4.9 Numerical Differentiation 163
4.9Numerical Differentiation
Atthebeginning ofthischapter westated thatderivative approximations arenotasfrequently
needed asintegral approximations .This istruefortheapproximation ofsingle derivatives ,
butderivative approximation formulas areused extensively forapproximating thesolutions
toordinary andpartial differential equations ,asubject weconsider inChapters 11and12.
Thederivative ofthefunction /atx pisdefined as
/'(*o)=lim
h— *0f(X Q+h)-f(X Q)
h
This formula gives anobvious waytogenerate anapproximation tof'(xo);simply compute
fjxp+h)-FJXQ )
h
forsmall values ofh.Although thismay beobvious ,itisnotvery successful ,duetoour
oldnemesis ,round -offerror.Butitiscertainly theplace tostart.
Toapproximate fixo ),suppose first thatxoe(a,b),where/eC2[a,b\,andthat
x\=xo+hforsome h^0thatissufficiently small toensure thatx\e[a,b\.Weconstruct
thefirstLagrange polynomial ,PQ.I,for/determined byXQandx\with itserror term
f i x)=Po.iOO+2! /(tW)
/(*>)(*-x o-h)+fjxo+h)j x-X Q)+j x-x0)(x-x p-h)-h h 2
forsome number £(x)in[a,b].Differentiating thisequation gives
f\x)=f t o+V-f t o )+D xh{x-x0)(x-x0-h)„
2 /(fto)
f i xo+h)-f i x0)+2(x-x0)-hy//(^(x^)h 2
SO
Difference equations were used
andpopularized byIsaac Newton
inthelastquarter ofthe17th
century ,butmany ofthese
techniques hadpreviously been
developed byThomas Harriot
(1561-1621 )andHenry Briggs
(1561-1630 ).Harriot made
significant advances innavigation
techniques ,andBriggs wasthe
person most responsible forthe
acceptance oflogarithms asan
aidtocomputation .f(x)*/.(*>.+ft
h
with error
There aretwoterms fortheerror inthisapproximation .Thefirstterm involves /"(£(x)),
which canbebounded ifwehave abound forthesecond derivative of/.Thesecond part
ofthetruncation error involves Ac/"(ICO)=/"'(ICO) |'C0,which generally cannot be
estimated because itcontains theunknown term|'C0-However ,when*is*o,thecoefficient
ofAc/"(ICO)iszero.Inthiscase,theformula simplifies tothefollowing :
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Two-Point Formula
I f/"exists ontheinterval containing XQand JC0+h,then
/'(*>)=fixo+h)— f(xQ)
h\f"«) 
forsome number $between xoand XQ+h.
Suppose that Afisabound on|/"(JC)|forxe[a,b].Then forsmall values of/i,
thedifference quotient [fixo+h)— fixo )]/hcanbeused toapproximate fixo)with an
error bounded byM\h\/2.This isatwo-point formula known astheforward -difference
formula ifh>0(seeFigure 4.20)andthebackward -difference formula ifh<0.
Figure 4.20
y- Slope/'(*o)
fixo +h)-f(x0)Slope
x0x0+h x
Example 1Usetheforward -difference formula toapproximate thederivative off(x)=lnxatxo =1.8
using h=0.1,h=0.05,andh=0.01,anddetermine bounds fortheapproximation errors .
Solution Theforward -difference formula
/(1.8+h)— f(1.8)
h
with h=0.1gives
In1.9-In1 .8
0.10.64185389 -0.58778667
0.1=0.5406722 .
Because fix )=— l/x1and1.8<f<1.9,abound forthisapproximation error is
\hfm
21*1
2H2<0.1
2(1.8)2=0.0154321 .
Theapproximation anderror bounds when h=0.05 andh=0.01 arefound inasimilar
manner andtheresults areshown inTable 4.9.
