View of On the Chebyshev method for approximating the eigenvalues of linear operators

REVUE D'ANALYSE NUMÉRIQUE ET DE ^THÉORIE DE L'APPROXII\IATION Tome 25, No' 1-2, 1996, pp. 43-56

ON THE CHEBYSHEV METHOD FOR APPROXIMATING

THE EIGENVALUES OF LINEAR OPERATORS

EMrL CATTNA$ and ION PÃVÃLOru (Cluj-Napoca)

I.INTRODUCTION

Approaches to the problem of approximating the eigenvalues

of

linear operators by ^the Newton method have been done in a series ofpapers

([l],

_{[3], [4],}

[5]). ^There is a special interest in using the Newton method because the operatorial equation to be solved has a special form, as we shall see. We shall stttdy

in

the

following the convergence of ^the Chebyshev method attached to this problem and we shall apply ^the results obtained for the approximation of ^an eigenvalue and of a corresponding eigenvector for a matrix of real or complex numbers.

It

is known that the convergence order of Newton method is 2 and the convergence order

of

Chebyshev method is 3. Taking into account as

well

the number

of

operations made at each step, we obtain that the Chebyshev method is more efficient than the Newton method.

Let9bea

Banach space overK, whereK= R

orK=

C, and T: E

+

E a linear operator. It is well known that the scalar À is an eigenvalue of ^?" if the equation

(l.l)

^Tx

-),x:

⁰

has at least one solution

î

^{+ 0} , ^where 0 is the

null

element of the ^space E, The elements x +

6

^that satisff equation ( I .l) are called eigenvectors of the operator 1, corresponding to the eigenvalue À.

For the simultaneous determination of the eigenvalues and eigenvectors of

I

we can proceed in the following way.

We attach to equation (l.l) an equation of ^the form

(1.2) ^Gx:

^I

whe¡e G is a linear functional ^{G :} E -+ K.

Consider the real Banach space

F:

^{E x K, with the} norm given by

44

(r.4)

(r.7)

where Æ =

Emil

t(;.)

Ion Pãvãloiu Eigenvalues of Linear Operators 45

2 3

(1

3)

_ll,ll

=-ax{lÞll,l!} , ueF, ^{u= witlr} xeE and )'eK

^wlrere

¡,: lf'(u,))-r.

Letu.eF¿urtlõ

>0,b>

0be two realruunben. Write ^S =

{' 'Flll'-'Oll

⁼

^{¡ }}

rr

n4 =ì:lllr"(,)ll, ^u'e,'

)*llr

^{(,)ll =}

llr'(,b)ll +',,a

",ø,,'l,lt@)1¡

^<

llr('t)ll.

*allf'(ø)ll

+ nt282 = zo. Consider the ^nunrbers m¡nxb2 (2.2)

x î.

In this space we consider the operator

f

^{: F -+ F given by}

Tx

-

^ì,x

Gx-l

If we denote

by 0,

₌

[;) ""

^null elemenr of the space F, then the eigenva- lues and the corresponding eigenvectors ofthe operator Tare solutions of the equatio¡

(1.5) _f@)=}t.

Obviously,

/

^{is not a linear operator.}

It

can be easily seen that the

first

order Fréchet deriyative

of/has

the following _{fomr [4]:}

(1.6)

_{-f,(uòh =}

(", ^-

^t"oht

^-

^Àr¡o

I

\ Går )'

"

⁼

^{r(t *} j*r*or,')

With the above notation, ^the following theorem holds:

THronBv ^2.1

If

the operator

f

^í.s three tintes differentiable ^v,i.l.h

f

^{"' (u)} = 0,

for all ^u e

S (0, being the ^3-li.near null operator) and íf, moreover, tltere ^exísf

u.

e F, ^ô

>

0, b

> 0

such that thefollou'ing relations hold

' _i.

_the operaÍor'.f ' (u) has a bounded inversefor all u e S, attd

llrr,i-'ll=,,

ä. the nwnbers ^p" and v given by (2.2) satisfi ^the relaÍiotts

po=Jlllr(r)ll .t

and

vPo_

_<

¡

J[(t-

^po)

th en the fo ^{IIov, i.tt g} Pt'operti ^{es ho I d} :

i.(u,),.0

given by (?,1) Ir convergent:

ä.¡f u-lirnu,,, ^thett ueS and ¡(ø)=et;

n-)û

jll.

