Centro de Matemática

da Universidade do Porto

Minicurso

Accelerated Spectral Approximation

Balmohan Limaye(Indian Institute of Technology, Bombay)

Horário:

. Segunda-feira, dia 16 de Maio 9h30 -11hOO(sala 1.21)

. Terça-feira,dia 17de Maio9h30 - l1hOO (sala 1.21)

Local:

. Edificio dos Departamentos de MatemáticaRua do Campo Alegre, 6874169-007 Porto

Accelerated Spectral Approximation

B.V. Limaye

Department of Mathematics

lndian lnstitute of Technology Bombay

In order to find an approximate solution of an eigenvalue problem T<p= À<pfor a

bounded operator T on a Banach space, one often considers a sequence (Tn) of bounded

operatars which converge to T in some sense and then one solves Tn<Pn= Àn<Pn.When

the convergence of (Tn) to Tis slow, one needs to accelerate the convergence of (Àn) to À

and of <Pn to <p. This can be accomplished by considering a higher arder spectral analysis

involving an approximate solution of a polynomial eigenvalue problem for T. Errar

estimates in the case of a nonzero simpIe eigenvalue of T will be given, and a scheme of

implementing this procedure by a suitable matrix formulation will be discussed.

1

.

Iterative Refinement

(a) Linear Systems: Ax == y, Ax == y.

For k == 1, 2, . . .

set r(k-l):== y - Ax(k-l),

solve Au(k-l) == r(k-l), ,

lo

set x(k):== x(k-l) + u(k-l).

(b) Eigenvalue Problems: Tcp == Àcp,

T~== À~ T*~* == À~* /~ ~* ) == 1cp cp, cp cp ,\cp,cp .

Recast the equation ,\lqJ<plq]= q"L,l(T - T)kT<p[q]

as follows: k=O (,\Iq])k

cpfq] T (T - T)T ...- 1-

. .. (T - T)q- T cpfq]

cpfq]

À[q]I o o cpfq]

À[q]

,X[q]-

o I o o

cpfq]

j l j l

cpfq]O ... O I O

(,\[q])q-1 (,\[q])q-1

Similarly, recast the equation Trp == Àrp as follows:

cp T o o cp

I 11 O . . . . ..O II '

.

ÀI = I

O I O . .. O

Àl I

lO O I O cp. .. -Àq-1

For z -I O in G, T - zI is invertible in BL(X) if and only

if T - z1 is invertible in BL(X), and in that case

R(T, z) o o

R(T,z) 1 o oz z

R(T,z) ==. .

...

o

R(T,z)zq-l

1 1 1zq-l zq-2 z

Hence

-1 I I~

--:- JR(T, z )dz O. .. O ~

21f1 C

-1

p"= -1 - JR(T dz

.2--:- J R(T Z )d _I 21fi C

' z)-

}fI C ' z - z

O ... O

-1 dz--:- JR(T,z) O ... O21f1 C zq-l

Errar Estimates

Let Tcp == Àcp, cp i- O and TnCPn == ÀnCPn, CPn i- O.

Then for alllarge n,

1. /Àn - ÀI < c I1(T ~ _T~)CPnl/= 0(1/(T - Tn)Tnl/),

...

PCPnis an eigenvector of T (corresponding to À) and

IICPn- PCPnl/ < C II(T - Tn)CPnl/ = O(II(T- Tn)Tn) I/).IICPn/l - IICPnl1

II(T - Tn)cpll2. I Àn - ÀI < C 11 - 11 == O ( 11(T - Tn) T /I) ,

PncPis an eigenvector of Tn (corresponding to Àn) and

IlPnCP- cPl/ < c 1/ (T - Tn)cpll = O( II(T - Tn)TI/).IIcpll - /lcpll

.

The operator T : X -+- X is given by

..

and the operator Tn : X -+- X is given byq-lL (T - Tn)kTnXk+1

Xl I I k=O

Hence

(Tn - T)

Tn

Xq

Xl

Xq

Xl. .

Xq-l

q-l

(Tn - T)XI + L (T - Tn)kTnXk+1k=l

O

O

.

XlI I TXI

I I

T'Xl

1.-

Xq

Xq-l

Xl

(Tn - T)T I :

2 q-l k-(T - Tn) Xl + 2: (T - Tn) TnXkk=2

O

XqO

and~

"

-(T - Tn)qTnx

Xl I I O q I

I Xl

(Tn - T)Tn : I == I I

for I : I E X.

Xq I I

OI I Xq

Theorem. Let À be a nonzero simple eigenvalue of T in

BL(X) with a corresponding eigenvector cp. Let (Tn) be a

sequence in BL(X) such that

(1ITnID is bounded and I/(Tn - T)21/ ~ O.

Fix an integer q with q > 2. Then there is a positive integer~

no (depending on q) such that for each n > no the following

holds:

Tn has a nonzero simpIe eigenvalue À~]with a corresponding

eigenvector 'P~]. Let cp~]and 1jJ~]denote the first components

of 'P~] and Pn'P respectively. Then cp~]i- O, 1jJ~]i= O and we

have the following:

----.