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Table 4.9
h /(1.8+h) /(1.8+/i)--/(1.8) \h\
h 2(1.8)2
0.1 0.64185389 0.5406722 0.0154321
0.05 0.61518564 0.5479795 0.0077160
0.01 0.59332685 0.5540180 0.0015432
Since fix)=l/x,theexact value of/'(1.8)is0.555 ,andinthiscase theerror bounds
arequite close tothetrueapproximation error .
Toobtain general derivative approximation formulas ,suppose that*o,*i,...xHare
(n+1)distinct numbers insome interval /andthat/C"+1(/).Then
m=£fWito +
forsomef(JC)in/,where L,(x)denotes theythLagrange coefficient polynomial for/at
JCQ,x\,...,xn.Differentiating thisexpression gives
n
f'(x)=J2f(*j)L'j(x)+Dx
j=0(X-X0)--(X-X„)
(n+1)! /("+1)(*(*))
(n+1)!
Again wehave aproblem with thesecond partofthetruncation error unless xisoneof
thenumbers x*.Inthiscase ,themultiplier ofDx[/(n+1)(£(*))]iszero ,andtheformula
becomes
/-(**)=E/(*/)ty**)+
j=of{n+l\Hxk ))
(n+1)! -X j).
j=0
j*k
Three -Point Formulas
Applying thistechnique using thesecond Lagrange polynomial atxo,=*o+h,and
x2=XQ+2hproduces thefollowing formula .
Three -Point Endpoint Formula
If{"'exists ontheinterval containing XQandxo+2h,then
fixo)=^[-3/(xo)+4/(*o+h)-f(x0+2h))+y/'"«).
forsome number £between XQand XQ+2h.
This formula isuseful when approximating thederivative attheendpoint ofaninterval .
This situation occurs ,forexample ,when approximations areneeded forthederivatives used
fortheclamped cubic splines .Left endpoint approximations arefound using h>0,and
right endpoint approximations using h<0.
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When approximating thederivative ofafunction ataninterior point ofaninterval ,itis
better tousetheformula thatisproduced from thesecond Lagrange polynomial atx0-h,
XQ,andxo +h.
Three -Point Midpoint Formula
If/'"exists ontheinterval containing xo— handxo+h,then
/'(*o)=L[/(Jo+h)-f{xo-A)]-^/"(*),
forsome number £between XQ-hand XQH-h.
The error intheMidpoint formula isapproximately half theerror intheEndpoint
formula and/needs tobeevaluated atonly twopoints whereas intheEndpoint formula three
evaluations arerequired .Figure 4.21 gives anillustration oftheapproximation produced
from theMidpoint formula .
Figure 4.21
y-
Slope f\xd^^
^-^^Slope2h[/(*>+A)-/(*o“ A)]
1 1 1
*0~h XQ XQ+h X
These methods arecalled three -point formulas (even though thethird point ,f(xo),
does notappear intheMidpoint formula ).Similarly ,there arefive-point formulas that
involve evaluating thefunction attwoadditional points ,whose error term is0(hA).These
formulas aregenerated bydifferentiating fourth Lagrange polynomials that pass through
theevaluation points .The most useful istheMidpoint formula .
Five-Point Formulas
Five -Point Midpoint Formula
If/(5)exists ontheinterval containing *o— 2hand*o+2h,then
/'(*<>)=7Ll/to-2/0-8/(*0-h)+8/(*o+h)-fixo+2h)]+^/(5)(f),12/i 30
forsome number £between JCO-2hand XQ-F2h.
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There isanother five-point formula thatisuseful ,particularly with regard totheclamped
cubic spline interpolation .
Five-Point Endpoint Formula
If/(5)exists ontheinterval containing xoandxo+4/z,then
/'(*>)=^r-25/(*o)+48/(*0+h)-36/(JCO+2h)
+16/(x0+3h)-3/(x0+Ah)]+j/»>«),
forsome number £between XQand JCQ+4/i.
Left-endpoint approximations arefound using h>0,andright-endpoint approxima -
tions arefound using h<0.
