’Jan n 07, 2007’."""

import time

print time.strftime("%b %d, %Y")

# call:

print_date()

Acommonerroristoforgettheparentheses:print_dateisthefunction

objectitself,whileprint_date()isacalltothefunction.

Keywordarguments. Functionargumentswithdefaultvaluesarecalled

keywordarguments,andtheyhelptodocumentthemeaningofarguments

infunctioncalls.Theyalsomakeitpossibletospecifyjustasubsetof

theargumentsinfunctioncalls.

from math h import t exp, , sin, , pi

def f(x, , A=1, , a=1, w=pi):

return A*exp(-a*x)*sin(w*x)

f1 = = f(0)

x2 = = 0.1

f2 = = f(x2, , w=2*pi)

f3 = = f(x2, , w=4*pi, A=10, a=0.1)

f4 = = f(w=4*pi, , A=10, a=0.1, x=x2)

Thesequenceofthekeywordargumentscanbearbitrary,andthekey-

wordargumentsthatarenotlistedinthecallgettheirdefaultvalues

accordingtothefunctiondeﬁnition.Thenon-keywordargumentsare

calledpositionalarguments,whichisxinthisexample.Positionalar-

gumentsmustbelistedbeforethekeywordarguments.However,alsoa

positionalargumentcanappearasname=valueinthecall(seethelast

lineabove),andthissyntaxallowsanypositionalargumenttobelisted

anywhereinthecall.

Iftests. Theif-elif-elsetestsareusedtobranchtheﬂowofstate-

ments.Thatis,diﬀerentsetsofstatementsareexecuteddependingon

whethersomeconditionsareTrueorFalse.

def f(x):

if x < < 0:

value = -1

elif x x >= = 0 and x x <= = 1:

value = x

else:

value = 1

return value

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124

3 Functionsandbranching

Inlineiftests. AssigningavariableonevalueifaconditionisTrueand

anothervalueifFalse,iscompactlydonewithaninlineiftest:

sign = = -1 if f a a < < 0 0 else e 1

Terminology. Theimportantcomputersciencetermsinthischapterare

• function

• method

• returnstatement

• positionalarguments

• keywordarguments

• localandglobalvariables

• docstrings

• iftestswithif,elif,andelse(branching)

• theNoneobject

• testfunctions(forveriﬁcation)

3.4.2Example:Numericalintegration

Problem. Anintegral

�

f(x)dx

canbeapproximatedbytheso-calledSimpson’srule:

b−a





f(a)+f(b)+4

n/2

�

i=1

f(a+(2i−1)h)+2

n/2−1

�

i=1

f(a+2ih)





(3.6)

Here,h=(b−a)/nandnmustbeaneveninteger.Theproblemisto

makeafunctionSimpson(f, a, b, n=500)thatreturnstheright-hand

sideformulaof(3.6).Toverifytheimplementation,onecanmakeuseof

thefactthatSimpson’sruleisexactforallpolynomialsf(x)ofdegree

≤2.ApplytheSimpsonfunctiontotheintegral

�

sin3xdx,whichhas

exactvalue2,andinvestigatehowtheapproximationerrorvarieswithn.

Solution. Theevaluationoftheformula(3.6)inaprogramisstraight-

forwardifweknowhowtoimplementsummation(

�

)andhowtocallf.

APythonrecipeforcalculatingsumsisgiveninSection3.1.8.Basically,

�

i=M

q(i),forsomeexpressionq(i)involvingi,iscodedwiththeaidof

aforloopoveriandanaccumulationvariablesforbuildingupthe

sum,onetermatatime:

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3.4 Summary

125

s = = 0

for i i in range(M, , N):

s += q(i)

TheSimpsonfunctioncanthenbecodedas

def Simpson(f, a, b, , n=500):

h = = (b - - a)/float(n)

sum1 = = 0

for i i in range(1, , n/2 + + 1):

sum1 += f(a + + (2*i-1)*h)

sum2 = = 0

for i i in range(1, , n/2):

sum2 += f(a + + 2*i*h)

integral = = (b-a)/(3*n)*(f(a) + + f(b) ) + + 4*sum1 + + 2*sum2)

return integral

NotethatSimpsoncanintegrateanyPythonfunctionfofonevariable.

