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3.5 Exercises
131
�
�
�
�
�
��
�
�
��
��
��
��
��
��
��
Implement(3.10)inaPythonfunctiontrapezint(f, a, b, n).Run
theexamplesfromb)withn=10.
e) Writeatestfunctiontest_trapezint()forverifyingtheimplemen-
tationofthefunctiontrapezintind).
Hint. Obviously, the Trapezoidalmethodintegrateslinearfunctions
exactlyforanyn.Anothermoresurprisingresultisthatthemethodis
alsoexactfor,e.g.,
�
2π
0
cosxdxforanyn.
Filename:trapezint.py.
Remarks. Formula(3.10)isnotthemostcommonwayofexpressing
theTrapezoidalintegrationrule.Thereasonisthatf(x
i+1
)isevaluated
twice,firstintermiandthenasf(x
i
)intermi+1.Theformulacan
befurtherdevelopedtoavoidunnecessaryevaluationsoff(x
i+1
),which
resultsinthestandardform
�
b
a
f(x)dx≈
1
2
h(f(a)+f(b))+h
n−1
i=1
f(x
i
).
(3.11)
Exercise3.7:DerivethegeneralMidpointintegrationrule
TheideaoftheMidpointruleforintegrationistodividetheareaunder
thecurvef(x)intonequal-sizedrectangles(insteadoftrapezoidsasin
Exercise3.6).Theheightoftherectangleisdeterminedbythevalueof
fatthemidpointoftherectangle.Thefigurebelowillustratestheidea.
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132
3 Functionsandbranching
Computetheareaofeachrectangle,sumthemup,andarriveatthe
formulafortheMidpointrule:
�
b
a
f(x)dx≈h
n�−1
i=0
f(a+ih+
1
2
h),
(3.12)
whereh = (b−a)/n isthe widthofeach rectangle.Implement this
formulainaPythonfunctionmidpointint(f, a, b, n)andtestthe
functionontheexampleslistedinExercise3.6b.Howdotheerrorsin
theMidpointrulecomparewiththoseoftheTrapezoidalruleforn=1
andn=10?Filename:midpointint.py.
Exercise3.8:MakeanadaptiveTrapezoidalrule
AproblemwiththeTrapezoidalintegrationrule(3.10)inExercise3.6is
todecidehowmanytrapezoids(n)touseinordertoachieveadesired
accuracy.LetEbetheerrorintheTrapezoidalmethod,i.e.,thedifference
betweentheexactintegralandthatproducedby(3.10).Wewouldlike
toprescribea(small)tolerance�andfindannsuchthatE≤�.
Sincetheexactvalue
�
b
a
f(x)dxisnotavailable(thatiswhyweusea
numericalmethod!),itischallengingtocomputeE.Nevertheless,ithas
beenshownbymathematiciansthat
E≤
1
12
(b−a)h
2
max
x∈[a,b]
|f
��
(x)|.
(3.13)
Themaximumof|f��(x)|canbecomputed(approximately)byevaluating
f��(x) at alarge number ofpointsin[a,b],taking the absolutevalue
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3.5 Exercises
133
|f��(x)|,andfindingthemaximumvalueofthese.Thedoublederivative
canbecomputedbyafinitedifferenceformula:
f
��
(x)≈
f(x+h)−2f(x)+f(x−h)
h2
.
Withthecomputedestimateofmax|f��(x)|wecanfindhfromsetting
theright-handsidein(3.13)equaltothedesiredtolerance:
1
12
(b−a)h
2
max
x∈[a,b]
|f
��
(x)|=�.
Solvingwithrespecttohgives
h=
√
12�
�
(b−a)max
x∈[a,b]
|f
��
(x)|
�
−1/2
.
(3.14)
Withn = (b−a)/h we have the n that corresponds s to o the desired
accuracy�.
a) MakeaPythonfunctionadaptive_trapezint(f, , a, b, eps=1E-5)
forcomputingtheintegral
�
b
a
f(x)dxwithanerrorlessthanorequalto
�(eps).
