185

8. LAVRENT'EV M.M., ROMANOV V.G. and SHISHATSKII S.P., Ill-posed problems of mathematical physics and analysis (Nekorrektnye zadachi matematicheskoi fiziki i analiza), Nauka,

Moscow, 1980. 9. STECHKIN S.B. and SUBBOTIN YU.N., Splines in computational analysis (Splainy v vychislitel'noi

matematikel, Nauka, Moscow, 1976. 10. PATSKO N.L., Approximation by splines in a segment, Matem. zametki, 16, No.3, 491-500, 1974. 11. PAKHNUTOV I.A., The stability of a spline approximation andthe re-establishment of mesh

functions, Matem. zametki, 16, No.4, 537-544, 1974.

12. SUBBOTIN YU.N., On the connection between finite differences a7d the corresponding derivatives, Tr. Matem. in-ta AN SSSR, Moscow, Vo1.78, 24-42. 1965.

13. SUBBOTIN YU.N. , The diameter of class W'L in L(0.2~) and the spline function approxi- mation,Matem. zametki, 7,No.l, 43-52, 1970.

14. CANNON J.R., and DUCHATEAU P., Approximating the solution to YTe Cauchy problem for Laplace's equation. SIAM J. Numer. Analys., Vo1.14, No.3, p.473-483, 1977.

Translated by W.C.

U.S.S.R. Comput.Maths.Math.Phys., vo1.25,No.l,pp.185-190,1985 0041-5553/85 $10.00+0.00 Printed in Great Britain Pergamon Journals Ltd.

AN APPROACH TO THE QUESTION OF CONSTRUCTING EFFECTIVE RECOGNITION ALGORITHMS*

A.A. ALEKSANYAN and YU.1. ZHURAVLEV

A general approach to the question of the effective realization of recognition algorithms for calculating estimates is proposed. The possibility of such a realization is closely connected with the 'symmetry' of supporting sets. The concept of a rank and of a symnetric rank, which describes the 'simplicity' and 'symmetry' of a system of supporting sets is introduced. The properties and behaviour of a system's ranks when set-theoretic operations are used and an isometric permutation group is on operation are studied.

1. Introduction. Consider the following model of recognition and classificaticn algorithms

estimates. The model is described in detail in /l/. The objects from a certain set MsM,X...XM., where MI is t_?e domain of

metric space with metric p,),i=l,2,....n, are arranged in 1 classes. The estimates ri(S) of the occurrence of the object S in the j-th class

by using training information, and the object is classified by using threshold

for calculating

features (a

are constructed functions.

Each specific algorithm A in the model is determined by the choice of system of supporting sets Q,,the proximity function B(o, S, S'), the weights of objects from the training informa-

tion, the weights of features and the decision rule. The system of supporting sets 0, is a certain set of subsets from (1,2,...,n}. We shall

identify the element 52 from Q4 with its characteristic vector o=(w,,..., o.),in which a,=1 for j=n, and 0 for j&P.

Let ,s=(a,,..., a.). S’=(b,...., 6,) be the objects from M, o++Q={i,,..., i,.}; the numbers

Q,. qtao and vector (E,,..., e.) where E,>O, i=l, 2,..., n, are fixed. Then B(o, S. S')=i if in

the system &,(a,,, bi,)Get,, , Pz,(G~, h,) GEM!: not less than q1 and not more than qI inequalities

are satisfied. Otherwise B(o, S, S')=O. We assign to each abject S of the training information a numter r(S), and to each feature

1=(1, 2,..., n) the weight p&O. The weight of the supporting set o++Q=(i,...., i*) is

defined as p(o) =pi,+...+p,,.

Let us denote by IV, the set of ob3ects of the training information which belong to the j-th class, i=l, 2,..., I. Then the estimate of the occurrence of :he object S in the class with number j is computed from the formula

(1.1)

where N is the normalizing multiplier, and 1.1 is the cardinal number of the set.

