The rate of convergence of projection methods in the eigenvalue problem for equations of special form

U.S.S.R. Comput.Maths.Math.Phys.,Vo1.25,No.4,pp.l-7,198s Printed in Great Britain

0041-5553/05 $10.00+0.00 Pergamon Journals Ltd.

THE RATE OF CONVERGENCE OF PROJECTION METHODS IN THE EIGENVALUC PROBLEM FOR EQUATIONS OF SPECIAL FORM*

A.G. ZARUBIN The equations Au=pR~+p-~Ku and Arr=pRn+p'Ku are examined from the point of view of finding estimates of the rate of convergence of the approximate eigenvalues p,, obtained using the method of moments and the BubnovGalerkin method. The eigenvalues

of the equations Au=@u+p-‘Ku,

(1)

Au=pRu+$Ku require

to be found

for a number of important

(2)

problems

(see, e.g., /l-4/).

1. The method of moments for Eq.(l). We shall use L(H,,H) to denote the space of the linear bounded operators acting from the separable complex Hilbert space H, into the separable complex Hilbert space H. We assume that H, is compactly and densely imbedded in H. Consider Eq.(l) , where A,R,K=L(H,,H). Suppose {u,),- is a coordinate A-total sequence in H (see, e.g., /5/J and v,=Aui. We denote the linear envelope of the elements v,,...,v,, by H”, and the linear envelope of the elements Suppose P, is an orthoprojector in H on to the subspace U1,.. .I J-&l from H, by H,“. H”. The method of moments for Eq.(l) consists of determining the approximate numerical values Ir., from the equations UEH,“.

Au=~P,Ru+~-‘P.Ku, Consider

(1.1)

first the equation

dll+%!?l=mlh (1.2) is a coordinate .&full sequence in a. We denote 26). Suppose (Q),” of the elements cp,....,‘pmby ar",and the linear envelope of the elements %~...do~si.-e:h;;,e e;u;fl;n - by .W. Suppose Fp. is an orthoprojector in % on to &?.

where &, 9E,%L(%‘,, the linear envelope

(1.3) assume that .& is positive We shall in a (see /6/), and %! and .?i!aresubordinate following into&with order s, OGsCl, and r, 0461, i.e. for any element 0 from a, the equalities hold: II .% Ilx
n33l II*

< Cl u64~ IId 110l/T>

Since &is positive, where c and c, do not depend on o. Suppose &‘, (.G+9t!)-1~L(%,Gtf,). Since %, is the fractional steps .S@ and tie',0<~<1 are determined (see, e.g., /6/). A-' also compactly imbedded in a, then &-' is fully continuous in a; the fractional steps have the property of full continuity. We make the substitution ~=.!$~+'?l in Eq.(1.2), where &$~-"-'o+$i&-~-"o= Then a=max(s,r), and e is an arbitrary fairly small positive number. ?,.%!.?8-"-'0. The operator Se'-"-'-!-%!&-"-acts from D(.&'-*-') into 1. Since (Se+%!-' exists, acts from sinto D(&'-a-") and is inverse to the operator the operator .&-'+~+'(E+~-')-' &+Z-*+ge&-,-e Finally the last equation is transformed to the form ,,~-L+"+'(E+~-')-'~~-'-~"=ITO. Suppose

3-. is an orthoprojector

Lemma

1.

Suppose &is

in 8

positive

on to

&"'-.@+g&%,n.

in 1and

g(n) -+o, n-3-m. II~-'(E--B")Ilx,xgg(n), Suppose 5? and the closed operator x are subordinate to &with order Then respectively. Suppose (&+9)-l, .5d-‘EL(Z,J6,).

s,OGs
IIT--~*l)ll
a==max (s,r), T,=~-‘+“+‘(E+~~~-‘)-‘~“IXP~-=-‘, on R.

(1.4) T,O~T<~ (1.5a)

( and the constants

eZ and

(1.5b) c, dc not

for fairly large n follows frctm of the operator (E-!-~~~-')-‘ of 5? and 2 to the operator d (see /7/). And what is more. where c, does not depend on R. Reasoning by analogy with the foregoing, II(E+9”9w-‘)-‘llGc,, we transform Eq_(1.3) into the equation *Zh.vychisl.Mat.mat.Piz.,25,7,963-972,1985 1 Proof.