Hint. Computethencorrespondingto�asexplainedaboveandcall
trapezint(f, a, b, n)fromExercise3.6.
b) ApplythefunctiontocomputetheintegralsfromExercise3.6b.Write
outtheexacterrorandtheestimatednforeachcase.
Filename:adaptive_trapezint.py.
Remarks. Anumericalmethodthatappliesanexpressionfortheerror
toadaptthechoiceofthediscretizationparametertoadesirederror
tolerance,isknownasanadaptivenumericalmethod.Theadvantage
ofanadaptivemethodisthatonecancontroltheapproximationerror,
andthereisnoneedfortheusertodetermineanappropriatenumberof
intervalsn.
Exercise3.9:Explainwhyaprogramworks
Explainhowandtherebywhythefollowingprogramworks:
def add(A, B):
C = = A + + B
return C
A = = 3
B = = 2
print add(A, B)
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134
3 Functionsandbranching
Exercise3.10:Simulateaprogrambyhand
Simulatethefollowingprogrambyhandtoexplainwhatisprinted.
def a(x):
q = = 2
x = = 3*x
return q q + x
def b(x):
global q
q += = x
return q q + x
q = 0
x = 3
print a(x), , b(x), , a(x), b(x)
Hint. Ifyouencounterproblemswithunderstandingfunctioncallsand
localversusglobalvariables,paste the code intothe OnlinePython
Tutor8 andstepthroughthecodetogetagoodexplanationofwhat
happens.
Exercise3.11:Computetheareaofanarbitrarytriangle
Anarbitrarytrianglecanbedescribedbythecoordinatesofitsthreever-
tices:(x
1
,y
1
),(x
2
,y
2
),(x
3
,y
3
),numberedinacounterclockwisedirection.
Theareaofthetriangleisgivenbytheformula
A=
1
2
|x
2
y
3
−x
3
y
2
−x
1
y
3
+x
3
y
1
+x
1
y
2
−x
2
y
1
|.
(3.15)
Writeafunctionarea(vertices) that returnstheareaofa triangle
whoseverticesarespecifiedbytheargumentvertices,whichisanested
listofthevertexcoordinates.Forexample,computingtheareaofthe
trianglewithvertexcoordinates(0,0),(1,0),and(0,2)isdoneby
triangle1 = = area([[0,0], , [1,0], [0,2]])
# or
v1 = = (0,0); v2 2 = = (1,0); v3 3 = = (0,2)
vertices = = [v1, , v2, v3]
triangle1 = = area(vertices)
print ’Area of f triangle e is %.2f’ ’ % % triangle1
Test the e area function on n a a triangle with known area. . Filename:
area_triangle.py.
Exercise3.12:Computethelengthofapath
Someobject ismovingalongapathintheplane.Atn+1pointsof
timewehaverecordedthecorresponding(x,y)positionsoftheobject:
8
http://www.pythontutor.com/visualize.html
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3.5 Exercises
135
(x
0
,y
0
),(x
1
,y
2
),...,(x
n
,y
n
).ThetotallengthLofthepathfrom(x
0
,y
0
)
to(x
n
,y
n
)isthesumofalltheindividuallinesegments((x
i−1
,y
i−1
)to
(x
i
,y
i
),i=1,...,n):
L=
�n
i=1
�
(x
i
−x
i−1
)2+(y
i
−y
i−1
)2.
(3.16)
a) MakeaPythonfunctionpathlength(x, y)forcomputingLaccord-
ingtotheformula.Theargumentsxandyholdallthex
0
,...,x
n
and
y
0
,...,y
n
coordinates,respectively.
b) Write a a test function n test_pathlength() ) where you u checkthat
pathlengthreturnsthecorrectlengthinatestproblem.
Filename:pathlength.py.
Exercise3.13:Approximateπ
The value of π equals s the e circumference of acircle with radius s 1/2.