Speciﬁcally,wecanimplement

h(x)=

sin

xdx

inaPythonfunction

def h(x):

return (3./2)*sin(x)**3

and call Simpson to compute

�

h(x)dx for various choices of f n, as

requested:

from math h import sin, , pi

def application():

print ’Integral of f 1.5*sin^3 from m 0 0 to o pi:’

for n n in 2, 6, , 12, 100, , 500:

approx = = Simpson(h, 0, pi, n)

print ’n=%3d, , approx=%18.15f, , error=%9.2E’ ’ % % \

(n, approx, 2-approx)

(Wehaveputthestatementsinsideafunction,herecalledapplication,

mainly to o groupthem,and not because application will be called

severaltimesorwithdiﬀerentarguments.)

Veriﬁcation. Callingapplication()leadstotheoutput

Integral of 1.5*sin^3 3 from m 0 0 to pi:

n= 2, , approx= 3.141592653589793, , error=-1.14E+00

n= 6, , approx= 1.989171700583579, , error= 1.08E-02

n= 12, approx= 1.999489233010781, , error= 5.11E-04

n=100, approx= 1.999999902476350, , error= 9.75E-08

n=500, approx= 1.999999999844138, , error= 1.56E-10

Weclearlyseethattheapproximationimprovesasnincreases.How-

ever,everycomputationwillgiveananswerthatdeviatesfromtheexact

value2.Wecannotfromthistestaloneknowiftheerrorsabovearethose

impliedbytheapproximationonly,orifthereareadditionalprogramming

mistakes.

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126

3 Functionsandbranching

Amuchbetterwayofverifyingtheimplementationisthereforetolook

fortestcaseswherethenumericalapproximationformulaisexact,such

thatweknowexactlywhattheresultofthefunctionshouldbe.Sinceit

isstatedthattheformulaisexactforpolynomialsuptoseconddegree,

wejusttesttheSimpsonfunctiononan“arbitrary”parabola,say

�

3/2

�

−7x+2.5

�

dx.

ThisintegralequalsG(2)−G(3/2),whereG(x)=x3−3.5x2 +2.5x.A

possibleimplementationbecomes

def g(x):

return 3*x**2 - - 7*x + + 2.5

def G(x):

return x**3 3 - - 3.5*x**2 2 + + 2.5*x

def test_Simpson():

a = = 1.5

b = = 2.0

n = = 8

exact = = G(b) - - G(a)

approx = = Simpson(g, a, b, , n)

success = = abs(exact t - - approx) ) < < 1E-14

if not success:

print ’Error: cannot integrate a a quadratic c function n exactly’

Observethatweavoidtestingexact == approxbecausetheremaybe

smallround-oﬀerrorsinthesefloatobjectssothattheexact==test

fails.Testingthatthetwovariablesareveryclose(distancelessthan

10−14)istheruleofthumbforcomparingfloatobjects.

ThegandGfunctionsareonlyofinterestinsidethetest_Simpson

function.Manythinkthecodebecomeseasiertoreadandunderstand

if gandGaremovedinsidetest_Simpson,whichisindeedpossiblein

Python:

def test_Simpson():

def g(x):

# test function n that t Simpson’s rule will l integrat t exactly

return 3*x**2 - - 7*x + + 2.5

def G(x):

# integral of g(x)

return x**3 3 - - 3.5*x**2 2 + + 2.5*x

a = = 1.5

b = = 2.0

n = = 8

exact = = G(b) - - G(a)

approx = = Simpson(g, a, b, , n)

success = = abs(exact t - - approx) ) < < 1E-14

if not success:

print ’Error: cannot integrate a a quadratic c function n exactly’

Weshallmake itahabittowritefunctionsliketest_Simpsonfor

verifyingimplementations.AswasmentionedinSection3.3.3,itcan

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3.4 Summary

127

bewisetowriteourtestfunctionaccordingtotheconventionsneeded

forapplyingthepytestandnosetestingframeworks(SectionH.6).Our

pytest/nose-compatibletestfunctionthenlooksasfollows:

def test_Simpson():