The existence

(1.4) and the subordination

2

(1.6) where

The boundedness of x&-"-' in .%follows from /6, Theorem 14.5/. The full continuity of in aiahence also follows (from the full continuity of the operator &-'+n+'. operators T and T, By definition, %'=.@+'%,". element from 1,". Then

We shall show that

%n=&g-'+a+'%'.

Suppose

E

is an arbitrary

f=&. j=l Hence Jz+eE=J-l+a+c

(2

ajqj),i.e. &+eSE

AR"+""P.

j=1

Similarly, &-'+a+'~ESka+*&%~n for any 5 from 1". identity is correct: (E-P.).@-'+"+'P"rl=o

By definition

We shall set @,=T.a,T

estimates

and

F,=T-9.T.

9,

and the following

VI@%.

The following

(1.7) follow from (1.7):

ilQ)ntlII-II(E-~~)~-l+a+e(E+~-‘)-’~~-=-’rl(J= 11 (E-~),)~-‘+~+‘(E-9,) @$-“““(E-9”)

From the moment

inequality

and

(/3+9i?s6-‘)-‘IXP&‘-‘q(l~ (E+S!sd-‘)-‘XL&=-‘qll.

(1.4) we obtain

(I&-'+'=+* (E-9,) n~~G,\~~-‘(E-9’,) The operators (E-k%%-')-' and (1.9). Obviously,

qi/‘-“-‘b#=+‘~

A!&$-"-' are bounded;

II(T-Td 4l~ll~-

Since

(1.8)

‘+a+c(E+Y”s!Sr’)

(see /7/J (Ef~,~-‘)-‘=E-~n(E+W~-‘~P,)-‘~~-’

c&=-e(n)

therefore

-’ @-9”)9w-’

and

(1.9)

ihi\.

(1.5a) follows

from (1.8) and

(1.10)

(E+

~~(E+~-'~,)-'(~~c~,

l,n
estimate (1.11)

c,g’-* (n) IIdis proved similarly. Therefore (1.5b) follows from (1.9). The estimate Jzn~c,og'-a-'(n)lloll follows from (l.lO), (1.11) and (1.5a). cr. is a set of eigenvalues of the equations &n+L%n=L%q, Suppose and ar is a set of characteristic numbers of the operator T. Further, we shall set ~,,={rl~~,:(~+.4e-h~)tl=O) and A%','=(oG%: (E-hoT)o)=O. ‘+a+‘(l(((E+~-‘)-‘Illl~~-“-‘oll~

Lemma

2.

If the conditions

of Lemma

c&‘=-‘(n)l~u\\.

1 hold, then

a.=ar

and

~a~c~~,=~~',~-a-'~~'=~,,.

Proof. _If &~a., then hp~by. Suppose TJ&%~,, then EO=.$+'no&&'. Suppose hpEoT and the corresponding eigenelement is Lo. We shall show that h&a. and Q%%~,. It follows from /8, 9/ and the inequalities (1.5) that ho is the limit of the characteristic numbers h,, But ~0n~hOn~-"afr(E+4)~~~-')-'~,~~-"-e~0n. where o,,=z@P+*(pj. Then (.@-a--I+ of Eqs.cl.6). &g?-=-*&” = .qm. ~"W~-"-')o,,~h,,~,X~-Q-'o,.. We shall put )lon=~~~@,. Then Therefore tin,, + ,"~~-"-c~-'+"+~(~ll~~) ~,.,~"~~-a-z~-'+a+e(~~o"). Using the properties of the fractional steps of the operators defined on all k% we obtain serlo,,+~"Wtlo""h,,~"~~~"i.e. ho, is an eigenvalue of Eq.(1.3), and r)on=&-"-'~on is an eigenelement. Suppose won is normalized in L%.It then follows from /$, 9/ and inequalities (1.5) that we can separate the subsequence Oon, converging in %, from aon. From the form of the operator T and the definition of @on we have A l-LTd(Oonj-Ooni)'~Pni~J1-a-eOoni-~nj~~-a-eoo"j+ Hence

Q 11 (Pq - s”Tlj)zU4-QdQni!I + II4 1%-E(o”nj- OonJ11 11 S~j$!AR-a-C (Q*j - b&J 11 + 1ho*j- ‘OniI IISIljXAR-a-‘ooIZj II + + I bn{(ll~~x~~-e(oolIj-~j~ll* I b,l II(9~i-.%jm”u -p-eWnjll Since Oemj are bounded in the set and the operators %$ (see /6, Theorem 16.3/J, the sets XA-(r-Lo~,j and II (Sni- sSnj) .ZJKa-eOOni11-+O. ll(Pni- tPnj) XARd-eo,jJl-O

5?.&-"-',.%d-a-' are fully continuous in zA-a-e oonj are compact in 1. Consequently,

as n,,nj-m.