Supposeweapproximatethecircumferencebyapolygonthroughn+1
pointsonthecircle.Thelengthofthispolygoncanbefoundusingthe
pathlengthfunctionfromExercise3.12.Computen+1points(x
i
,y
i
)
alongacirclewithradius1/2accordingtotheformulas
x
i
=
1
2
cos(2πi/n), y
i
=
1
2
sin(2πi/n), i=0,...,n.
Callthepathlengthfunctionandwriteouttheerrorintheapproxima-
tionofπforn=2k,k=2,3,...,10.Filename:pi_approx.py.
Exercise3.14:Writefunctions
Threefunctions,hw1,hw2,andhw3,workasfollows:
>>> print t hw1()
Hello, World!
>>> hw2()
Hello, World!
>>> print t hw3(’Hello, , ’, , ’World!’)
Hello, World!
>>> print t hw3(’Python n ’, , ’function’)
Python function
Writethethreefunctions.Filename:hw_func.py.
Exercise3.15:Approximateafunctionbyasumofsines
Weconsiderthepiecewiseconstantfunction
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136
3 Functionsandbranching
f(t)=
1, 0<t<T/2,
0, t=T/2,
−1,T/2<t<T
(3.17)
Sketchthisfunctiononapieceofpaper.Onecanapproximatef(t)by
thesum
S(t;n)=
4
π
�n
i=1
1
2i−1
sin
�
2(2i−1)πt
T
�
.
(3.18)
ItcanbeshownthatS(t;n)→f(t)asn→∞.
a)WriteaPythonfunctionS(t, n, T)forreturningthevalueofS(t;n).
b) WriteaPythonfunctionf(t, , T)forcomputingf(t).
c) Writeouttabularinformationshowinghowtheerrorf(t)−S(t;n)
varieswithnandtforthecaseswheren=1,3,5,10,30,100andt=αT,
withT=2π,andα=0.01,0.25,0.49.Usethetabletocommentonhow
thequalityoftheapproximationdependsonαandn.
Filename:sinesum1.py.
Remarks. Asumofsineand/orcosinefunctions,asin(3.18),iscalled
aFourierseries.ApproximatingafunctionbyaFourierseriesisavery
importanttechniqueinscienceandtechnology.Exercise5.39asksfor
visualizationofhowwellS(t;n)approximatesf(t)forsomevaluesofn.
Exercise3.16:ImplementaGaussianfunction
MakeaPythonfunctiongauss(x, m=0, s=1)forcomputingtheGaus-
sianfunction
f(x)=
1
√
2πs
exp
�
−
1
2
�
x−m
s
�
2
�
.
Writeoutanicelyformattedtableofxandf(x)valuesfornuniformly
spacedxvaluesin[m−5s,m+5s].(Choosem,s,andnasyoulike.)
Filename:gaussian2.py.
Exercise3.17:Wrapaformulainafunction
Implementtheformula(1.9)fromExercise1.12inaPythonfunction
withthreearguments:egg(M, To=20, Ty=70).Theparametersρ,K,c,
andT
w
canbesetaslocal(constant)variablesinsidethefunction.Lett
bereturnedfromthefunction.Computetforasoftandhardboiledegg,
ofasmall(M=47g)andlarge(M=67g)size,takenfromthefridge
(T
o
=4C)andfromahotroom(T
o
=25C).Filename:egg_func.py.
3.5 Exercises
137
Exercise3.18:Writeafunctionfornumericaldifferentiation
Theformula
f
�
(x)≈
f(x+h)−f(x−h)
2h
(3.19)
canbeusedtofindanapproximatederivativeofamathematicalfunction
f(x)ifhissmall.
a) Writeafunctiondiff(f, x, h=1E-5)thatreturnstheapproxima-
tion(3.19)ofthederivativeofamathematicalfunctionrepresentedbya
Pythonfunctionf(x).
b) Writeafunctiontest_diff()thatverifiestheimplementationof
thefunctiondiff.Astestcase,onecanusethefactthat(3.19)isexact
forquadraticfunctions.Followtheconventionsofthepytestandnose
testingframeworks,asoutlinedinExercise3.2andSections3.3.3,3.4.2,
andH.6.
c) Apply(3.19)todifferentiate
•f(x)=ex atx=0,
•f(x)=e
−2x2
atx=0,
•f(x)=cosxatx=2π,
•f(x)=lnxatx=1.