_ll,r,,*,

_-

",lls

v

=

|,r,Zt r(t+1

4

hr

?u1

th

where _{and h ,}

For

the second order derivative

of/we

obtain expres

the

.f" (us)ltk =

-7,2\ -

^X1h2

0

l4

?t2

The Fréchet derivatives of order higher than2

of/are

riull.

Considering the above fomts of the Fréchet derivatives

off,

weshall study

i¡

the following the convergence of the Chebyshev method for the operaton havilg the third order F¡échet derivative the null operator.

2. THE CONVBRGENCE OF CHEBYSHEV ]\{ETHOD

The iterative Chebyshev niethod for solving equation (1.5) corrsists

in

the successive constructiorr of the elelnents of the sequerr"" @)rr,-given by

(2'l)

_un+t

_=u,, -f,f(un)-)r,f,,çu,,)(r,,f(u,))', lt=

0,1,...,,uo

*F,

Proof, Denote by g : S

+

F the following ^mapping

s(,) = -r(u)¡(u) - Lrr(")t"

çu¡lrlu)¡(u))'z ^, s,here

f(u) : _lf

' _@))-l

n

= 0rl,

. r- ,,

^vP3"

¡".

lp

- u,,llt6trÐ,

^/,

⁼

^o,r,

(2.3)

46 Ernil C Ion Pãväloiu

4 5 of Linear eratols 4'7

It can be easily seen that for

all

u

e

_^s the folrowing identity holds

.f(") * f'(u)s(u)

⁺

)f,'(u)sr(u)

⁼

=

:r" ø(lr'øl)-' r(u),|r,(,)l-'/' (ù{lr,(òl-' r(,)}').

. !

^r',

al{Lr, (u)l-' r,, (ù{lr, (,)l-' _r (,)}' }

and from relation

ilk*t - ^uo:

^S(u¡)

(2.8)

ll"o*,

- "rll. ^,llt@)ll.

The inequalities

b)

and (2.7) lead us to

(2,e) I

J' (Jtll/(,')ll)'',, ^{= t,k.}

^I

We have that øo*,

e

S:

k+1

llro*,

- r'll< Illrr ^{- ur-'ll=}

i=l

llr(,,)ll .

whence we obtain (2,4)

or

k+1

I "llr(,,-,)ll

=

l= I

J'

lløl. r ^(u)s(u). )r (")r,(ùll=

|,a0,(, ^* !,n,,ul) _r(u)'

_{, Now} we shall prove that the ^sequence

(u,),ro is

Cauchy. Indeed,

for

all tn'

n

e N we

have

,--r

^,n-1

llu,*, -

",,11 =

Ellu,*,*, - ^,,*,11. uI _llf(r,.,

_{¡ll =}

(2.10) i=o

^í=o

nt-l n-l

v \l^3n*t_ v ^3,T^3,.'-3".

^vPó

<: ) oi, =:oi, )

^t:,,

= Jl ,.*'o ^{- JfYo} /¿Yo =.F(t-p¿')'

whence, taking into account that po

<

1,

it

follows tl':.a;t(ur)r20 converges. Let

i ₌ ^lim

u,,. Then, for nr -> ^æ in (2.10) it follows

jv.

The consequence

iìì

follorvs from (2.8) and

(2.9). tr

3. THE APPROXIIÍATION OF THE EIGEN\/¡ILUES AND EIGEN\/ECTORS OF THE il{ATRICES

(2

s)

llrø ^{. r'(u)s(u)} * i¡" ^{@)r,(òll= d[(,)ll', ror ar}

^{u e}

^s

similarly, _{by (2.3) and taking into acount} the'otation we made, we get

(2,6) ll"(,)ll

₌

,ilr(,)ll, _{ror an}

_u e s.