Therefore

(since

i,,j--).""._ 0

3

and

I/~o,,-~o,,~(/-o) the sequence

tlonj - r)Onillx,
&=-'O#",

is a fundamental in .%. But 5&-"-co0.=~0.; therethe sequence nanj is 11 and this means no is its limiting element, then

II1

(?Ollj-

PhiJl( =cloil~1-ar-+bni-02000nj)

II A% + mo- Lmo II< II64 + 54- aox) ho - ?onj) II + II CA + 57 - bx) llOVljII Q II A + s - ‘OX II II 90 - rlOrtjIIX,+ )I~"lljXflo7Lj-~OX?oTtj I(<'cl1 IIrlO- rlOnj(IX,+ / bnjl‘0 111 XllO*j Letting ~l,+m, we obtain 90G%!,~O"O.. It is known (see /El, 9/J, that we can derive converging subsequence has the eigenelement o0 show that ~-=-'o~=n~. We have

Hence

/I.

a converging subsequence from won and each We shall from ZO, using its own limit.

II “P’%o - 00 II < /IAa+e(rlo - rlonj) II + IIAa+evhj- oonjII+ IIO”- %lj IId ‘12II1 (rl0- 'lO*j)Ila+’II?a- ‘lhj II’*-” + I/00- %j II’ ~~fti~+'n~, i.e. Q=&5&“-‘0,. The lemma is proved.

We return to Eq.Cl.2). Following /lo/, we shall introduce the concept of adjoint elements forthe linear mean L(h)=&+9?-hlkP.Suppose i& is an eigenelement which corresponds to the eigenvalue ho. The element cP is called an adjoint element of order p to the eigenelement co, if bP is obtained by solving the chain of equations @,,+Se~,-&.X'~,=Z'&,, t=l, 2, .... p. If m is the maximum order of elements adjoint to co, then the number m-t1 is called a multiplicity of the eigenele,nent cO. From the general theory of completely continuous linear operators we have. Lemma 3. If the conditions of Lemma 1 hold and the kernel of the operator only of zero, the multiplicity of each eigenelement f;O from a,, is finite.

.%!consists

Note. Under the conditions of Lemma 3 the multiplicity of the eigenelement E. from &, does not exceed the range of the characteristic number lo and 02. Since T is completely continuous, then, by virtue of Lemma 2, the subspace & is finitedimensional. Suppose dim%&=v, and q',...,q’ is the basis in a,,. It follows from Lemma 3 that each n' has a finite range. Suppose o*, q,',...,q!,,*, t=l, 2,..., v is a chain of adjoint elements. This system is canonical /lo/ and, as mentioned in /lo/, the numbers m,,...,m. do not depend on choosing a canonical system. m-mar We shall call the number mi-i, where aol. (ml,..., mV), a multiplicity of the subspace Theorem 1. Suppose & is a positive operator in aand l~sk-'(E-3',,)Ilx+&g(n), g(s)+o, n-c=, and 5? and the closed operator jkpare subordinate to &with order s, OGs
The validity of Statements 1) and 2) follows from Lemmas 1 and 2 and the results Proof. in /B, 9/. We shall proceed to Statement 31, It follows from the definition of the multiplicity zQ,that n0&f!?0,is found, forwhich the maximum length of the chain of adjoint elements By virtue of Lemma 2, the number h@eT; suppose its rank for the operator T equals m+i. By determining the root subspace equals 1. According to the note, m+iGl. Suppose m+i-d. we find E,G%,' and &e%L-', We shall set %0'

EL_,= -

(-

Lo)~ TP-‘E1 -ho&.

