Some iterative methods for finding the eigenvalues of self-conjugate operators

SOME ITERATIVE METHODS FOR FINDING THE EIGENVALUES OF SELF-CONJUGATE OPERATORS* F.

KUNERT

(Leningrad) (Received

23

February

1963)

Various iterative methods have been proposed to find the eigenvalues of self-conjugate operators in Hilbert space. They require different amounts of computation and have different rates of convergence. Certain methods often give identical results although the calculation formulae are not externally similar. This paper proves the equivalence of eigenvalue [I]) for findi ng an isolated the iterative method proposed by Holdt that the Boldt method also has the rate

M.K. Gavurin’s LB2 process (see of self-conjugate operators and (see [21)*. This by itself proves of convergence of the LB2 process

(i.e.

Aoldt

the

order

of

qsn,

p < l),

whilst

could

only

Prove

cubic

con-

vergence.

Let A be a self-conjugate operator with a closed range of definition A has an isolated D(A) in Hilbert space 8. We assume that operator simple eigenvalue h subject to determination with a normalized eigenvector 9. Let us assume that we know a certain normalized approximation ~c E D(A) to ‘p. We denote h, = (ALIT q~,,), u,, = Ape - hop,,. Obviously (90,

l

l

00)

Zh.

= 0.

vych.

amt.,

4,

No.

1,

1964.

143-145,

* The Holdt method was proposed by the author sional case; the application of calculation operator

problems,

formulae

(see

generally

[11I) used

speaking,

in the

operator

194

is

for the finite dimenformulae for arbitrary

impossible. case,

are

The calculation given

below.

Eigenvalues

Let us introduce

of

se If-conjugate

operators

CO; (I E 0).

CfJz= (2,gkJ co + 2 (2, co) %I Then C,‘x = (x, self-conjugate

195

oO)qo + 2(x, q~~)ue. In addition, operator JO:

we shall

Let us denote C, = G; = A - B, and Ho = A - C,, For all x E D(A) it is true that (2, a01 p,

examine the

(2 E b).

Boz = (2, cpo)oo + (2, co) 90

H,,x = C,z -

(9

and H; s = Coz -

(2) then Ho’ = A - Co’.

(2, ‘p. )cr,.

(3)

It can easily be verified that hu’and ‘p. are an eigenvalue and eigenvector respectively for operators H, and Cc, If q~s approximates Q eigenvalue of these satisfactorily enough, A,, is the simple isolated operators. Let us denote system

by R,

a bounded self-conjugate

operator

defined

by the

(2 ED (A), (x,0 = O), (2 E 8, (2, 90) = 0).

and we shall observe tionship is valid

that

(Go-

for

It

follows

C&-Q%= We shall ator Hi.

from formula

yO =

(3)-(5)

(CO- kJ (cpo+ Rod -

rela-

Q.

f

(5) Roao

are eigen

for

the

that

(q. + Roco, 90) a0 = a0 - =o =

assume that A, is the simple

Let us introduce

the following

&,I Roz = a~-- WQOIQO.

Lemma 1. The number A, and vector operator ii;. Proof.

z EQ

an arbitrary

(4)

isolated

eigenvalue

0.

of oper-

the notation: VII *O= II=

TO+ Roco V ~+IIkz-’

In the LB2 process the approximate lated from the formula

vector

(5) Q1

fOllOWing

Q.

iS

CalCU-

196

F.

Kunert

where

80= Qo - Rouo+ and z. = Rio,,

Here y. = R,,u,, systems

*(~~~o$2 qoo.

are the main essentials

(Co - &J Yo = oo* C(Yo.Po) = 0;

(7) for

solving

(Co- %I) 20 = Yo, i (20,cpo) = 9.

(3)

The sequence of vectors ‘po, 91, . . . , qn, . . . , obtained according LB2 process converges to the vector 9 at a rate 9sn, Q < 1. if IIQoII

(P = distance

As an approximation

from h to the remaining

following

the

to the

part of A operator

T,,. Holdt proposed

to select

the vector

(9) where u0 is the solution

of the system

(H~--(H,--nduo-(H~--)u,,

(uo,Qol = 0.

(10)

Examples can be given for which (Ho’ - ?+,)u, has no meaning and conIn these cases the following sequently these formulae are inapplicable. system should be examined

(Ho- LJ uo=

80 -

which can always be solved, systems are equivalent.

and if

Let us examine the limited system P, (Ho (Ho-&jPoz

Lemma 2. for

all

hot $J $09

hl,f+d = 0,

system (10)

operator

P,

has a solution

which is determined

Ao) x = z (z ED (4, (2, cpo) = 0). = a! (z E 4, (2, $d = 01, PO*, = 0.

y E @

it

is correct (Ys

(11) then both

by the

(12)

that

$0)

POY= Rby - 1 + U&Jo IP R&. Proof. Let x E D(A) and (x, Q,,) = 0. Then from (12) we obtain P,(H, - A,)% = x. On the other hand, by virtue of (3) and (4) R,(H, A,,)% = Ro(G,, - h,)x - (x, a,,)R,q~, = X, and consequently P,(H, - h,)z = If we denote (Ho - A,)z = y. then y E .$ 63 ($J~} and R,(H,, - $+

Eigcnvalues of se If-conjugate

197

operators

POY = ROY. The

vector

arbitrary

(Y, Y~)YJ

+

POY= Po(y-

y E.fj

be presented

can

in the form Y = [Y -

(Y, Y,,)Y~, then (yt~,o)So)~(Y,0a)P,~,=R,(Y-(Y,0d90)

In particular,

when y = ou, P0a0

=

=

-

then

(Roao,ad

Roao -

(13)

1 -j- IIRoaop Riaw

System (lo)

or (11) has a single solution uu and as can easily be seen, But then it follows from formulae (7) and (9) that qI = ‘pl uO and thus the equivalence of both methods is established. =

Poao.

Remark

f . A parameter

CO(4

2 =

t can be introduced

b, cpo)a0 + t (2, 00,cpov

in formula

(1):

c, (f)

Ho (t) = A -

and the iterative procedure can be considered as dependent on t. t = 0, it again appears to be equivalent to the LB* process. Renark 2. We note

that

the formulae

(y. see (8)) can be replaced

derived

for

If

the LBI process

by the formula (Ho’ (G)- ho 1% = 0,

which may provide

some advantage.

Translated

by

E. Semere

REF’FRWCES 1.

Gavurin,

2.

von Holdt,

M.K., Zh. R.E.,

vych. J.

Assoc.

mat.,

1, NO. 5, 757-770,

Comput.

Mach.,

3,

No.

1961. 3.

223-238,

1956.