ANumericalinlefrationisiastadler/na18/material/Notes19.pdfYo 2 z IIT Last class Lagrange...
Transcript of ANumericalinlefrationisiastadler/na18/material/Notes19.pdfYo 2 z IIT Last class Lagrange...
Last class Lagrange interpolation
Hermite interpolation finish
ANumericalinlefrationisia34 Kut IR distinct
Yoy yutR20,21 2n412
Find Pan EBane such that
Pantaleo YiRin Xi z
9 n
pau.ciHntk2iismatphofedit'mnmipkynauid
What properties of Huand Ka do we expect
Hubli to IIT Kalki
t.i.io finna iii
Hula Lucia ll 2 Lien H xD E Gant
Kxtx LalxD Attn c Gant satisfy the
aboveconstraints
I76247
Theorem no f ab R f ant is continuous
then the v Hermiteinterpolant off
satisfiesunique
fGa Pan.la f fIIa TutiHD2
Mnt he x ta
12ht2Htt x
2
thx PantGc f Manta
ProfileExample For n I construct a cubic polynomial
B such that
Bait Balti pjco.tt B'a It
Xo O K I
BbcHoldyo
t ko 2o t Hildy t k 127211o
lol l
Kola t Hile
Local I K L Ix 3C for 26 024 1
Kola7432 x ko s l xp K
Hilal Lila I 241k Ge xD243 2x
Bce231 K2 K
i ego
computeapproximate
derivatives to f
How accurately does Prix approximate flex
Numerical differentiation
Recall fent sixfix Pma Tnt Knit
we can try todifferentiate the
above
except it's not clear that fuelis continuous
or evendifferentiable
If Pu is Lagrange interpolant off on aidwehave
f let pricey e pbMnet Cfa proof see
Siili
where Mnt man flat xRecap
If himlbnt.IM O then Rief
uniformly ou laid
Numerical Integration
The definite integrals
e'die andcoscaldic
are easy toevaluate
analytically
Unfortunately most integralscannot be
evaluated
analytically with for examplea table of integrals
For example
Idid and fcos da
Now for generalintegralsb
fun dua
ince polynomials are easy to integratethe
s f with a polynomial
idea is to replace
and integrate the polynomial exactly
Ne estReplace
fix with its Lagrangeinterpolaatof degree n
all thatis repaired
is the functionvalues fix
fabfun de I Rhodewhere the
interpolationpoints
ti at Andi i 91,2 n
are equidistantn Kai
Pala II Luck funLula Ito xn xD
Recall it
b
fix die I Lula fun da
II feed Latade
can be easilyprecomputed exactly
Earyacijweights
we
with above choices weobtain a Newlon
Cotes formula
of ordern
fi wgn Hanrem
For n I o Trapezoid rule
fix DX I PCH de
k at i xo a X b
P Ix Local ffa t L a fcb
xa bfeat t 3 fad
b Afca t x a FID
a
Realdie fzffiafbxdxtfcbbf.sc adx
bzIffca t f lbs
Jara ofthebtrapezoidi
j in
b j
Xo a 74 bwoped
Wo BIA W BI Wo ki2
For n 2 Simpson's Rule
a at i Xo a K BE x b
fields fled Lo t fail FIX d
w InadaI
du
evaluate theintegral with
the change ofvariables
Nb2Itt beta2that baa dt.by
wo Wao by symn.net
Remark youcould use the change of variables
or more simply expand the product
III H Gutta Ktx as
constant
and integrate eachterm separately
tbafabfinds Palmde bitfealtyffatty
FID
Since the weightsare
independent of f
they can beprecomputed
in advanced
ERRORESTIMATES
we seek tostudy the
error
Enif Iaflatde II Waffen
i e what is the wife of the error that
is beingcommited by integrating the Lagrange
interpolaert
ten I Ifda II we fun I
If da Procida
If Predictfix Pala Idn
RECAU
I HuipnhaltfnY.TT
Tntilx7l
facMnIjTTntilxYdseMna faf
tf GH
Now we can use thisestimate to determine
an
upper bound on the einer commitedwhen using
the trapezoid mule Crel
IE I E MI bfathisaldic
ME I all b Ida
May µ a b a die
b a 3Mr
12For Simpson's rule D
tea I E fable a x latbI ex b Ida
196
when f eThisistmetofatbe
NOTE Ei El D shown
Eoff 0 when f EPs fE z
n odd Newton Coles is dead for polynomials