Therefore O= (E-hoT)‘~,=(E--h,T)6i-Ter-, and &,=8:-r. Similarly, we can construct the element

and ~+-2G%~-a etc. for which (E-h,Z')&l=TE,_, constructed, for which the equations

, until the chain of elements

j=l, 2 ,..., (E-M)I,-Rx-,. L According to the construction, hold, and the relations &&%,' are correct. therefore c,G% are found, such that E,=&-'+a+'&. By virtue of (1.12),

~-I”+~S,_h”~-I~+~(~+~~-l)-‘~~-“-”~-’+”+’f,e~-~+~+~(~+~~-‘)-L~~-’“-‘~-‘+D+~~,, whence b,-h,(E+~-‘)-‘X~-‘b,-(E+~-‘)-‘~~--(~,_, and, consequently, j=l. 2.. 1. (~+se)~-'S,--h~~-'b,=~~-151_,,

El,...,&

is (1.12)

E@(5d’-a-e). a.?

, 1 1,> ‘2:

4

J$-~~~G%'. Hence and @G&’ then &-a-'EOG%~, by virtue of Lemma 2 and, therefore, from (1.13) it follows that Lola. and the multiplicity zO, is greater than m+i. Thus m+l=l. Then (see, e.g., /9, Theorem 18.4/J the estimate In~"-hol~cll(ll~'.ll+llF.ll)"'"+" holds for the eigenvalues ho, and ho, by virtue of which Statement 3 follows from (1.5). The lemma is proved. Besides Eqs.(l) and (1.1) we shall consider the "vector" equations

Since

JQ+serl=hxpn,

(1.14)

5Q"+B.sEq.=i~.x~",

(1.15)

where

(1.16)

In &'(like 8,) we shall introduce We shall put %=HXH, &‘,=HiXH,, .%‘s-H”XH”, i%‘,n~H,“XH,“. the scalar product (E,$x=(z, u)s+(y, u)=, &=(z, y)',q=(u, u)< Then 9. is an orthoprojector in 3 on to an. o, is a set of eigenvalues of Eq.(l) and 15 is a set of eigenvalues Suppose of Eq.(1.14). Lemma

The number

4.

h belongs

to

oz when and only when

h=p+p-'

and CL is from

cr.

Suppose hmol and g=(r, y)' is the corresponding eigenelement. We take p such Proof. that h-_~l+p-'. We shall show that ~,~o,. For h and E we have the identities As+(R-K)y= Hence we obtain Az=Ky+p-‘Rr+R(p.z-y), A~-‘yaKy+y-‘Rx+IL-‘K(IL-‘~-z). If AAx, Ay+ (K-R)x-dKy. Suppose CL=& and u is an eigenwe set u=x--~-'y, then (1) holds identically, i.e. pm&. Eq.(l) holds identically for the pair element. (K u). Hence An+ (R-K) p-k= (p+y-‘) Ru, Ap-‘u+ (K-R)u-(p+p-‘)K&u. pLfp-'=A If we set and t=(u, k(-'u)' in these identities, we obtain the identity &~-l-5?~-A.J!~. The lemma is proved. When proving the lemma, the connection between the eigenelements of Eqs.(l) and (1.14) is established incidentally. Similar statements also hold for Eqs.(l.l) and (1.15). Theorem

2.

Suppose A is positive

and

IIA-'(I-P.)IlQ(n), and the closed operators R and K are subordinate respectively. Suppose the equation

gibd-+o, to

A

n*m,

with the order

(1.17) s,OGs
[Z+A-‘(R-K)A-‘(R-K)]u=O

and

r, O
(1.18)

