Orthogonal polynomials ou o B w x I4,4 yay Chek

4 GI I

YCH I4264 x x I43Gt N Z x't X Io

To find an orthogonal family of polynomials or Eaib w

r b w we can use a linear transformation

b a x ta then the resultingpolynomials are orthogonal onGib

Fa Gib El I these polynomials are called Legendre polynomials

Y I These are scaled

H siennaCould Sealethem

such that

The giver f Cab IR there 114ill Ice da

exists a unique polynomialRER thenthey would besuch that orthonormal

Hfpullz min Hf91kohCPu

Phoefi Yo 4h family of orthogonalpolynomials andwe

normalisethem guy 48114yd 1 01 in

Every q cPu is of the form qlxkfbo4CAT tbhYukBie IR

Goal choose Bi such that on mimmies Hf ohh overall REPDefine ECG ibn If q f q f oh

the 25ha Lanelift 2 picky t fBsBuGn4D

riZuo Ba Chas 5tMk okhq

Minimum will be attained for indepsofpsi

Bg Cf xD jeo in

RCA pothole t t Bfydx is the uniqueminimize

Thin ThePu is the best ft polynomial forD

f Cab IR if and only if f pin is orthogonal to

everydhePu i e f ph q D for all

IFFP

P

Practical computationof ph for giver fyo en orthogonal 4g iffy Bj Chula

pulxk BoyG t it Bn4hCx

p4Gy

t t Ba k1144711

go4 G t 1 JonYuk

8s Shaggy Hid a o inLazier

Examples Beat fit polynomial in 12 of f x ex overo D with wkle I

4 Gl l

i E 9ir f eI de

to Ge

k 210 e 570G Ifk

Pdx e l t 18 Ge x E t 210 e 570 tf

loNumerical integration quadrature

f a b IR continuous diffable

wetflak w x O

Newton Ghs allowed to compute integralsexactly forpolynomialsup to degree n i we fixed nodes to as being uniform

In Gauss quadrature we allow thosepoints to change and

hope to find more accurate rules

wklflxtde E.tw Skiµ fquadrahmudes

weights

we assume to be determined Xo in and instead ofLagrange interpelation let us try Hermite interpolation

Parti EgotthG fkn tEKaCx fkxuHermite interpolationof f using

Hu xj 1 if heyMdb Xo a

else HIGH O

Kala's o Kiley l k j0 else

fabwcxtfklden fawklpznticxlde

Zflxnlfbh.CHHdx1det fYxuawHkuk

we don't want to involve flag Wh K

so can we find quadraturepoints such that Yeo heon h

n l Genus points integrate polynomials

q to b up todegree Intl exactly

me 2 Y tb comparedtoNewton Coles whenca is n anti