Gene Golub: for the construction of a 3-term conjugate gradient like (CG) method for nonsymmetric real matrices

SPD

IA

AAA

A

one. degree of in polynomial aas written becan if normal(1) is

:Def

T

*

Remark:1)A symmetric??2)A is normal matrix ???

TT AAAA

IA

AAA

A

one. degree of in polynomial aas written becan if normal(1) is

:Def

T

*

IaAaAa

AAA

ss

ss A

s. degree of in polynomial aas written becan if normal(s) is

:Def

01

1T

*

)3( CGA

normal(1)A

CG(3)=matrices for which iterations canbe carried out using 3-term recursion

Faber&

M

IA A*

def

Cin line aon lie eigs+

A normal

F&M

slides

H,.a in A is s

D s.t H SP

H A H AT w

athen

s.t , )3(CGA

Any idea???Any idea???

+A normal

If eigs not line

1)( Ad

H,.a in A is s

D s.t H SP

,n,,iR

XXA

i 21

1

,n,,iR

XJXA

i 21

1

)3( CGA

normal(1)A

IA A*

H A H AT

1--TΘXXH Use

1--TΘXXH Use

wathen

s.t , )3(CGA

3)( Ad Faber & M

CG(3)=matrices for which iterations canbe carried out using 3-term recursion

wathen

wathen

Faber&

M

Cin line aon lie eigs+

A normalF&M slides

def

F&M

slid

es

+A normal

If eigs not line

1)( Ad

GGr

GIieA ri

,20,0

)(2

A is the translation and rotation of a -self adjoint matrix

TH

if A is B-self

adjoint

0r

BA

Hermition

for HSPD