with the operator A-‘(R-KjA-‘(R-K), fully continuous in H,, only has a trivial solution. Suppose A-‘=L(H, H,). Then the following statements hold: 1) each eigenvalue p0 of Eq.(l) is a limit of the eigenvalues po,,of Eqs.Cl.1) and, conversely, each limit point p0 of the sequence of eigenvalues lo. of Eqs.(l.l) is an eigenvalue of Eq.(l); 2) from each sequence of normalized eigenelements of Eqs.(l.l), corresponding to the eigenvalues pa,,pLpn*pO,n*-, we can derive a subsequence convergent in H,, and each convergent subsequence has the eigenelement of Eq.(l), corresponding to the eigenvalue pO, by its own limit; 3) if the kernels of the operators R and K consists only of zero, pLois an eigenvalue lo,, equal to m+i, of Eq. (1) and 50-~0+~0-' is an eigenvalue of Eq.(l) with multiplicity where the constant C,S does not depend on n, and P~"+-)L~ as n-+m, then Ipan- paI
cc-max Is,7). It follows from the conditions of the theorem that the operator .@,which is Proof. defined by Eq.(1.16), is positive in %, and the inequality (1.4) with g(n)=g,(n) holds for The operator 9 and the closed operator 2 are subordinate to S$ with order a-max (S,7). iknce A-‘=L(H, H,), then &~~‘EL(%, a,). We shall now show that the operator (&-%)-' exists and that it belongs to L(a,%,). The equation .@+$?e=h is equivalent to the system Ax+(R-K)y=f, Ay-(R-K)x=q. Since A is reversible, we obtain the equation s+A-‘(R-K)A-‘(R-K)z=A-‘f from this system which, by virtueof (1.18) andthe Fredholm alternatives has a unique solution I from H,. Thus, the conditions of Theorem 1 hold for Eqs.Cl.16) and (1.17). Therefore the statements of Theorem 2 follow from Lemma 4. The theorem is proved.

2. The method Consider moments:

of moments

for Eq.(2).

Eq. (2) and the following

approximate

Au=pP,Ru+pZP,Ku, Besides

(2) and (2.1) we shall consider .!++W,rl=hX,rl,

equations,

constructed

using the method

u=H,“.

of

(2.1)

the "vector" equations sen.+~"%q"=w,,~,n",

rl"fX,",

(2.2)

where

by

We denote the set of eigenvalues of Eq.(2) by a, and the set of eigenvalues 06. The following is proved in a similar way to Lemma 4:

of Eq.(2.2)

5

when

Len?na 5. Suppose the kernel of the operator and only when h=P', and P=oS. From Theorem 1 and Lemma 5 we have

A consists

onlv of zero.

The number

>.E(I.

Theorem 3. Suppose A is positive and jlA-‘(I-P,)Il
3.

The

method for Eq.(l).

Bubnov-Galerkin

Suppose the positive-denfinite selfadjoint operator B, similar to A, is found (see /g/J, whilst the operator B-' is fully continuous. V,,..., V”,... are eigenvalues of the Suppose operator B, O
(3.2)

+%4,".

Theorem 4. Suppose the similar operators &and A%'form an acute angle in a (see /11/j. Suppose J and the closed operator X are subordinate to&with order s, O&
of the operator

&.

We shall (3.3)

Then we shall transform Eq.(3.2), using Lemma where 2' is completely continuous. to the form o,=~~-"=+~(~~-')~(~"'~)-'[E+~,'~(~,'~)-']-'X~,'X~-"-'o,=~T"(I~,,

1 from /12/,

T, is completely continuous. Here andin (3.3) we have o,=.&=+c'n.,o=s$"+'n. We shall denote the orthoprojector in % onto %'s"=.@+'%sn by n,'. Then #+*rI"'E=O for all E from A'. We shall set F.=T,-n.'?', (D,==?'-n,'T. Obviously,

(3.4)

where

1a&,511 = u (E -

ll(.E- i=I,;),-*+a+‘(E + fl’n-l)-l X”u-y fi,,‘) c,4a+c(E -

!I“p”p-” Let us examine

!III04l

(3.5)

I<

ll(E + flL”n_‘)_’ I!I!X”4-a-L llPnza+Ccl*llSII< c1o,;l:;=* II 5 II.

the difference

T-T,.

The following

inequality

holds:

(E+~-‘)-‘(E-n”‘)X~-“-‘ll+

IIT-T_llQ?i-‘+“+’

ild-i+“+*.d[

11 =

n,l) 1-l (E + .%‘&‘)-’ x”K=-‘E

(E-n"')

(3.6)

