Summary QR algorithm

A triadiagonal through Householder Givens

A A for h 944A2µhI QQRK QR fectoritahe

Acht RAHIMImalt in reverseorderend

shifted version µhEIR approx

Acht zQUEACH

to eigenvalue

Cast k a

D Q Q FA I am ID diagonalmodern

CGT ChQ Q

we find the eigenvalues Howabouteigenvectors

Start with Be 1kWh BIB hidagonal

PIBPu AK flops Nn

afhogonal productsof Householder orgirusmatricesIs IFTA 0 Dpdiag.walk n'PaQRQRalso iteration

98 EarputBPuOTH D CannB Roth That D

B FIFTHP Bv dire where rei is thei thcolumnof THEM

How expensive is one stepof QR alga

A

ta menu

D uppertriangular

profdofGivensrotations

Alk futsal R ol

17 F 1Since A is symmetric x mustvanish and A is kidnagemel

Each QR iteration is cheap since we are only working withtridiagonal matrices

6.2 Polynomial Interpolation

Approximate functions uning polynomialsthat coincide with the

function values andfoederivatives at wide pointsPK

iii y t.EEQ Xa b

Lagrange interpolatier pay 3rdorder polynomial

onlyfunctionvalues p Xi Ski i O l

p i ft is0,1Hermite interpolationfunctionvaluesand derivatives

funInterpolationpnoblemI

M 1 Xo a distinct xitxy for it gYo yaC IR i Find Pn E Ph suchthatpuki gi s

0 in

Pn Spaceof polynomials of degree Em i e

Pu auxht ta Xiao I au lowC K

Lemmai m 1 Thereexistpolynomials La ER 6 0 n such

thatLf f if i Lagrangepolynomials

Moreover puca ZaoLawyn is the interpolatingpolynomial

I

ii cn

Praed x qtT x xi since Luka Iizoith polynomial

caiq xaxiJ

DLuCxkiFhGHgigi kand0ehe

Fkn xiinnumber

putti Luca Yu Yi interpolatingpolynomialD