Obviously, the fast operation of algorithm A depends essentially on the speed of Calculat-

ing the estimates T,(S). A 'straight' calculation by formula (1.1~ is clearly ineffectIVe,

even if practically possible. In fact, only the number of terms XI the inner sum of (1.1:

*Zh.v~c,~isl.Mat.mat.Fiz., 25,2,283-2?1,1385

USSR 25 : 1-I.’

186

can have an exponential order, therefore an effective realization of the algorithm requires additional specific efforts in each particular case.

We propose below a fairly general approach to the problem of an effective calculation of the estimates I',(S), based on the following observation: it is sufficient to know how to evaluate rapidly the expression

in formula (1.1). However,

P(0)=xP, and x (S(O,~S,S')~~~)= lWQ,Q* iEQ

B(o,S, S’)=

Thereforewehave the formula

where Q,(S, s') is the number of the supporting sets REQ., containing features i for which B(O, S, S') =I.

Thus, the number of terms in (1.2) is already not greater than n. Calculating numbers

Qi(S, S’) may prove to be time-consuming. However, if the number of different values of Q,(S, S') is finite, or is small compared with n, formula (1.2) is simplified considerably, and consequently this is an effective way of computing estimates r,(s).

It follows that an effective realization of the algorithm for computing estimates depends

on the choice of a system of supporting sets.

2. The rank and the symmetric rank. Let us attempt to answer the following question: for what kind of systems of supporting

sets is the number of different values of Q*(S, s') small? We denote by (/a[/ the number of units in the binary vector c&and by a+8 the component-

wise addition in absolute values of two binary vectors a and p; then $a+$/; defines the Hamming metrics for binary vectors, and equals the number of coordinates in which these vectors can be distinguished.

Let S=(a,,..., a.), S'=(6,,. ,., b,,)~lv. We shall construct a system of inequalities, p,(a,,

~,)GE,, . . . . pn (a., ~.)GE., and compare with it the binary vector 6=(6,, ,.., 6,) as follows: &=I when and only when the inequality ~,(a,, O,)<E, is satisfied. Therefore the number of unities in 6 equals the number of satisfied inequalities, and the number of zeros the number of those

unsatisfied. Let 6,,=...=6,,=1, j,<...<im, and let the remaining coordinates be zeros. We assume A=

(il.. . . , id. Now any n-dimensional binary vector a can be represented in the form (a', a"), where a'=(a,,,..., aj,), and a' consists of the coordinates with the indices from (1, 2,..., n)\A taken in increasing order. Clearly,

B(o,S,S’)= B(o,&A,g,,gi)= (2.1)

I

1, if 1) 6’ + d I/ .< IA 1 ---qI, !/ 6' -i o1 jl 'q2. 0 otherwise.

Thus, the function U(o, 6, A, qt. qr) is defined for all the quintets (w.6, A,q,.q~), where 0 and 6 are binary n-dimensional vectors, A~(l,Z,...,n), and qI, q1 are the natural numbers.

We introduce the following notation: Q,(b, A, p,, ql) is the number of supporting sets

sl4.4 containing the feature i, for which B(o, 6, A. q,, qz)=l, and qi(S, A, pi, qz) is the number of supporting sets REnr not containing the feature i, for which B(o,h A,Q,,Qz)=~. Because Qs(6, A, qt, df@@, A, 41, qd equals the number of those o++%% for which B(o, 6, A, p,, qi)=l, the numbers of different values assumed by Q,(6. A, q,. q3) and QS(6, A, Q1, q,), are

the same. If Q* is a system of supporting sets such that the number of different values of

Q,(6, A, 4,. qz) for all (6, A, q,, q2) is small, then so is the number of different values of Q,(S, S') for all S, S'EM.

Let us classify all systems of supporting sets.

Definition 1. The system of supporting sets is called a system of rank k, if the number of different values of Q,(6, A, q,, q2) does not exceed k for all (6, A, qI, q2) and if for at least

one set of four (6, A, q,,q.) the number of values of Q,(6, A, q4,qr) is k.

We denote the rank of % by R(S2*). It appears that the rank and the 'symmetry' of a system are closely connected.