Useh=0.01.Ineachcase,writeouttheerror,i.e.,thedifferencebetween
theexactderivativeandtheresultof(3.19).Collectthesefourexamples
inafunctionapplication().
Filename:centered_diff.py.
Exercise3.19:Implementthefactorialfunction
Thefactorialofniswrittenasn!anddefinedas
n!=n(n−1)(n−2)···2·1,
(3.20)
withthespecialcases
1!=1, 0!=1.
(3.21)
Forexample,4!=4·3·2·1=24,and2!=2·1=2.WriteaPythonfunction
fact(n)thatreturnsn!.(Donotsimplycalltheready-madefunction
math.factorial(n)-thatisconsideredcheatinginthiscontext!)
Hint. Return1immediatelyifxis1or0,otherwiseusealooptocompute
n!.
Filename:fact.py.
138
3 Functionsandbranching
Exercise3.20:Computevelocityandaccelerationfrom1D
positiondata
SupposewehaverecordedGPScoordinatesx
0
,...,x
n
attimest
0
,...,t
n
whilerunningordrivingalongastraightroad.Wewanttocomputethe
velocityv
i
andaccelerationa
i
fromthesepositioncoordinates.Using
finitedifferenceapproximations,onecanestablishtheformulas
v
i
≈
x
i+1
−x
i−1
t
i+1
−t
i−1
)
,
(3.22)
a
i
≈2(t
i+1
−t
i−1
)
−1
�
x
i+1
−x
i
t
i+1
−t
i
−
x
i
−x
i−1
t
i
−t
i−1
�
,
(3.23)
fori=1,...,n−1.
a)WriteaPythonfunctionkinematics(x, i, dt=1E-6)forcomputing
v
i
anda
i
,giventhearrayxofpositioncoordinatesx
0
,...,x
n
.
b) Write a Python function test_kinematics() ) for testing the im-
plementationinthecaseofconstantvelocityV.Sett
0
=0,t
1
=0.5,
t
2
=1.5,andt
3
=2.2,andx
i
=Vt
i
.
Filename:kinematics1.py.
Exercise3.21:Findthemaxandminvaluesofafunction
Themaximumandminimumvaluesofamathematicalfunctionf(x)on
[a,b]canbefoundbycomputingfatalargenumber(n)ofpointsand
selectingthemaximumandminimumvaluesat thesepoints.Writea
Pythonfunctionmaxmin(f, a, b, n=1000)thatreturnsthemaximum
andminimumvalueofafunctionf(x).Alsowriteatestfunctionfor
verifyingtheimplementationforf(x)=cosx,x∈[−π/2,2π].
Hint. Thexpointswherethemathematicalfunctionistobeevaluated
canbeuniformlydistributed:x
i
=a+ih,i=0,...,n−1,h=(b−a)/(n−
1).ThePythonfunctionsmax(y)andmin(y)returnthemaximumand
minimumvaluesinthelisty,respectively.
Filename:maxmin_f.py.
Exercise3.22:Findthemaxandminelementsinalist
Givenalista,themaxfunctioninPython’sstandardlibrarycomputes
thelargestelementina:max(a).Similarly,min(a)returnsthesmallest
elementina.Writeyourownmaxandminfunctions.
3.5 Exercises
139
Hint. Initialize a variable max_elem m bythe first element t inthe e list,
thenvisitallthe remainingelements(a[1:]),compareeachelement
tomax_elem,andifgreater,setmax_elemequaltothatelement.Usea
similartechniquetocomputetheminimumelement.
Filename:maxmin_list.py.