Example 2Values for/(x)=xe1arcgiven inTable 4.10.Usealltheapplicable three-point and
five-point formulas toapproximate /'(2.0).
Table 4.10
x f i x)
1.8 10.889365
1.9 12.703199
2.0 14.778112
2.1 17.148957
2.2 19.855030Solution Thedata inthetable permit ustofindfour different three-point approximations :
Three -Point Endpoint Formula with h=0.1:
-L-3/(2.0)+4/(2.1)-/(2.2)]=5[— 3(14.778112 )+4(17.148957 )-19.855030 )]
=22.032310 ,
Three -Point Endpoint Formula with h=— 0.1:
-^[-3/(2.0)+4/(1.9)-/(1.8)]=— 5[— 3(14.778112 )+4(12.703199 )
-10.889365 )]=22.054525 ,
Three -Point Midpoint Formula with h=0.1:
-L/(2.1)-/(1.9)]=5(17.148957 -12.703199 )=22.228790 ,
V/.Z
Three -Point Midpoint Formula with h=0.2:
^[/(2.2)-/(1.8)]=
^(19.855030 -10.889365 )=22.414163 .
The only five-point formula forwhich thetable gives sufficient data isthemidpoint
formula with h=0.1.
Five-Point Midpoint Formula with h=0.1:
^[/(1.8)-8/(1.9)+8/(2.1)-/(2.2)]=— [10.889365 -8(12.703199 )
+8(17.148957 )— 19.855030 ]
=22.166999 .
Ifwehadnoother information ,wewould accept thefive-point midpoint approximation
using h=0.1asthemost accurate .Thetruevalue forthisproblem is/'(2.0)=(2-fl)e2=
22.167168 .
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Round -OffError Instability
Itisparticularly important topayattention toround -offerror when approximating deriva -
tives.When approximating integrals inSection 4.3,wefound thatreducing thestepsizein
theComposite Simpson ’srulereduced thetruncation error,and,even though theamount
ofcalculation increased ,thetotal round -offerror remained bounded .This isnotthecase
when approximating derivatives .
When applying anumerical differentiation technique ,thetruncation error will also
decrease ifthestepsizeisreduced ,butonly attheexpense ofincreased round -offerror.To
seewhy thisoccurs ,letusexamine more closely theThree -Point Midpoint formula :
/'(*o)=-U/(*>+h)~f(x0-*)]-In o
Suppose that,inevaluating /(xo+/i)and/(XQ H),WCencounter round -offerrors e(xo+h)
ande(*o-h).Then ourcomputed values/(xo+h)and/(xo-h)arerelated tothetrue
values/(xo+h)and/(XQ-h)by
/(*o+h)=f(xo+h)+e(xQ+h)and/(x0-h)=/(x0-h)+e(x0-h).
Inthiscase,thetotal error intheapproximation ,
x/(*>+W-/(*o-W e(xo+h)-e(xo-h)h2/(*b)2*2h~
6f
isdueinparttoround -offandinparttotruncating .Ifweassume thattheround -offerrors ,
e(xo± h)tforthefunction evaluations arebounded bysome number e>0andthatthe
third derivative of/isbounded byanumber Af>0,then
f\xo)-fixo4-h)-/(x0+h)
2hEh\.
Toreduce thetruncation portion oftheerror,hrM/6,wemust reduce h.Butashisreduced ,
theround -offportion oftheerror ,e/h,grows.Inpractice ,then,itisseldom advantageous
tolethbetoosmall ,since theround -offerror willdominate thecalculations .
Illustration Consider using thevalues inTable 4.11 toapproximate /'(0.900 ),where f(x)=sin*.
Thetruevalue iscos0.900 =0.62161 .Theformula
/'(0.900 ).^(0-900+ft)— /(0.900— h),2h
with different values ofA,gives theapproximations inTable 4.12.