Thus, we have obtained an unambiguous correspondence o*aEQ, between the binary vectors

of length n, and the subsets from 0,. We also use the notation a., for the aggregate of these vectors. It is obvious that a, is a subset of the vertices of an n-dimensional unit

cube E" which is a set of all binary vectors of length n. Consider the following transformations of E": Od permutates the i-th and j-th co-

ordinates of all vectors in E",and n, replaces the i-th coordinate of each vector in E"

by its negation (0-i,l+O). Clearly, by the multiple application of the transformations aif we can realize any

permutation of the coordinates. Let US close the set (oi,),~_,U(nr),~, regarding the operation

of composing the transformations, and obtain a group of permutations on E". We denote it by n. such a group is referred to as a group of the isometric permutations of E". This group

covers all permutations cp of the vertices of the cube E", such that liQ+?II=II~(a)+(p(B)il.

Consider the set G(Q,,)=((PI(PE~, (F(a)@& v a62,), that is the maximum subgroup in n which leaves the set n., immobile.

Definition 2. The subgroup C(8*) is a symmetry group of set 52,. Let us describe a special family of subgroups of group n. Let N,U...UN,=(l, 2,..., a) and N,DN,=@, if i+j. We denote by Sh.( the subgroup in n

generated by the set (a,,),,,.,,, and construct the straight product S,,X...XS+. We say that

the group SN,X...XSlk is induced by the partition (N,,....N,) of length k.

Definition 3. The set 0,i.s a set with a symmetric rank equal to k, if the Symetry

group G(%) contains a subgroup induced by the partition IV,...., N, of length k, and does not contain any group induced by the partition whose length is less than k.

The synrnetric rank is denoted by SR(Q,). The following theorem expresses the connection between the rank of a set and its symmetric rank.

Theorem 1. The inequality R(n*)<4SR(n,) holds for all n,.

Proof. Let &SE" and SR(&)=k. This mean that the group G(S1,) contains a subgroup induced by a certain partition, of length k, of the set (1, 2,..., n}. Let this partition be N,U...UN,=(i, 2,..., a). We fix the quartet (6. A, Q,, Q~), and introduce the notation A'*=-(il6,'= p), 6=0, l,a=l, 2,A,ab=A"bDN,,t=l, 2,..., k. ,and we shall show that Q,(6, A, G,, qd-f?i(& A, PI, qd, if i, j=A,"@, a=l, 2, p=O, 1. Let us partition the set Q, into three non-intersecting parts %'",

%'I, 8,O': R,aA=(~/~~R,, o,=a, o,=& B(o, 6, A, q,, q2)=1),

a, F(O, I).

Obviously, Q,(6, A. q,, ql)=jSL”l+lQ.,‘Ol, Q,(6, A, q,, q,)===lQ,“~+PAO’(. Therefore, it is sufficient to

show that JO.OL(-+=IO*'oI. Let OES~.,'O then oi=l, o,=O and B(o,6, A, q,, q,)-i. We compare the vertexou(o),with (~i.e., we interchange the position of m1 and 0,. Clearly, oe(u)e&, since

i, j=N,, and therefore a,,~&,, a,,=G(52,,). Further, B(o,6.A, q,,qz)=l when and only when )I8'+

o'lidlA1 -q, and j)b2+o*(l<p, (see formula (2.1)). However, i, j=A,=O; that is 6,a=6,a-B, and the vectors o and s,,(o) differ only in the coordinates with numbers i and j. Now it becomes

clear that I16a+lo=ll=lla+a~~~)II, ae((2, 1) ; then also B(o,~(o). 6, A, q,, qt)-1, which proves the membership e,,(m)m&". the bijection between 51," and Q.,"' is constructed, and

therefore Qt(6, A, q:, d=Qj(k’A, qt. qz). At each N, the quantity Ql(6, A, q,, qs) takes not more than four values, and since there are k subsets K,,..., I\'~, this quantity takesnotmore than 4k different values. The theorem is proved.

Definition 4. A system of supporting sets is described as a A-rank k system (notation: R,(&)) if it satisfies the conditions of Definition 1 in which A-(iIS,-1).

Clearly, Ra(QA)<R(Q.,).