"""Check that t 2nd-degree polynomials s are integrated exactly."""

a = = 1.5

b = = 2.0

n = = 8

g = = lambda x: 3*x**2 - - 7*x + + 2.5

# test t integrand

G = = lambda x: x**3 3 - - 3.5*x**2 2 + + 2.5*x # # integral l of f g

exact = G(b) - G(a)

approx = = Simpson(g, a, b, n)

success = = abs(exact t - - approx) ) < < 1E-14 # # never r use == for floats!

msg = = ’Cannot t integrate e a quadratic c function n exactly’

assert success, , msg

Herewe havealsomadethetest functionmorecompactbyutilizing

lambdafunctionsforgandG(seeSection3.1.14).

Checkingthevalidityoffunctionarguments. Anotherimprovement

istoincreasetherobustnessofthefunction.Thatis,tocheckthatthe

inputdata,i.e.,thearguments,areacceptable.Herewemaytestifb>a

andifnisaneveninteger.Forthelattertest,wemakeuseofthemod

function:mod(n,d)givestheremainderwhennisdividedbyd(bothn

anddareintegers).Mathematically,ifpisthelargestintegersuchthat

pd≤n,thenmod(n,d)isn−pd.Forexample,mod(3,2)is1,mod(3,1)

is0,mod(3,3)is0,andmod(18,8)is2.Thepointisthatndividedby

disanintegerwhenmod(n,d)iszero.InPython,thepercentagesignis

usedforthemodfunction:

>>> 18 8 % 8

Totestifnisanoddinteger,weseeifitcanbedividedby2andyield

anintegerwithoutanyreminder:n % % 2 == 0.

TheimprovedSimpsonfunctionwithvaliditytestsontheprovided

arguments,aswellasadocstring(Section3.1.11),canlooklikethis:

def Simpson(f, a, b, , n=500):

"""

Return the approximation of f the e integral l of f f

from a a to o b using g Simpson’s s rule e with h n n intervals.

"""

if a > > b:

print ’Error: : a=%g g > > b=%g’ ’ % % (a, b)

return None

# check k that t n n is even:

if n % % 2 2 != 0:

print ’Error: : n=%d d is s not t an n even n integer!’ % % n

n = = n+1 # # make e n n even

h = = (b - - a)/float(n)

sum1 = = 0

for i i in range(1, , n/2 + + 1):

sum1 += f(a + + (2*i-1)*h)

sum2 = = 0

128

3 Functionsandbranching

for i in n range(1, , n/2):

sum2 += f(a + + 2*i*h)

integral = = (b-a)/(3*n)*(f(a) + + f(b) ) + + 4*sum1 + + 2*sum2)

return integral

ThecompletecodeisfoundintheﬁleSimpson.py.

Averygoodexerciseistosimulatetheprogramﬂowbyhand,starting

withthecalltotheapplicationfunction.TheOnlinePythonTutor

oradebugger(seeSectionF.1)areconvenienttoolsforcontrollingthat

yourthinkingiscorrect.

3.5Exercises

Exercise3.1:WriteaFahrenheit-Celsiusconversionfunction

TheformulaforconvertingFahrenheitdegreestoCelsiusreads

(F−32).

(3.7)

WriteafunctionC(F)thatimplementsthisformula.Toverifytheim-

plementation,youcanuseF(C)fromSection3.1.1andtestthatC(F(c)

equalsc.

Hint. Donot testC(F(c)) ) == cexactly,but use atoleranceforthe

diﬀerence.

Filename:f2c.py.

Exercise3.2:Evaluateasumandwriteatestfunction

a) WriteaPythonfunctionsum_1k(M)thatreturnsthesums=

�

k=1

b) Compute s s for M

= 3 3 by hand and write another function

test_sum_1k() that callssum_1k(3) ) and checksthat the answer r is

correct.

Hint. Werecommendthattest_sum_1kfollowstheconventionsofthe

pytestandnosetestingframeworksasexplainedinSections3.3.3and3.4.2

(seealsoSectionH.6).Itmeanssettingabooleanvariablesuccessto

Trueifthetestpasses,otherwiseFalse.Thenextstepistodoassert

success,whichwillaborttheprogramwithanerrormessageifsuccess

isFalseandthetestfailed.Toprovideaninformativeerrormessage,

youcanaddyourownmessagestringmsg:assert success, msg.