Table 4.11X sinx X sinxTable 4.12Approximation
0.800 0.71736 0.901 0.78395h to/'(0.900 ) Error
0.850 0.75128 0.902 0.784570.001 0.62500 0.00339
0.880 0.77074 0.905 0.786430.002 0.62250 0.000890.890 0.77707 0.910 0.789500.005 0.62200 0.00039
0.895 0.78021 0.920 0.795600.010 0.62150 -0.00011
0.898 0.78208 0.950 0.813420.020 0.62150 -0.000110.899 0.78270 1.000 0.841470.050 0.62140 -0.00021
0.100 0.62055 -0.00106
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Theoptimal choice forhappears toliebetween 0.005 and0.05.Wecanusecalculus
toverify (seeExercise 13)thataminimum for
e h2
e(h)=-+— Af,n o
occurs ath=y/3e/M,where
M= max |/'"(*)|= max |cosx \=cos0.8^0.69671 .
jt[0.800.1.00 ) x[0.800.1.00]
Because values of/aregiven tofivedecimal places ,wewillassume thattheround -off
error isbounded bye=5x10-6.Therefore ,theoptimal choice ofhisapproximately
/1=3(0.000005 )
0.69671%0.028 ,
which isconsistent with theresults inTable 4.12.
Inpractice ,wecannot compute anoptimal htouseinapproximating thederivative ,
because wehave noknowledge ofthethird derivative ofthefunction .Butwemust remain
aware thatreducing thestepsize willnotalways improve theapproximation .
Wehave considered only theround -offerror problems thatarepresented bytheThree -
Point Midpoint formula ,butsimilar difficulties occur with allthedifferentiation formulas .
The reason fortheproblems canbetraced totheneed todivide byapower ofh.Aswe
found inSection 1.4(see,inparticular ,Example 1),division bysmall numbers tends to
exaggerate round -offerror,andthisoperation should beavoided ifpossible .Inthecase
ofnumerical differentiation ,itisimpossible toavoid theproblem entirely ,although the
higher -order methods reduce thedifficulty .
Keep inmind that,asanapproximation method ,numerical differentiation isunstable ,
because thesmall values ofhneeded toreduce truncation error cause theround -offerror
togrow.This isthefirstclass ofunstable methods thatwehave encountered ,andthese
techniques would beavoided ifitwere possible .However itisnot,because these formulas
areneeded inChapters 11and12forapproximating thesolutions ofordinary andpartial -
differential equations .
Methods forapproximating higher derivatives offunctions using Taylor polynomials
canbederived aswasdone when approximating thefirstderivative orbyusing anaveraging
Keep inmind thatdifference technique thatissimilar tothatused forextrapolation .These techniques ,ofcourse ,suffer
method approximations can from thesame stability weaknesses astheapproximation methods forfirstderivatives ,but
beunstable . they areneeded forapproximating thesolution toboundary value problems indifferential
equations .Theonly onewewill need isaThree -Point Midpoint formula ,which hasthe
following form.
Three -Point Midpoint Formula forApproximating f"
If/(4)exists ontheinterval containing XQ— handx$+h,then
/BC*0)=^[/(*o-h)-2f(xo)+f(xo+/.)]-^/(4,«).
forsome number fbetween XQ— handx$+h.
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EXERCISE SET 4.9
l.
2.
3.
4.
5.
6.
7.
8.Usetheforward -difference formulas andbackward -difference formulas todetermine each missing
entry inthefollowing tables .
a.x fix) fix ) b. X fix) fix )
0.5 0.4794 0.0 0.00000
0.6 0.5646 0.2 0.74140
0.7 0.6442 0.4 1.3718
ThedatainExercise 1were taken from thefollowing functions .Compute theactual errors inExercise 1,
andfinderror bounds using theerror formulas .
a./(x)=sin*b./(x)=e*— l x2+3x— 1
Usethemost accurate three-point formula todetermine each missing entry inthefollowing tables .