Corollary 1. The inequality &(8,)<2SR(%) holds. This is obvious if we observe that Q,(6,A, qt.qz) takes not more than two values on each

block N,. Now we shall show that the estimate in Theorem 1 is attainable. We construct a system

of supporting sets g, with SR(QJ=s and R(RJ=4s. Let us assume

n= i (p,+3). a-1

We partition the set {1,2,...,n) into s consecutive blocks N,,...,N., where INf1==p,+3. MY n-dimensional vector a canbe representedinthe form (cc',..., a’), where a1 is the vector composed of the coordinates of the vector a, which correspond to block N+ We define

S~~=(CC=(CZ',..., a')jamE" and IJa'lJ=l, j=l, 2,..., s).

In other words, aEc1, when and only when exactly one unity is present in the coordinate block

A', . Obviously, SN.X...XSN,~G(52_,) and SR(&)<s. Therefore by Theorem 1 we have R(R.,)G4s. We shall show that R(Q,)=4s. Hence SR(Q,J =s.

thus, we construct a vector 6, set A, and numbers q, and qt so that Q,(b, A, q,, qz) takes 4s different values.

Let 6=(6',..., 6'). We assume 6'=(6,',..., a:,+), where 6,J=6,'=0, and the remaining 62-1,

that is &=(I, 0, 0, l;..., 1). The set A is obtained by the union of all the elements of all blocks N,,..., N. with the exception of the first two elements of each block. Therefore,

‘

ILy==C (p,H)=n-2s. ,-I

Further, q,=S-2, qz=3.

(&',.... G,+d. we present the vector 6, in the form (6a', So'), where a,'-(6,', a,>), 6,'=

Then 6=(6,, 60), where 6*=(6*‘,. ., 6&‘), 60=(60’,. ., 60’). Any vector a=(a’, , a”) cm be expressed in the form (aa, ao), where aa=(aa', . . . . a,‘), c%=(a~‘, . . . . ao'), and ad=(a,', . . . .

ai+, ad=(a,‘, a,‘), j=l, 2,. . ., s. In our choice of 6, A, qi, 4% condition (2.1) will take the form

B(a. 6, A, ql, q&=1 when and only when 1)6rta~((dr~-3s+2 and llbo+a&3.

Consider block Nj. Let us number its elements in increasing order from 1 to p,+3, that

is N,=(i,,..., iPPJ). By Theorem 1, if m, lE(i,,..., b,+a), then Qm(6, A, q,, ql)=Q1c6, A, ql. %). There-

fore, Qm(6, A, ql. qd for mdi can take not more than four different values. Let us compute them: .

Q,AW,q,,q,)=l+~ (p,+i)+ z (pr+l)(p,+l)+

QJ6, A, qi, qd=1+ z (Pt+i),

Q&i A, PI, qzl =-I-+ (pr+l)+b-i)+

z 1<*

(pc+l) (pk+i) - (s-2)f-1)

Q-WLwd=~+I: (p,+i)+(s-i)+z (p,+~)(p,+~), d-i

m=(i,, , i,j+l}. t<* i+, i.L+i

Obviously, for p,<...Cp. and s>2 the quantities Q,<, Qi,, Q,,, Qm of block N, are different. What is more, if we assume p,=s'+(j-1)s. je{i, 2,.... s), then all Q,,, Q,,, Q%,, Qm. which are *e functions of the block number j, are also different for all j~(l,2....,s}. Therefore, for our choice of (6, A, 41, qa), we obtain exactly 4s values of the quantity Q,(6, A, q,, q,)? which means that R(n,j=4s. The attainability of the estimate from Corollary 1 can be established

in a similar way. we can now feel satisfied about the conciseness of the concept of the synm=tric rank

regarding supporting sets.

Theorem 2. Let the group SN,X...XS.~6 be induced by the partition {N,,..., N,) of length

k. Then a system of supporting sets 51, exists such that SR(P,)=k and SN,X...XS,,., cc&).

