is suf¬cient to show the property for the monic polynomials .

T y

¢

The proof is by induction. The property is true for the polynomial . Assume that !

X R ! R y

¢

it is true for : T T

§ R ¢ ¦ X § X

¬

¢ R ¦

¢

§

Multiplying the above equation by on both sides yields

T T

X X

§ R¢ § ¬ ¦ §

¢ R ¤¦

¢

"U

W

²

If the vector on the left-hand side belongs to , and therefore if the above

wE ¢

!

y y

equation is multiplied on both sides by , then ¤

T T

¢

§ R ¢ T § ¥¦ X § "U R ¢ X

¤¬ ¤¦

¢ ¢

W

X

X T ¦ X§ R¢T

Looking at the right-hand side we observe that belongs to . Hence,

T ¢ ¢

X

§ W"U R ¢ v¦ § R ¢ ¤ ¨§ ¤ ¬ ¦ § W"U R ¢

¬ §

(¤¦ ¢ ¢ ¢

Tw E

² ! X y ¢ X T y¢ w E

which proves that the property is true for , provided . For the case

y

¬ § ¬y¦ § ¤

, it only remains to show that , which follows from

wE ¦ ¢ ¢ ¢ ¢

!

y

¨ ¡¡µ£ „ ¢

§|5¥ yq¢| ¢ £ ¥§

„ 5| j C¦£¥

5§

tC

9 "–© ¡"

§

1B

T T

¬ ¦X § X

§ §

¢ ¢ by simply multiplying both sides by .

¤

¦

¢ ¢ ¢ ¢

W“ W“

¦ Q— “ ˜m

©™ ¡

¡

Arnoldi™s method [9] is an orthogonal projection method onto for general non- ¢

Hermitian matrices. The procedure was introduced in 1951 as a means of reducing a dense

matrix into Hessenberg form. Arnoldi presented his method in this manner but hinted that

the eigenvalues of the Hessenberg matrix obtained from a number of steps smaller than

¤ could provide accurate approximations to some eigenvalues of the original matrix. It

was later discovered that this strategy leads to an ef¬cient technique for approximating

eigenvalues of large sparse matrices. The method will ¬rst be described theoretically, i.e.,

assuming exact arithmetic, then implementation details will be addressed.

!$6¨

§ I H BA ©' & 8 ) 7F P% 50A

H

B8

3

¢

Arnoldi™s procedure is an algorithm for building an orthogonal basis of the Krylov subspace

. In exact arithmetic, one variant of the algorithm is as follows:

¢

˜Q¤ v ¢

¡¦ P¥ B 76 8

D

Q

4DA

3C

' 7$

%# ) 2

1

1. Choose a vector of norm 1 ¦

TW

}

¬

2. For Do: #V¡™(

!( ( ˜

XR u vR £ y

§¬

¬

u

3. Compute for E u ¦( © ¦

V(Q˜ (

(

R y ¦ vR £ £R £ u ¦§ 3 u ¥ ² (¬

§

u

4. Compute

£u ¥

¬ W u u£

5.

u u W’U

¬

6. If then Stop Y £

u u ¤ u ¥ W"U

¬ u¦

7. £

W"U

W"U

8. EndDo