X fix) fix )
1.1 9.025013
1.2 11.02318
1.3 13.46374
1.4 16.44465X fix) fix )
8.1 16.94410
8.3 17.56492
8.5 18.19056
8.7 18.82091
X fix) fix )
2.9-4.827866
3.0-4.240058
3.1 — 3.496909
3.2-2.596792X f i x)/'(*)
2.0 3.6887983
2.1 3.6905701
2.2 3.6688192
2.3 3.6245909
ThedatainExercise 3were taken from thefollowing functions .Compute theactual errors inExercise 3,
andfinderror bounds using theerror formulas .
a.f i x)=e2x b./(x)=xlnx
c. f i x)=xcosx— x2sinx d. f(x)=2(lnx)2+3sin*
Usetheformulas given inthissection todetermine ,asaccurately aspossible ,approximations foreach
missing entry inthefollowing tables .
X fix) fix ) b. x fix)
2.1-1.709847 -3.0 9.367879
2.2-1.373823 -2.8 8.233241
2.3-1.119214 -2.6 7.180350
2.4-0.9160143 -2.4 6.209329
2.5-0.7470223 -2.2 5.320305
2.6-0.6015966 -2.0 4.513417
ThedatainExercise 5were taken from thefollowing functions .Compute theactual errors inExercise 5,
andfinderror bounds using theerror formulas .
a.f(x)=tanx b. f(x)=e+x2
Let/(x)=cos 7i x.UsetheThree -Point Midpoint formula forf"andthevalues of/(x)atx=0.25 ,
0.5,and0.75 toapproximate /"(0.5).Compare thisresult totheexact value andtotheapproximation
found inExercise 7ofSection 3.5.Explain why thismethod isparticularly accurate forthisproblem .
Let f(x)=3xe*-cosx.Usethefollowing data andtheThree -Point Midpoint formula for/"to
approximate /"(1.3)with h=0.1andwith h=0.01.
X 1.20 1.29 1.30 1.31 1.40
f i x)11.59006 13.78176 14.04276 14.30741 16.86187
Compare your results to/"(1.3).
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9. Usethefollowing data andtheknowledge thatthefirstfivederivatives of/were bounded on[1,5]
by2,3,6,12,and23,respectively ,toapproximate /'(3)asaccurately aspossible .Find abound for
theerror.
X 1 2 3 4 5
f i x)2.4142 2.6734 2.8974 3.0976 3.2804
10.
11.Repeat Exercise 9,assuming instead thatthethird derivative of/isbounded on[1,5Jby4.
Analyze theround -offerrors fortheformula
fixo)=/(*0+ft)-/(*o)
h\f"(So).
Find anoptimal h>0interms ofabound Mfor/"on(x0,x0+h).
12. Allcalculus students know thatthederivative ofafunction /atxcanbedefined as
fix)
h-*0 h
Choose your favorite function /,nonzero number x,andcomputer orcalculator .Generate approxi -
mations f'n(x)to/'(-*)by
13./»=f i x+10-")-f(x)
10"
forn=1,2,....20anddescribe what happens .
Consider thefunction
e h2'(*)=h+1M'
where Misabound forthethird derivative ofafunction .Show thate(h)hasaminimum aty/3e/M.
14. The forward -difference formula canbeexpressed as
/'(*o)= +h)~/(*»)]- -j f'(xo)+0(h3).
Useextrapolation onthisformula toderive an0(/i3)formula forf\xo).
15. InExercise 7ofSection 3.4,data were given describing acartraveling onastraight road.That problem
asked topredict theposition andspeed ofthecarwhen t=10s.Usethefollowing times andpositions
topredict thespeed ateach time listed.