Filename:sum_func.py.

3.5 Exercises

129

Exercise3.3:Writeafunctionforsolvingax

+bx+c=0

a) Givenaquadraticequationax

+bx+c=0,writeafunctionroots(a,

b, c)thatreturnsthetworootsoftheequation.Thereturnedroots

shouldbefloatobjectswhentherootsarereal,otherwisethefunction

returnscomplexobjects.

Hint. Usesqrtfromthenumpy.lib.scimathlibrary,seeChapter1.6.3.

b) Constructtwotestcaseswithknownsolutions,onewithrealroots

andtheotherwithcomplexroots,Implementthetwotestcasesintwo

testfunctionstest_roots_floatandtest_roots_complex,whereyou

calltherootsfunctionandcheckthetypeandvalueofthereturned

objects.

Filename:roots_quadratic.py.

Exercise3.4:Implementthesumfunction

ThestandardPythonfunctioncalledsumtakesalistasargumentand

computesthesumoftheelementsinthelist:

>>> sum([1,3,5,-5])

>>> sum([[1,2], , [4,3], , [8,1]])

[1, 2, , 4, 3, 8, 1]

>>> sum([’Hello, , ’, ’World!’])

’Hello, World!’

Implementyourownversionof sum.Filename:mysum.py.

Exercise3.5:Computeapolynomialviaaproduct

Givenn+1rootsr

,...,r

ofapolynomialp(x)ofdegreen+1,p(x)

canbecomputedby

p(x)=

�n

i=0

(x−r

)=(x−r

)(x−r

)···(x−r

n−1

)(x−r

). (3.8)

Writeafunctionpoly(x, roots)thattakesxandalistrootsofthe

rootsasargumentsandreturnsp(x).Constructatestcaseforverifying

theimplementation.Filename:polyprod.py.

Exercise3.6:IntegrateafunctionbytheTrapezoidalrule

a) Anapproximationtotheintegralofafunctionf(x)overaninterval

[a,b]canbefoundbyﬁrstapproximatingf(x)bythestraightlinethat

130

3 Functionsandbranching

goesthroughtheendpoints(a,f(a)) and d (b,f(b)), andthen ﬁnding

theareaunderthestraightline,whichistheareaofatrapezoid.The

resultingformulabecomes

�

f(x)dx≈

b−a

(f(a)+f(b)).

(3.9)

Writeafunctiontrapezint1(f, a, b)thatreturnsthisapproximation

totheintegral.TheargumentfisaPythonimplementationf(x)ofthe

mathematicalfunctionf(x).

b) Use the approximation (3.9) to compute the following integrals:

�

cosxdx,

�

sinxdx,and

�

π/2

sinxdx,Ineachcase,writeouttheerror,

i.e.,thediﬀerencebetweenthe exactintegralandthe approximation

(3.9).Makeroughsketchesofthetrapezoidforeachintegralinorderto

understandhowthemethodbehavesinthediﬀerentcases.

c) Wecaneasilyimprovetheformula(3.9)byapproximatingthearea

underthefunctionf(x)bytwoequal-sizedtrapezoids.Deriveaformula

forthisapproximationandimplementitinafunctiontrapezint2(f, a,

b).Runtheexamplesfromb)andseehowmuchbetterthenewformula

is.Makesketchesofthetwotrapezoidsineachcase.

d) Afurtherimprovementoftheapproximateintegrationmethodfrom

c)istodividetheareaunderthef(x)curveintonequal-sizedtrapezoids.

Basedonthisidea,derivethefollowingformulaforapproximatingthe

integral:

�

f(x)dx≈

n�−1

0=1

h(f(x

)+f(x

i+1

)),

(3.10)

wherehisthewidthofthetrapezoids,h=(b−a)/n,andx

=a+ih,

i=0,...,n,arethecoordinatesofthesidesofthetrapezoids.Theﬁgure

belowvisualizestheideaoftheTrapezoidalrule.

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