^.. ^.uring

the hypotheses

of

the theorem, inequalify (2.5) and the fact trrat -f "'

(u):0r,

we obtain the following inequality:

ll¡@)ll=llrø,1 -

^.f @ù

-

-f,(uo)s@")

- |f,,(uòs,(*,)ll .

.llrt^l

il + r,(us)g(uo)

* ir' ^{nlr,{ùll

⁼

piir(,,,)|,

Since z,

- uo: g@),by

(2,5) we have

ll",

_,oll=

"llr(r)ll

⁼

"Jlllr("'

^vpo

Jrr(r -

^{po) ô}

p

In

the

following

we shall apply the previously studied method

to

the approximation of eigenvalues and eigenvectors of ^matrices with elements real or complex numbers,

Let p e N

and

the matrix l=("r),,r=;,

rvhere

a, eK ,i,i =\,p.

Using the above notation, we sliall corrsider E

:

_{Kp and}

_F :

_{Kp x}

_K.

_Any

solution of equation (3

r) _r(i)

=

[ï ^_i)

⁼

^(:) ,

^{io e}

{r, ^,p}

^{being r-rxed,}

wherex = (x,, _{..., x o)} e Ã? and 0

:

(0,,.., _{0) e} KP,

will

^lead us to an eigenvalue

of

A andto a correspoirding eigenvector. For the sake of sinrplicity write

À .Þt,

so that equation ^(3,

l)

is represented by the system

whence

it

follows that

z,

e S.

Suppose now that the following properties hold:

a) u, e S, ¡

=[¡.

¡r _ilr(,,)ll

=

pii/(,,-,)ll'. r=

By the fact that ur

e

S, using (2.5)

'* it

follows

(2

7) ll¡@r.)ll=

^{pllr@òll' ,}

48

(3.5) r'(x)h

₌

0

(3.7) _î, = p,,(r,)u,1

₌

Cãtinaç, Ion pãvãloiu

Enil

6 7 Eigenvalues ofLinear Operators 49

(3.2) f(rr,..,,xp,xp+t)=0, i=l,p*l

where

(3,3) .f,(*r,...,xp+t)=

aitxt+...*(o,,

-

xr*1)x¡+...*aipxp,

i =\p

and

(3.4) _fr*r(r)

=

",;

^_

l.

Derrote byP the mapping

p

_:

Kt*t

-+

KÈt

defined by relations (3.3) and (3.a),

Let Í,, _{= (xl' ,'}

'

',

^{x')+t) e I<}

r+t,

_{Trren trre first}

order Fréchet derivative of the operator P at

Ì,,

^has the following _fonri

ott

- x'lr+t

^a12

a2t or, -

^x')+t

From tliis ¡elation \\'r can easily deduce the equalities vu

=-2u,)*fl!,, í=ú;

v')*1=0'

Writirig w,

₌

(wl',w!,...,*i,*r) ₌ f(¡,)4,

and supposing

that

x-,, is an approximation of the solution of system (3.2), _{then the} next

approxi'ratio' !,*,

given by method (2.1) is obtained by

(3.9)

_Ìn+t

_{=i,, -}

_ù,,

- !r,,,

^ri

⁼

^o,l,.,.

consider Kp with the norm of an erement.r: (r¡, ..., -rr) given by the equality

(3'10)

il"il =

,ig{i",|}

^,

and consequently (3,11)

T

j=1 Itcanbeeasilyseenthat

ffP"(-r-,,)lf

=2, for

a1l x,, eKp+I.

Let ïo

e Kp+r be

a'i'itial

approxi'ration of the solution of system (3.2). consider a rear number

. wrire _{rue =}

jjr(;1)ll+,.llr'1r¡)l+zr2

.

where ¿

=.rpllr(")ll , r¡";

being rne

Taki'g i'ro

accounr rhe resulrs

"#Ï:í;1)s."tio,,2

^{rhe following conse-}

quence holds:

CoRoLLaRy 3.1

If

_xs

_eKp+t

_and

_{r e R r.>}

_0, are chose, suclt r,lzat the ntatrix aflached to the operator p,(x) _is nonsíngularþr

all.r _eF,

and rheþllov,ing ittequalities hoÌd:

n,,

.JFllr(;¡)lJ.r

=Y.P'= .