Time 0 3 5 8 10 13
Distance 0 225 383 623 742 993
16. Inacircuit with impressed voltage £(t)andinductance L,Kirchhoff ’sfirstlawgives therelationship
„ di£(t)=L-+Ri,at
where Ristheresistance inthecircuit andiisthecurrent .Suppose wemeasure thecurrent forseveral
values oftandobtain :
t1.00 1.01 1.02 1.03 1.0
i3.10 3.12 3.14 3.18 3.24
where tismeasured inseconds ,iisinamperes ,theinductance Lisaconstant 0.98 henries ,andthe
resistance is0.142 ohms .Approximate thevoltage £(t)when t=1.00,1.01,1.02 ,1.03,and1.04.
17. Derive amethod forapproximating /"(x0)whose error term isoforder h2byexpanding thefunction
/inathird Taylor polynomial about XQandevaluating atXQ+handx0-h.
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4.10 Survey ofMethods andSoftware
Inthischapter weconsidered approximating integrals offunctions ofone ,two,orthree
variables andapproximating thederivatives ofafunction ofasingle real variable .
The Midpoint rule.Trapezoidal rule,andSimpson ’srule were studied tointroduce the
techniques anderror analysis ofquadrature methods .Composite Simpson ’sruleiseasy to
useandproduces accurate approximations unless thefunction oscillates inasubinterval of
theinterval ofintegration .Adaptive quadrature canbeused ifthefunction issuspected of
oscillatory behavior .Tominimize thenumber ofnodes andalso increase theaccuracy ,we
studied Gaussian quadrature .Romberg integration wasintroduced totake advantage ofthe
easily -applied Composite Trapezoidal rule andextrapolation .
Most software forintegrating afunction ofasingle realvariable isbased ontheadaptive
approach orextremely accurate Gaussian formulas .Cautious Romberg integration isan
adaptive technique thatincludes acheck tomake sure thattheintegrand issmoothly behaved
over subintervals oftheintegral ofintegration .This method hasbeen successfully used in
software libraries .Multiple integrals aregenerally approximated byextending good adaptive
methods tohigher dimensions .Gaussian -type quadrature isalsorecommended todecrease
thenumber offunction evaluations .
The main routines inboth theIMSL andNAG Libraries arebased onQUADPACK :
ASubroutine Package forAutomatic Integration byR.Piessens ,E.deDoncker -Kapenga ,
C.W.Uberhuber ,andD.K.Kahaner published bySpringer -Verlag in1983 [PDUK ].The
routines arealso available aspublic domain software ,athttp://www .netlib .org/quadpack .
The main technique isanadaptive integration scheme based onthe21-point Gaussian -
Kronrod rule using the10-point Gaussian rule forerror estimation .The Gaussian rule
uses the10points JCI,. . . ,X\oandweights w j,. ..,vt>iotogive thequadrature formula
Wi/fo)toapproximate f(x)d x.Theadditional points x\\,...,*21andthenew
weights i»i,...,V2\arethen used intheKronrod formula , v,-/(*, ).Theresults of
thetwoformulas arecompared toeliminate error .Theadvantage inusing x\,.. .,*10in
each formula isthat/needs tobeevaluated atonly 21points .Ifindependent 10-and21-
point Gaussian rules were used ,31function evaluations would beneeded .This procedure
also permits endpoint singularities intheintegrand .Other subroutines allow user specified
singularities andinfinite intervals ofintegration .Methods arealso available formultiple
integrals .
Although numerical differentiation isunstable ,derivative approximation formulas are
needed forsolving differential equations .The NAG Library includes asubroutine forthe
numerical differentiation ofafunction ofone real variable ,with differentiation tothe
fourteenth derivative being possible .AnIMSL function uses anadaptive change instep
sizeforfinite differences toapproximate aderivative of/atxtowithin agiven tolerance .
Both packages allow thedifferentiation andintegration ofinterpolator cubic splines .
Forfurther reading onnumerical integration ,werecommend thebooks byEngels [EJ
and byDavis and Rabinowitz [DRJ .Formore information onGaussian quadrature ,see
Stroud andSccrest [StSJ .Books onmultiple integrals include those byStroud [Stro]and
bySloan andJoe[SJ].
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