DIFFERENTIATION - Oregon State Universitysites.science.oregonstate.edu/~restrepo/MTH452/... ·...
Transcript of DIFFERENTIATION - Oregon State Universitysites.science.oregonstate.edu/~restrepo/MTH452/... ·...
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NUMERICALDIFFERENTIATION
f4xo hhmo flxothyfx.li
oElaiDfEC4abTFinite difference Approximations
six
Assumefix issufficientlysmooth oh Xoth
Sixth ftp.hflxdtthYYxd ffh3f Gift
ForwardDifference Approximatiatoffxo
ftxothl f d.fi xo tfhf45 3Efxoxothh
BackwardDifference Approximation error
ftp.flxo.hI
a sf teth143 Elko hid
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CenterDifferenceApproximationtoffxo
ffxoth7 ffxo.tn2
fExo tfhf 3h Terror
The center difference approximation
ftp.flxothl f xo h2h
is 01h21 approximationwhereas
f'M fCxuth 7 forwardh
tcxdflxo.tl backward
1hare 01h approximations
ex Using derive a finitedifferenceapproxinahu
to f Cxo
I
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use ff fGoth 2Mt Ho h
t fh f 13 error
tle OCK center difference approximationRush highorder approximationcouldbefoundfromTaylorseries
GENERALPROCEDUREFORDERIVATIVES BYFINITE DIFFERENCES
Issue f E Cnt Cab lattice xi3i.no
fix g that lady t iIcxxi7Terror
Lagrange polynomials
Differentiate both sides
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ntl
fix EHaddidnt H GI axilforany Xfl aid Now confineto X Xi is
this thadduki t if Gftp.xila
errorLossofPrecision in Computing DerivativesCautionarytale computingderivatives usyfiniteprecision Suppose approximate Hx bycenterdifference
X 4h Cflxoth throh YfGlet faith Ftxth t eExoth
theM 4 PrusIerror
W FCxoth khekothzge
x.nl
hj f 45
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assure lea.IM cE
f4D cMTleutletnncataapbesGnitepseeisiaerror yields
114 1 Flinch 3Eq thatElm's
y
EAµ
El isemininel
ii
hi
NUMERICALINTEGRATION QUADRATURE
163mostConnon strategiesn Fixxi findZiemann dx re Eciflxi naunknownsciSum iso
Gaussian 11 dx.ggqffxi seFndxiicis.tQuadrature II Hismmnised2h unknowns
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Montecarlo MC ffcxdxn.im tmsiosthmemguessessiaceepLMoftlese
RIEMANNSUMS
Assume Lxi distinct on Cab
bflxidxsfb.ESxi liCx7dx
f Icx.oilfmY5Ddxhtt
If dx Ecifcxi t Error f Terroris 0
G Gdx is91,2 in
EXAMPLES RIGHTHAND8LEFTHANDSUMS eex Piecewise arstatholxPide Xosa X sb hex Xo
Xo X
rec 1 lies I
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NI
LAR feokddxfcxdshfk.io h X XzTo
fati error is
underestimate
µr i for fkoLHR funchies
f Exactfor Cx
Xoh x
Emory dx LARI Gkixddx.ch f433
CoMPosiTEhHRff Yfa X Yz Xm Xn b
n l Xo Xu
bfhddxeiqoxiflxi oxi xiti
xigpyp.fi'eRcxdxftxDh fjem
oerrorisover
asxo.ba 7 77467 PHR functions
x
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COMPOSITE PHR
fabfcxsdxe.EOxiflx.jpJti Iitixihy
ta X Yz Xn Tn byu
TRAPEZOIDAL Xosa X sb h X Xo
Xl Xl
Stands t MdxXo Xo
f G Cx xxx x d
Terror we'll usethemeanucleetleoren
x
fftddx.bz flxdtfCxi
f g ixxDCx a dx
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ffcxdtflx.tl B I hz xi.mx
Xi
ffhddxshz fflxdtf.CH Y3zf4s
Xu Terror01h37
f i giftf no error
Xo XlCOMPOSITE TRAPEZOIDAL
fakddxz.tn 8xifCxi tOxzfho t0xngflxn
ftp.yykxsxaoxz x.xbn
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Composit UNIFORMGRID NEWTON COATES
I dx re tho 2fix t2fGd
2fanD t fanwith Ox ban
RudiNewton Gates quadrature refers toquadratureformulas on equallyspacedintends
Simpson Uses 2ndorder Lagrangepolynomialsand this requires atleast 3points
Xo a X at h Xz bI
h s b
ffabfixdxsflolxlfcxdtl.CMdxDtlzCxlfCxzDdx
f ftp.xokx.xkx.xdfmfscxddxE error
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Xz
ffcxjdx flolxlfcxdtl.CMdxDtlzGdfCxzDdx
01h fIt turnsoutthat theerror isoverlypessimistic
see
printednotes Byusingan integral oftheTaylor
series of fad about X one can find
Iaadx hglfcxdt4flxdtflxdfghgf.MG
SAMPSON
error errorerrorOchs
tN exactforfadSIMPSON FERG
i
Xo X Xz
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COMPOSITE SIMPSON Newton GatesVersion
f dxqoxgfkxdt4flxdtlfcxd.tlfCxDtt2flxyt 2fCxn.jt4flxn.ptflxn
nhustbeeoeuaxsbn
affxft.snXxz x.xbn
ADAPTIVE QUADRATURE
Reg uses Richardsonextrapolationfeelater
Adhoc use a finergridwhere fG charges moreri e where f Nae large
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GAUSSIANQUADRATURE
ChooseExitsin and fi 3in so that
ECsx It dx a flail ftis minimized There are In unknowns
Ci and 2x
Thetraditional strategy is to differentiate fwith respectto Xi's and Ci's and set these all
to zero The 1 3 ki that minimize Efix theseunknowns
Traditionally pick fad CPen Gand then find Xi is Eh that lead
to C so i
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ex Take aid EH and n 2
Ci Cz X Xz are unknowns M
R f dx.in ciffxi7
if f Epyx then
froth xtazx4 azxfdx.aedxta
dxtaz.fx2dxtazfx3dx
where ai are arbitrayarstats suffices to
show that issatisfied when aistis91,213
a gxo.tczx.ISdx
2cixitczxis.ixdx0
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cxitcaxis dx
cix.twf'x3dxsoSohrgforeiixi
G Cz lxi zV3 Xz Bg
i f k dxxffr tffBg