(~+~)-‘-(n,‘~+n,‘~)-*]n,‘~~-~-‘~l~

~~~-““+‘(E+~~-‘)-~(~-~,‘)~~-=-~~~+ j;~&‘+~(d+Se) Since

-* (l-I,‘-@

(.$+W)

(e-t~-')-'=E-(E+Se~-')-'~-'

(n,‘~+n,‘se)-‘w~-a-e,,=J~~+J,“.

then

-l+a+eghz-e II ,/xA-z-c ,Ip;l$+c+ Jan< II“4 Il ll (Ef s&A-y--1/I11 .Zh?- (e- n,y XA-“-e 11. II“6 l+a+e The operators 3 and .X'are subordinate to d, therefore Ijl~-‘tlllc~*~II~-‘llll’-‘lltlla, It follows

from the inequalities Jsn
Let us now consider

the operator

(3.6) and

(3.7)

IIx~-‘~Il~c,~ll~-‘~ll’-~ll~ll~~.

(3.7) that (3.8)

+ Cea!I .A!-% J/r-' 11 X.A*-' Iypit:,
li(~+se)(n"'~+n,'a)-lli=

acting from

%a"

into

%'.

We have (3.9)

6 (E IIJ- l+OAc

.!Z,d- (E - ll,l) (“4 + 9) x rI,‘X.d-a-e 11< 11“4-1W+e~31-a-E 1111 x”4-p+a+c 1111(E + zK’)-’ 11 cm /IA-% Iv- x

+ _!LPp

+ rI”l.%y

(rI,‘J

-1+a+e

+

~dh+1

II

pi::’ 1 XJz-From

(3.51,

(3.61,

(3.8) and

11<

IIx

c*8p;~a+e.

(3.10) it follows

that (3.11)

UF,I]rr-.n< cs$nz=.

The statements of Theorem 4 follow from inequalities (3.5) and (3.11) : it is sufficient to use reasoning similar to the proof of Theorem 1 given above. Using Eq.(l) we shall conWe shall proceed to the Bubnov-Galerkin method for Eq.(l). struct the "vector" equations

(3.13) Theorem 5. Suppose the related operators A and B make an acute angle and the closed Suppose the operators R and K are subordinate to A with order s,O
Note that for (3.12) the conditions

The rest of the

argumentisthe

of Therem

same as when proving

4 hold if we set

Theorem

2.

4. The Bubnov-Galerkin method for Eq.(2). Retaining

the notation

of Sect.3,

let us consider

Eqs. (2)

and

n,Au.=~n.Ru,+~*n,Ku,. Using

Eqs. (2)

and

(4.1) we construct

(4.1)

the equations n.'~~~+n~'4e,ll"=h~"'~,~".

dn+sE,11=w*I,

(4.2)

Theorem 6. Suppose the similar operators A and B make an acute angle and the closed A-‘=L(H, operators R and K are subordinate to A with order s,Ods
0 B

I I

We shall show that the conditions

as 9.

Then the similar

operators

of Theorem

in a-

.?# and

4 hold

for Eqs.(4.2).

We shall use

9 - form an acute angle.

Consider operator Se,. It is easy to see that )19e,rll/CC,,lj~~llrllf111'-'. It follows from the form of operator I]Ep, that 2, is closedin % and ~~X,~ll~cs,(ll~~ll~ll~~~'-7+~~l'-'+~~~~i~~~'-"~. Hence ad from the fact that the operator .& is positive definite we obtain ~~3ip,~jj~~~;I~~~~~j"ilrlII'-~, i.e. X, is subordinate to &with order 8,). a=max(s,z). It is easy to see that .s&‘EL(%, .@+%!,E=h. 1t It remains to be verified that (.&+97)-‘~L(%‘,X,). Consider the equation is equivalent to the system Ax-Ry=f, Ay=cp. Since ‘p=H and A-‘=L(H,H,), the second equation R is subordinate to A, therefore RA-‘+I. has the solution y=A-‘9 from H,. The operator .&E+%!,E=h has the unique solution (A-‘RA-‘cpf Then for any element h from %,the equation A-If, A-‘~)‘&%,. Thus, all the conditions of Theorem 4 hold and we are left to cite Lemma 5, The which establishes the connection between the eigenvalues of Eqs.(2), (4.11 and (4.2). theorem is proved. REFERENCES 1. KREIN S.G., The oscillations of a viscous fluid in a vessel. Dokl. AN SSSR, 159, 2, 262265, 1964. 2. KREIN S.G. and LAPTEV G.I., The problem of the motion of a viscous fluid in an open vessel. 2, 1, 40-50, 1968. Funkts. analiz i ego prilozheniya, of weightlessness. Moscow: Nauka, 1976. 3. BABSKII V.G. et al., The hydrodynamics 4. KOLLATS L., Problems in eigenvalues. Moscow: Nauka, 1968. 5. MIKHLIN S.G., Variational methods in mathematical physics. Moscow: Nauka, 1970. 6. KRASNOSEL'SKII M.A. et al., Integral operators in the spaces of summed fractions. Moscow: Nauka, 1966. 7. ZARUBIN A.G., The rate of convergence of projection methods for linear equations. Zh. vychisl. Mat. mat. Fiz., 19, 4, 1048-1053, 1979. 8. POL'SKII N.I., The convergence of some approximate methods of analysis. Ukr. matem. zh., 7, 1, 56-70, 1955. 9. KRASNOSEL'SKII M.A. et al., Approximate solutions of operator equatlcns. Moscow: Nauka, 1969. 10. KELDYSH M.V., The completeness of eigenfunctions or some classes oi non-adjoint operarors. Uspekhi matem. Nauk, 16, 4, 15-41, 1971.