<,

Jr, ^(t

- ^pî)

then thefollou,ing propet.ti.es are lrue (3.8)

_

,,)*

olio Cl¡:

i

Qpio

I

h1 h2

;,

atp a2p

_x,1,

-xi

, -ri

0

h,.

hp*t

4

h"

p

apl _Q^¡

0

-kp+l

,kp*t)

^e

KP+l

flren for

0 k p*t

0 -u'1,+t

;

0

0 0

-np+lt- 0

ow

-kl

-k2

-uj'

0 A rnax aü

<i<

where

ì

=

(rr,..,,lrr*r)

e

Kp+|, If we wnte

_{E = (kt,}

the second order Fréchet derivative _{we get}

(3 .6)

-ke

0

Denote

by f

(;;,

)

^{trre inverse of the} matrix attacrred to the opera tor p, _(t:,)

and

u,, =

l(rn) r(¡,)

_{= (ui,} ,ur,. ..,u,i,*t)

. Let

_{v,, =}

p,,(x,,)(r(xr)r(x,,))2

=

= P"(x,,)i,1. We obtain the following representation

P

(x,,)E

h

=

0 0 0

0

0 hp*t

-tt)+r

0

0

I

ui ui

ui ,')*

-ui

;

0

-t/)t -ui

00

50 Ernil Cåtina5, Ion Pãváloiu ⁸ ₉ Eigenvalues of Liuea¡ Operators 5l

i,

^{the sequence} (xr)n>, generated by (3.9) is convergenl;

ärtf ¡=

^limx-¡ , ^t.hen

P(x)-0r

=(0,,.,,0) eKp+t;

,l-)€

lljr

_llx-,,*1

-

"-,,11

< -+,

, = 0,t,...

,

r/

t'

in, _llr - _",ll < -r"rd=r- ,

^t1

=

^0,1,,,,

r/ttlt - o'

_,

Retnark.

If

the radius r of the ball

5

^{is given and there exists} [P '(.r,,)]-l ^and

z,]l[r' t",)J-'ll . r,,n"n

systen, ^So, we shall adapt the Gauss method in order to perform as few as possible multiplications ^and divisions. When comparing the two methorJs we shall neglect the number of addition ^and subtraction operations.

The solving of a given linear system

Ax:

^{b, A} e

M^(n,

^{b, x e Kn, (where}

we lrave written

m: p +

1) using the Gauss method consists in two ^stages. In the first stage, the given systen

is

transformed into an equivalent one but

with

the matrix of the coefficients being upper triangular, In the second stage the unknowns (*,),=r,,n are detennined by backwarrl substitution'

The

first

stage. There are performed m

-l

^{steps, at each one vanishing the}

elements on the ^same column below the main diagonal.

We write ^the initial system in the form

al ol,, xl bl

"'(', ^)]-'ll

oln

^a,t ,n xnt b,t,,

llt" r"rf'll

p'('o)]-'

^{Suppose that}

^{al, +}

^0,

^al,

being called ^the first pivote. The first line in tlie system ^is multiplied

by

cr

= - þ

unais added to the second one, which becornes

aìt

1-

for

all

.x e ,S and in the above Corollary, taking into account the proof

of

^Theo-

retn2.l,

we can take

0,oîr, ú.r,,..,ú.n , ü.,

after performing ^¡z

+ I

multiplication or division (MID) operations. After ^m

-l

^such transformations, the system becomes

l["'t",1]-'ll

L_ il tl

tr--

1 _

rllt",r"ll-'ll ol, olz

^ol,,

0 o3,

^o?.,,

.f1 bl

ü ii,

X2

4. A, COÀ{PARISON \I/ITH THE NEWTON N{ETHOD

0

^{o'rr. a?,,n xrn}

Note that

if

in (3.9) we neglect the term

- |r,,

^then we get the Newton method:

Ìn+t

= Í, _-ún - ^Ín -f(1,)f$,),

ⁿ

⁼

^0,I,...

In

order to cornpare the Newton method and the Chebyshev method, we shall consider that the linear systems which appear iri both metliods are solved by the Gauss method.