Proof. Let N,U...UN,=(& 2 ,..., R). N,nN,=D for all i,j~{l, 2 .I.., k). we set n,=jNil. i-l,

2,... 1 k. Then

We determine B,,=E": &=={a', . . . . a'), iEN,b...iiN,, i=l. 2,..., n.

where a'=(a,',. .a;i~')thememoc~=l Clearly, SN,X_..XS~k~G(RA).

when and o;LYs;te; SR(Q,,)<k.

satisfy ourselves that SR(&)=k. Notice that (la'(j=n,+...+n,.

Let M,u...uM~=(I,~,...,~), M,fM,=0, i, j=(1,2,....1), Z<k, and let S~,x...xSyl be a grous of rearrangements induced by the partition of length l<k _ Since ltk. there exists a trio

of indices (m, i, j) such that M,flN,#O,M,ilNIPO. i+j. We assume p~M,flN~, qdf,nN, (p#q, since N,nN,-0), and separate in SH,X...XSY( the permutation u,,. Let i>j. Consider the vertex

alEO*. Clearly, qJ=l, a,‘-0 8 and therefore in ars(a') we have 1 at the p-th place, and 0 at

the q-th place, and \\~.~(d)\\=\\a'\\, op,(az)+a'. But j/a'/\< <\\a’\\ and therefore OP,(a')e% Hence

it follows that Sw,X...XSMIGG(nA) and SR(QA)=k. The theorem 1s proved.

3. Behaviour of the ranks under the action of group n We will investigate the variation of the ranks of the system QA in its transformation

by isometric rearrangements.

We fix (6, A, qL, q2), for which the maximum number of different values Q<'=Q-(6, A, qI.9.)

and QP=Qi(6, A, q,, q,f is achieved, and construct the system of equalities

Qn,"'= , =0x.“-. cc?={% I), :._

i+(l. 2,..., n), the quantities Q," from different equalities being different. Obviously, P.e number of different values of Ql(6, A, q,, qz) equals the number of equalities 1:: 1.1 Yri c re

189

there exists among a, at least one unity. If there is the equality Q8~"*=...=Q~~a-r in (3.11, then either in this equality together with each Q_' there is also Q,'. or there is present in (3.1) equality Q,,%=...-=Qsh-, where a=1 for a=0 and a=0 for a=l. For this reason R(&) does not exceed the number of equalities, and is not less than half of the equalities

in (3.1). Letup be an isometric permutation on E", that is an element of the gro'up n. We consider

the set cp(G?,)=(cp(a)~ad2,) and construct a system of equalities (3.1) for R,, and an analogous

system for m(9,), fixing ((P(6), cc(l), Q,. q2). Because q is isometric we have B(o.6, A, q,,qz)=l

when and only when B(cp(o),m(6),~(1), ql,qz)=i. Let 'p realize only the permutation of the coordinates (without imposing negations), that

is (pES,,,l... ,“,. Then the number of equalities in systems of type (3.1) for R* and ~(a,) be-

comes identical. Only the permutation of subscripts will be realized. Therefore, R(R,)G R(v(Q,)). Since q is reversible, we have R(!A)=R(~J(Q,)). If #S,,,,. ,nl. the number of equalities in systems of type (3.1) does not change, but the superscripts vary. Let r, be the number of equalities containing Q," with a-l for R., in a system of type (3.11, and r2 the number of equalities with a=f for system cp(Q*). Then r,=a+2c+b, where a is the number of equalities which contain Q,". together with 0%' ,2b is the number of equalities in which all

Q!" have the same superscripts, and 2c is the number of remaining equalities. Clearly, a+c+b<r~<u+2c+26, and therefore 05<r*/r,<2. On the other hand, naturally, r,<R (Q,) and

r,Gfl(cp(g,)). We fix (6, A, qt. qd. for which r,=R(Q,). Then 0.5r,sr, and 0.5R(9?,)<R(~(Qr)). Similarly, O.~R((F(&,))<R(Q,). Thus, the following theorem has been proved.

