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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
conventionsinthepytestandnosetestingframeworksforPythoncode.
(SeeSectionH.6formoreinformationaboutpytestandnose.)
Theseconventionssaythatthetestfunctionshould
6http://pytest.org
7
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
b
a
f(x)dx
canbeapproximatedbytheso-calledSimpson’srule:
b−a
3n
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
3
2
π
0
sin3xdx,whichhas
exactvalue2,andinvestigatehowtheapproximationerrorvarieswithn.
Solution. Theevaluationoftheformula(3.6)inaprogramisstraight-
forwardifweknowhowtoimplementsummation(
)andhowtocallf.
APythonrecipeforcalculatingsumsisgiveninSection3.1.8.Basically,
N
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)=
3
2
sin
3
xdx
inaPythonfunction
def h(x):
return (3./2)*sin(x)**3
and call Simpson to compute
π
0
h(x)dx for various choices of 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
2
3/2
3x
2
−7x+2.5
dx.
ThisintegralequalsG(2)−G(3/2),whereG(x)=x33.5x+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
2
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
C=
5
9
(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=
M
k=1
1
k
.
b) Compute 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
2
+bx+c=0
a) Givenaquadraticequationax
2
+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])
4
>>> 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
0
,r
1
,...,r
n
ofapolynomialp(x)ofdegreen+1,p(x)
canbecomputedby
p(x)=
n
i=0
(x−r
i
)=(x−r
0
)(x−r
1
)···(x−r
n−1
)(x−r
n
). (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
b
a
f(x)dx≈
b−a
2
(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:
π
0
cosxdx,
π
0
sinxdx,and
π/2
0
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:
b
a
f(x)dx≈
n−1
0=1
1
2
h(f(x
i
)+f(x
i+1
)),
(3.10)
wherehisthewidthofthetrapezoids,h=(b−a)/n,andx
i
=a+ih,
i=0,...,n,arethecoordinatesofthesidesofthetrapezoids.Thefigure
belowvisualizestheideaoftheTrapezoidalrule.
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