Wúle the Newton rnethod requires at each step the solving of a system

l.r:

^D,

the Chebyshev method requires the solving of two linear systems Ax -- ^b, Ay

:

_ç

witlr the same matrix

A

and the vector c depending on the solution.r

of

the first

Hence at the first ^step there were performed ^(m

- ^l)

^{(m + I)M I D operations}

In the same nìalllter, ^{at the} k-th step we have the system

"h _o

^"1,

o3r.

olr

"3r

ol,r*t

o3.,r*t

olu,

o3,

a'fu ot*r,,,

o!,,,n

x1 bl

X2 bî

0 0

0 0

olt oÍq,t

uk,k+l-.k

t-

"k+l,k+l

x¡

Ík+l

bf bt*,

0 0

"kt

^{afn**t X,,} b!,,

52 Emil Ion Pãvãloiu l0 ₁₁ Eigenvalues of Linear Operators 53

supposi'g rhe Æ-thpivote

"f* *

0 and

perfonning(nt-k) (m- _k+

_z)

M/D

operations we get

At

this stage there are perfonned I + Z+...+tn

-

^nt(tn+

¹⁾ M/D

^opera-

2 tions. In both stages, there are totally perfomred

ol,

^ol,

o

^oïz o3t,

^"h

o'fr 0

ol,t

*t oîft*t

oL, o3u,

X1

)í2

bl

b] ttt3 ¡

^lrl

-+ntL

33 __

^M

^lD

^operations

0 0

0 0

o|,t

*t

"[Tl,t

*,

a"'L

K'il| )t¡

x k+r

bI

bÍil

In the case when we solve the systems

Ax:

^b, Ay

:

c, where the vector c deperrds on the solution x, we first apply the Gauss metúod for the system

Ax:

b

a'd

at the first stage we keep below the main diagonal the coefficients by which the pivotes were rnultiplied.

Then we apply to the vector c the transformations perfonned _{to the} vector /,r

wlren

solvingAx:

b.

Write c

=

_{G/).\ /}

I=l,nt

At

the first step of1l.,,,

0 0 0

b!;+t xrt

At

each step Æ, the elenents below the È-th pivote vanishing, they are riot needed any more in the solving of the system.

"""r'l,tlirïrresponding

memory

in ihe

computer

is

used keeping

in it

the

-ú-Lt

^af,,t,

a?,KIC

c]:=

arrcl _{+ c\}

"It

u,lriclr, of course, will.be.needed only for solving _{another system}

Ay: c,with

c depending _{on -r:, the} solution oî Ax ='b.

At the first stage there are performed

(m

-

l)(nr+ l)+.,.+1. ₃

= ^Znf

^{+ hn2}

-

^Stn

?-

^M

lD

operations.

The second stage. Given the system

cln:= ^a,,,rc] + c),

At

the Æ-th step

c[¡l:= at,*t,F[ +

ct*t

"k*

^art

cf

+c!,

oL olz

ol,,

o o3z

a3,,

x1 b|

b3

b,;i

At

the

rl-th

step

c!]:=

a,r.rr-rcä_l + c,fi-t ^.

Tlrere wereperform ed

m-l+ nt-

2+...+l

- ^{nún-- l)} M I D

operatio¡s 2

Now the second stage of the Gauss method is applied to x2

0 0 aß^ _xnt

the solution.-r- is computed in the followirrg way xo,

=

^b#

I

a'fl^,

"lt "ï,I':,)

[;]

xo

₌

(tt[ - (o[¡,*r*r*r+...+o[,,x,,,)),

"[o ^{, 0}

In addition to the case of a single linear system, in this case were perfornietl nt(m

-

^1\

ï ^M/D

operations, getring

tr

=

(ói

-(o]rx2+...+ol,,,r,n))

/

ol,

54 Emil Cätinq\, lon Pãväloiu ₁₂

13

Eigeuvalues of Linea¡ Operarors

E(C) ₌

I -l

55

tn3

³

25

326

^end;

^qlt[k]l::

^auxi;

for

i :: k+ I

to m do

c[q[i)]:: clqfil)+ A[qli], kl+

clq[k]l

end;

{the solution y is now conputed}

for

i ::

^{¡z down} to 1 do begin

sum::0;

^.

forj

:= ¡

* I

to m do

sum::

sutn + A[plí1, í] *

yUl;

_{now

p:

q}

ytil ::

("[p͡)]

-

^{sunt) I}

^Alp[i], il;

end,

We adopt as the efficiency neasure

of

an iterative method

M

the number

E(M) -

^trrq _'

,'

where q is the convergence order and s is tlie number of M / D operations needed at each step.