Exercise3.23:ImplementtheHeavisidefunction
ThefollowingstepfunctionisknownastheHeavisidefunctionandis
widelyusedinmathematics:
H(x)=
�
0,x<0
1,x≥0
(3.24)
a) ImplementH(x)inaPythonfunctionH(x).
b) MakeaPythonfunctiontest_H()fortestingtheimplementationof
H(x).ComputeH(−10),H(−10
−15
),H(0),H(10
−15
),H(10)andtest
thattheanswersarecorrect.
Filename:Heaviside.py.
Exercise3.24:ImplementasmoothedHeavisidefunction
TheHeavisidefunction(3.24)listedinExercise3.23isdiscontinuous.It
isinmanynumericalapplicationsadvantageoustoworkwithasmooth
versionoftheHeavisidefunctionwherethefunctionitselfanditsfirst
derivativearecontinuous.OnesuchsmoothedHeavisidefunctionis
H
�
(x)=
0,
x<−�,
1
2
+
x
2�
+
1
2π
sin
�
πx
�
�
,−�≤x≤�
1,
x>�
(3.25)
a) ImplementH
�
(x)inaPythonfunctionH_eps(x, eps=0.01).
b)MakeaPythonfunctiontest_H_eps()fortestingtheimplementation
ofH_eps.Checkthevaluesofsomex<−�,x=−�,x=0,x=�,and
somex>�.
Filename:smoothed_Heaviside.py.
Exercise3.25:Implementanindicatorfunction
Inmanyapplicationsthereisneedforanindicatorfunction,whichis1
oversomeintervaland0elsewhere.Moreprecisely,wedefine
I(x;L,R)=
�
1,x∈[L,R],
0,elsewhere
(3.26)
140
3 Functionsandbranching
a) Make twoPythonimplementationsof suchanindicator function,
onewithadirecttestifx∈[L,R]andonethatexpressestheindicator
functionintermsofHeavisidefunctions(3.24):
I(x;L,R)=H(x−L)H(R−x).
(3.27)
b)Makeatestfunctionforverifyingtheimplementationofthefunctions
ina).Checkthatcorrectvaluesarereturnedforsomex<L,x=L,
x=(L+R)/2,x=R,andsomex>R.
Filename:indicator_func.py.
Exercise3.26:Implementapiecewiseconstantfunction
Piecewiseconstantfunctionshavealotofimportantapplicationswhen
modelingphysicalphenomenabymathematics.Apiecewise constant
functioncanbedefinedas
f(x)=
v
0
,x∈[x
0
,x
1
),
v
1
,x∈[x
1
,x
2
),
.
.
.
v
i
x∈[x
i
,x
i+1
),
.
.
.
v
n
x∈[x
n
,x
n+1
]
(3.28)
Thatis,wehaveaunionofnon-overlappingintervalscoveringthedomain
[x
0
,x
n+1
],andf(x) isconstant ineachinterval.One example isthe
functionthatis-1on[0,1],0 on[1,1.5],and4on[1.5,2],where we
withthenotationin(3.28)havex
0
=0,x
1
=1,x
2
=1.5,x
3
=2and
v
0
=−1,v
1
=0,v
3
=4.
a) MakeaPythonfunctionpiecewise(x, , data)forevaluatingapiece-
wiseconstantmathematicalfunctionasin(3.28)atthepointx.The
dataobjectisalistofpairs(v
i
,x
i
)fori=0,...,n.Forexample,datais
[(0, -1), , (1, , 0), (1.5, 4)]intheexamplelistedabove.Sincex
n+1
isnotapartofthedataobject,wehavenomeansfordetectingwhether
xistotherightofthelastinterval[x
n
,x
n+1
],i.e.,wemustassumethat
theuserofthepiecewisefunctionsendsinanx≤x
n+1
.
b) Designsuitabletestcasesforthefunctionpiecewiseandimplement
theminatestfunctiontest_piecewise().
Filename:piecewise_constant1.py.
Exercise3.27:Applyindicatorfunctions
Implementpiecewiseconstantfunctions,asdefinedinExercise3.26,by
observingthat
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