Theorem 3. Let cp=n. mer. If ‘FES,,,~,. .,n,, we have R,(‘L)-R,(q(O,)). R(B,)=R(rg(Q,,))

and if @S,,,Z.. .N, we have O,S<R(Q,)/R(cp(Q,))<2. Further, let SR(B,,)=li. We again consider the system cp(Q*) where ~'EII. Obviously,

C(g:(52*))=rpG(g,)m-'. If S-S,,X...X&,'=G(R,), then ~S~-'&G(~(9,,)). Hence we arrive at the

following corollary.

Corollary 2. Let the group S=SN,X...XS,, be induced by the partition iV,U.. .UN~=(l, 2,

. . , n) 1 and let there exist a permutation of cp6sN su& that r$~-'~c(s,). Then R, (Q,) <2k,

R(Q.,) <4k, if q=S,,, l,. , "J? and R(QA)<8k if +s,,, 1.. n,. Theorem 4. Let the group S=S,,X...XS,, be induced by a partition of length k, and let

there be (p=rI such that cp=no where CIES and n only superimposes the negations. Then SR(BJG2k follows from ~&J-‘=G(Q,).

Proof. Because each SNjis a normal subgroup in S, we have aSr,O-'=SNj.. Hence, since

n-'=JC we have

cps~-'-(~s~,~-')x...x(cps~,cp-')=(xos~,o-'x-')x...

x (naSs, r’x-‘) - (n&,n) x I . . x (xSh-, n) )

we set M,-(iliEN,, n replaces the i-th coordinate by negation}, and T,=#,\:M,. As a result

we have the partition {l, 2,..., ~)=.I~,UT,U...UM~UT,, imbedded in N,U...UN,, ar.c! S~,XS~,C-SN,, i=l, 7 _,I.., k. On the other hand, it is easy to see tha: .'~Sy~x==S~r~ and nsr s=s, ( and

therefore a~(S~,XSr,)n=(ffS~,n) (;~S,,~)=S~~XS+,EXSK)X. Consequently,S,~,XS,,X . ..XSN.XSr,+Sq-'~ G(QJ and we have SR(Cl,,)<Zk.

4. Rank and set-theoretic operations. Let us investigate the behaviour of a symmetric ra?:k when applying set-theoretic opera-

tion to a system of supporting sets. Let the systems 0, and 61e have symmetric ranks m and k, N,U...'J.V, an2 III,>. ..,.I[* being

the partitions which induce groups S*=zS,X . ..XS.,-C;((.!,) and S"=S,,,X...XS,,.,~G(a,). The

reduced form of these partitions can be constructed by separating the commcr. parts of the

initial partitions. We denote this form by I<,U...UK,. Clearly, pG!im and s=:sK.x.. XSapE S”, SB. Therefore, Sc(G(o,Uo,), G(a,nn.)) In additicn, G(O,)=G(E^\<L>) hence we have the

following theorem.

Theorem 5. The relations (.SR(!&U~2e), SR(n,nn,))~SR(R,)SH(Re), SH(!!,)=SR(E”\Q+) hold.

Corollary 3. The rank of systems R,Uo,, %nn, is not higher than 4SR(Q,)XSR(Qe). The rank of the complement of a sYstem is not higher than its quadruplicate symmetric rank

(for a A-rank not higher than ZSR(R,)SR(%)). Now consider the task of ;rsjecting on an intervalin E'. Let !!.,sE", and the interval

N'l "-GE", where II ,rn o,E(O. 1). (i,,.. !.,}&(I, L.. II}; i.e. ir follows from a=!\'; "1 that x=01,

.__, a,,=%. We denote the result of projecting on the lnrerval by pr co,), :.e. pr(R,) con- sists of all aEn* for which 3 =o,,..., a,,=o,. Let SJ?(Q,)=k, and the required subgroup

be induced by the partition iv,. U;V,=(l, 2...., ,,}=.1. We introduce the notarlon &f=(i,, . . . . i,). M"=(ijlij&f, o,=a), a=O, I , and construct the partition .V=WUM’U (~%‘,\.lf) U - (.V,\M). ObVlOUSlY,

the product of the groups induced by this partition is contained in G(pr (0,)). and therefore

the projection symmetric rank is not greater than k"2.