We obtain

E(N)

₌

3ln2

m3 + 3nt2

-

^ttt

for the Newton method and

-+_nl

^tn

M/D

operations,

and taking

into

account (3.8) we add

Qn-l)

^ntore

^MID

operations,

obtai'i'g

M/D

operations

nf3"nl -+-rnL*ã-t

Remark.

At

the first stage,

if

for some Æ we have afo

x

0, then an element

4,0 * 6,

^rio

e _{k+

1,...,'n} _{must be found, and the lines} io and kin

A

^and b be swapped.

In order to avoid the error accumulations, a partial or total pivote strategy is recomnrended even

if

_onr

* 0 for

Æ

= \ ^{tn_ l.}

For

pafial

pivoting, the pivote is chosen such that o!,,0 =

,ù,þfrl

The effective interchange of lines can be bypassed by using a pemrutation vectorp

:

(P),=r.,^, which is first initialized _so thatp,

:

_{i. The eleme¡ts in}

_A

_{and tt}

are tlren referred ,o

^ :, .::

oo!q,,ulrd

!,,:

^{brr,¡, and swapping} tlie lines Æ and

i,

is done by swapping the Æ-th an,i ?i+n elernenirìirp

For the chebyshev methoã, the use of the vecto r

p

carltbe avoided by the effective interchanging of the lines, because we r,ust keep track for the pemruta_

tions made, in order to appry them in the same order to the vector c.

Moreover we need two extra vectors

t

and q, in r

stori'g

the transpositions applied to the lines in

Ax:

b, and which are successively applied to

q.

Atthe first

stage of the Gauss method, when the ft-th pivote is c¿f

t

^{and i,}

^*

^{k, the k_thand i} ,_th elements

inp

are swappe.d, a'!_we assign to:

:

ioto

iiiäi.ut"

ìhut u, the Æ-th step we applied to

p

the transpositio tt (k, iç).

After computing the sorution of Ax

:

_{ó, we} iriitial ize the vector c by (3.7), tlre pen'utation _vector

qbr Qi:: i, i:= r,t,î,a'd

then we successively apply the transforms _operated to ó, taking into accout the eventual transpositio's.

The algorithm is as follows:

for/c:ltom-ldo

begi

ift[k]<>k

then

_{at the Æ_th step the trarrsposition}

begin

_{(k, tlkl) has been applied to

p}

auxí

::

^q[k];

qtkJ:= q[tLk]);

for the Chebyshev method.

It ^can be easily seen that we have

E(Q

^>

E(N

for n > 2, i.e. the Chebyshev method is lnore efficient than the Newton method.

5. NUMERTCAL EXAI\{PLE Consider the real matrix

A_

I I I

I I I I I I

1

I I I

1

which has the followiug eigenvalues and eigenvectors Lt,z3

= 2, rl =

^(1,

1,0,0), xz

₌ (1,0, l,

o),

x3

L4

= -2,

^l;¿

^{= (1,- 1,-} l,- l)

=

^(1,0,0,

l)

arrd

56

:::ri RE\¡UE D'ANALYSE NUI\{ÉRIQUD ET DE THÉOruE DE L'APPROXIN'IATION

Tomc 25, ^Nos 1-2, 1996, PP.57-61 Taking the

initial

value r0

-

^(1, -1.5,

-Z;

-1.5,

l),

and applying the two

methods we obtain the following results:

Nervûon method

ON THE APPROXIMATION BY FAVARD-SZASZ

TYPE OPERATORS

-1.0000000000 -1.6000000000 -20500000000 -2.4006097561 -2.0000000929 -2.0000000000 -t.5000000000

-0.90000000000 -1.0125000000 -t.0001524390 -10000000232 -1.0000000000 x!