7

11. SOBOLEVSKII P.E., Equations with operators making an acute angle. Dokl. AN SSSR, 116, 5, 754-757, 1957. 12. ZARUBIN A.G., The rate of convergence of pro3ection methods in the eigenvalue problem. Zh.vychisl. Mat. mat. Fiz., 22, 6, 1308-1315, 1982. Translated

U.S.S.R. Comput.l4aths.Math.Phys.,Vo1.25,No.4,pp.7-13,1985 Printed in Great Britain

by H.Z.

OO41-5553/85 $lO.GO+O.OO Pergamon Journals Ltd.

THE EFFECT OF ROUNDING ERRORS IN ITERATIONALPROCESSES*

E.I. FILIPPOVICH The effect of rounding errors on the accuracy of the result in iterational processes with an ultralinear convergence is examined. The well-known result for processes with linear convergence follows from the estimates obtained. It is shown that in the case of ultralinear convergence the overall effect of rounding errors on all the steps of the iterations is equivalent to the effect on several of the last steps.

Introduction. z&=X is defined in the metric space Suppose an iterational process for the quantities X with the distance p(s, 5') for the elements 2, z'=X, and it is established that the rate of convergence of the approximation Q+, to the exact value z' is given by the relation .Y (1) z')Gqp(t,,2:) 1 P("r+s, where g and Y are certain constants which do not depend on k. When v=l, q>I it is called ultralinear /l/. Because of rounding errors, the approximate values z",,,,rather than the accurate values &+,, are obtained in the calculations. Suppose at step k the value 2", is obtained and Q+, is the accurate value which would be obtained from zr if there were no rounding errors, while I,+1 is the correspondingvaluedistorted by the rounding errors. We shall further assume that for all k>l (2) p cc, 4 a the rounding errors at all steps have an upper limit set by the quantity U. Under these i.e. conditions the following questions are of practical importance: I. What is the asymptotic form of the behaviour of the quantities p(Z,,z') as k-m? II. Since the value 5' remains unknown, and only the values s", are known, can we obtain estimates for the quantity p(.z"*,z') using the values &) under conditions (1) and (2)? P(x",+,, III. Usually, when implementing the iterational process, it is required that the final approximation P, for Y should satisfy the relation

andtheprocess

stops

P(l*,r')Q if for some fairly large k P(f,,i)l+,)Gb

(3)

(4)

where E and 6 are certain small numbers determining the accuracy with which t* can be found. The quantities E and 6 must obviously be consistent with each other and with the quantity 0. It is clear, for example, that it is absurd to seek the solution x* with the accuracy or to assume 6<0 since in the sphere (s:p(z,z*)
as

p(t,, 2’) < o/(1---4) + qkf("o*5*)*

(5)

lim sup p(i'k, Z')
(8)

k--m

The corresponding estimates for ultralinear processes are established below, and estimates (5) and (6) follow from the general theory as a special case. The whole complex of questions The value Y is usually relatively easily obtained in the theory of the I-IV is examined. specific iterational method, whereas it is often difficult to establish an estimate of the 25,7,973-902,1985 *Zh.vychisl.Mat.mat.Plz.,