Finally let us consider the operation of adding fictitious variables. III thrs case, a

set of m new coordinates is add& to each o=%, and we obtain vectors (0. 7). where r runs

through a certain m-dimensional interval. However, this interval is invariant with respect

to any permutation of the coordxates of vector r.ConsequentlY, if Sm(<l,) =k. tne sYxmetric

rank of the new system is not rl_rner than ki+1.

190

Theorem 6. The operation of projection increases the symmetric rank by no more than two. The operation of adding fictitious variables increases the symmetric rank of a system

by no more than unity. The following conclusion may be reached from the theorems above. Proceeding from the

sets with a symmetric rank not higher than k, by using unions, intersections, complements,

projections, additions of fictitious variables, and by the action of isometric permutations we can construct systems of supporting sets of ranks considerably less than n.

Concluding, let us consider some important examples of the systems of supporting sets, which have a limited symmetric rank and can constitute a basis for more complex systems.

Symmetric sets: from a&,, I]all=#jI it follows that fi&,, that is a system of numbers

(r,,..., r*). /cSzn+i is given, and Q.,=(a~~~a~~~(r,, . , rh)). Obviously, G(%)-s,,, *. It7 and R(Ql)G4, and Ra(QA)B2.

Sphere: a=E” and the number r are fixed, and Q,=(BlIla+Wr). ot~i~usly, SN,)(SN,~G(nA)-~S(I.I,, .,,&, where &=(iJa,=~). j-0, 1, and o is an isometric

permutation such that (p(a,)-a, for iEN, and (I, for iEN,. Therefore, &(S2.4)C&R(Q*)<8.

Interval: O.,-N:.:,"-, I II 0,={0,1).

We have SssG(%), where N--(l,2,...,n}\(i,,..., f,), and therefore the symmetric rank is not higher than 3, Ra(QJQ6,R(RA)GL2.

It is also easy to compute the quantities Q,(6, 3, Ql? a) for all the above sets. This list can be completed by sets obtained from symmetric sets, spheres, or intervals, using set-

theoretic operations and isometric permutations.

REFERENCES

1. ZHURAVLEV Yu.I., Algebraic approach to problems of recognition and classification. In: Problems of cybernetics (Probl. kibernetiki), No.33, S-68, 1978.

Translated by W-C.

U.S.S.R. Comput.Maths.Math.Phys., Vo1.25,No.l,pp.190-192,1985 0041-5553/[35 $lo.CQ+o.OC

Printed in Great Britain Pergamon Journals Ltd.

DETERMINAT 'ION OF THE EXTREMAL EIGENVALUES BY THE MIN OF SPECIAL FUNCTIONALS'

SHORT COMMUNICATIONS

IMIZATION

G.A. SAVINOV

The determination of the extremal eigenvalues of definite matrices by minimization of functionals

symmetric positive of a special type is

proposed. An analysis of such functionals, and the results of test

calculations are given.

Let the matrix A of order nXn be symmetric and positive definite. We denote the eigen-

values of this matrix by A,>A+L>...>L,, and the corresponding eigenvectors by U,.....‘n_ It

is well-known that the extremal eigenvalues of the matrix A can then be determined by the minimization or maximization of the following functional

/l/j: p(rl-(.%4/(~d. L-mini.

Z.R"

where R, is the n-dimensional real space with the scalar

Let us compute the derivative of p(r):

known as a Rayleigh quotient (see

A,-maxp(4. I.&

product

Obviously, the solutions of the characteristic equation p(vJ-L*. Minimizing r(z) we Ir(V,)')il.

the equation p(r)-0, which, apart from a factor, is identical with

for the matrix A, will be I-u;, i-1.2, . . . . n. and at the same time

obtain I=",, and M(u.)-A., and maximizing p(z) we have ~=UI and

In the present paper we propose to determine the extremal eigenvalues by minimizing those functionals which are different from the Rayleigh quotient. Thus, we shall determine the minimal eigenvalues of symmetric positive definite matrices by minimizing functionals of the

*Zh.vychisl.Mat.mat.Fiz.,25,2,292-295,1905