-2.0000000000 -080000000000

-r.0250000000 -1.00030487S0 -1.0000000465 -t0000000000 -1.5000000000

-0.9000000000 -1.0t25000000 -t.0001524390 -1.0000000232 -1.0000000000

ALEXÄNDRA CIUPA (Clu¡-Napoca)

Chebyshev method

hr 1969, A. Jakimovski ^arid D. Leviatan [4] introduced ^a Favard-Szasz fype operator,

by

^{ureans of Appell} polynomials, One considers

S(") = Lo,,"'

^arr

arralytic function in ^{the rJisk}

lrl. ^n,

^.R

^>

^1, where 8(1)

+

0 ' It is

m,ll"

that the Appeil polynornials ¿i.(¡)

,

^{lc >-}

0

^can be defined by

(1)

^g(rr)e,,*

=tou|)uo

k=0

To a ^functi

on f

^:[0,

*) -+

-R one associates the Jakirnovski-Leviatarr ^operator

(z) (p,f)(*):äËrrOør(i)

B.

Wood

[6]

^has provetl that the ^operator

{,

^{is positive}

ⁱⁿ

^[0,oo)

^if

^and

^o¡ly if

!+ ,

^0, It

=

^{0, l, . . . The case g(z)}

:l

yielcls the classical operators of Favard-Szasz

8(l)

(s,,/Xr) ₌ e-'r'ì'; l"t)o ,¡{"¡

?^ ^{kt '\n )}

I¡

_[4] A. Jakiuroyski and D. Leviatan have obtained sever¿l approximafion propefiies of these operators. Let us mentiou some of thesc.

We

will

^{de¡ote by,E the class} of functious of exponential ^type, rvhicli have tlie properfy rhat

l/(/)l ^<

^eAt

^,¡or

^{each t. > 0 antl sonre finite}

^numberl.

^{Their basic}

tlreorenr can be state<l as follows:

If .f

^{e C[0, æ)}

î E ^then

,t1n (,9,/)(-u) =

/(t)

_'

l.he convergence behtg

tnifonn

^{ín each} conlPacl l0, ol.

xs=1Y -1.0000000000 -1.8880000000 -1.9998000075 -2.0000000000 -1.5000000000

-0.97200000000 -0.999950001s9

=1,0000000000 .t!

-2.0000000000 -0.94400000000 -0.99990000377 -1.0000000000

x2

-1.5000000000 -0.97200000000 -0.99995000189 -1.0000000000 x)

1.0 1.0 1.0 1.0

tt 0 I 2 3

REFERENC ES

l. Anselone,

M

_rye S?lutlon of Characteristic Value-l/ector Problems by Newton

Method,

(1968),3345,

2' Ciarlet'

P.G.

l'analyse numërique matricíelle el ò l'optímisafioz, Mason, paris

Milan

B

1990.

3. chatelin,_F., Yaleurs propres de matríces. Masou, paris Milan Ba¡celone Mexico,lggg.

4. Collatz

L,

Funclionalgnølysis und NunterÍsche Mathematik, Berlin-Göttingen-Heidelberg, Springer-Verlag, I 964.

5' Kartîçov, V. S., Iubno, F.L., O nekotoryh h ModiJ tsíøh Meroda Níutona tllea Resenia Nelineinoi spelaralnoí zadací, J. vîcisl. matem. i matem. fiz. (33) 9 (1973),1403.1409.

6. Pãvãloiu, 1., Sur dqs pro_ydés iteraifs ò un ordre étevé'dá coivergánce, Matbematica (Cluj) ₁₂ (35) 2 (1970), 309-324.

7. Traub, F , I ', Iterølive Methods

þr

^the Solutíon ofEquations, Prentice-Hall Inc., Englewood Clifß, N. J.,1964.

Received 07.03.1996 Romønian Academy

" T. P opovìciu " In stitute oJ Nu merical _{Analys is} P.O. Box 68 Cluj-Napoca

I

RO_3400 Ronvnía