Accelerated Errar Estimates

11

,,\[qJ - ,,\1

< c II(T- TnFTnip~JII = O(II(T - To )qTo 11). n - q /I <pWJ 11 n n ,

P<p~J is an eigenvector of T (correspondingto ,,\) and

lIip~] - PipWJII< c II(T - Tn)qTnip~JII= O(II(T- TnFTn) 11),1/<pwJII - q II<pwJII

2.IÀ~J- ÀI < Cqll(T -"T~!qTipll = O(II(T- Tn)qTII),

111jJ~J- ipll < c II(T - Tn)qTipll = O(II(T- Tn)qTII). ,11<p11 - q 11<p1!

r w

Matrix Formulation

Let a finite rank operator T : X -7 X be given by

Tx :== (x, fl)Xl + . . . + (x , fn)xn, X E X.

Define

K : X -7 Cnxl by

\:: (x , fI)Kx :== x E X,

(X, fn)

and

L : Cnxl -7 X by

Lu :==U(l)Xl + ... + u(n)xn, u E Cnxl .

Then T == LK.

Define A : Cnxl -7 Cnxl by A :== K L. Note that A is

represented by the n x n matrix

A :== [( x j , fi)] , 1 < i, j < n.

. I(

Lemma

Let o ~ À E C, x E X and u E Cnxl. Then

Tx == Àx and K x == u {::=:::? Au == Àu and Lu == Àx.

In this case, x ~ O ~ u ~ o.

Proposition

Let O ~ À E~ C. If À is an eigenvalue of T, then À is an

eigenvalue of A, and if u is a corresponding eigenvector of

A, then L(u) :== U(l)Xl + ... + u(n)xn is a corresponding

eigenvector of T.

. "] n

The operators T and T can be represented by q x q matrices

as follows:

T o o

I o o

o o I o

and

T (T - T)T (T - T)q-lT

I o o

T :== o I o o

o o I o

T :== 10 I O . .. O<

..

. . . .. .

Identify (CnXl)qXl with Cnqxl and define

A : Cnqxl -+ Cnqxl by

q-l- -k-L K(T - T) LUk+l

k=OUl Ul

AUl

for E ([}nqx1..

Uq .. Uq

Uq-l

Note that A is represented by the nq x nq matrix

A(O) A(1) .'. . A(q-1)

In On On

A :== I On In On On

. . .

On ... On In On

where for k == O, . . . ,q - 1,

A(k) := [((T - T)kXj, li)] ,

l<i,j<n,

Inis the n x n identity matrix, and On is the n x n zero matrix.

-

Define K : X ~ djnqxl by

Xl KXI Xl

KI: . for E X.

Xq KXq Xq

For O =I À E dj, define L(À) : djnxl ~ X by.'

~ k-- -~ q-l (T - T ) Lu I

L(À)U :== 2: - for U E djnxk=O À k

and 1:,(À) : djnqxl ~ X by

L(À)UI UI

L:(À)U :== for U == E djnqxl .

L(À)Uq Uq

-

Lemma

LetO i- À E C, x E X and u E Cnqxl. Then

Tx == Àx and Kx == u ~Au == Àu and L(À)U == Àx.

In this case, x i- O ~ u i- O.

Proposition

Let Oi- À E C. If À is an eigenvalue of T, then À is an eigen-

value ofA, and if Ul is the first component of a corresponding

eigenvector ofA, then

L:(À)Ul := q'Ll(T - T)k LUlk=O

is the first component of a corresponding eigenvector of T.

x = C([O,1]), Tx(s) = foI cos(5st)x(t) dt

À : third largest (simple) eigenvalue af T

(Knaw: À = .132657, correct upto 6 decimal places)

À - Àn À - À~J

4 10-5

9 1Ô4 4 10-6

10-6

2 10-4 10-7

10-4

6 10-5

-

n

11

21

31..

41

61

81

Size of the Errar in the Error in the

matrix e.v.p. usual method accelerated method

0J 20 9 10-4 4 10-5

0J 40 2 10-4 4 10-6

0J 60 10-4 10-6

0J 80 6 10-5 10-7

Bibliography

[1] - R. Alam, R. P. Ku1kami, B. V. Limaye, Acce1eratedspectra1approximation, Mathematics ofComputation vol.67, n. 224 (1998), pp 1401-1422.

[2] - R. Alam, R. P. Ku1kami, B. V. Limaye, Acce1erated spectra1 refinement. Part I: Simp1eeigenva1ue,J. Austral. Math. Soc. (Series B) 41 (2000), pp 487-507.

[3] - R. Alam, R. P. Ku1kami, B. V. Limaye, Acce1eratedspectra1refinement. Part lI: C1uster ofeigenva1ues,, J. Austral. Math. Soc. 42 (2000), pp 224-243.

[4] - M. Ahues, A. Largillier, B. V. Limaye, Spectra1 Computations for Bounded Operators,Chapman & Hall/CRC, Boca Raton (2001).

[5] - D. Dellwo, M. B. Friedman, Acce1erated spectra1 ana1ysis of compact operators, SIAM J.Numer Anal., 21 (1984), pp1115-1131.