Numerical canonical factorization algorithms and their application

U.S.S.R. Comput.Maths.Math.Phys.,vol.3o,No.2,pp.i-12,1ggo Printed in Great Britain

0041-5553/90 $10.00+0.00 01991 Pergamon Press plc

NUMERICAL CANONICAL FACTORIZATION ALGORITHMS AND THEIR APPLICATION* R.P. TARASOV

A study is made of numerical algorithms for canonical factorization, of two types: a) polar decomposition of an arbitrary bounded operator in H-space, and b) factorizationof an operator-valuedfunction of the unilateral shift operator in the space A. Two types of canonical factorizationplay an important role in the calculus of bounded operators in Hilbert space: a) the polar decomposition of an arbitrary bounded operator A (see /l/j: A-U(A’A)“= (AA’)‘“U, (1) where the operators T-(A-A)“, T’-(AA’)‘” are fractional powers of the positive selfadjoint operators A;4 and AA’, and U is 'a partial isometry; b) the factorization of an operator-valuedfunction u(T)=&of a completely nonunitary contraction T into interior and exterior factors n<(T) and a,(T), respectively /2/: n(T)- z aJ", L>O u.(T)-

&l, *>0

a,(T)-r, c$F, *>(I

Tk,

u(T)=u<(T)u.(T),

(2s)

(2b)

where u(A),&(A),u,(k) are functions of the Hardy class i? defined in the unit disc of the complex plane, with Ia,(&)1<1. In problems of applied mathematics, canonical factorization enables us to investigate the structure of a specific operator and use the resulting structural features to construct efficient numerical schemes for solving operator equations. In this paper we shall consider algorithms for the canonical factorizationof operators which in the case of the polar decomposition (1) yield procedures to compute the sequences {UJ, (T,), {T,‘) approximating the operators U, (A-A)“‘, (AA*)‘” in the strong operator topology: U=s-lirnU.,(A-A)* -s- limT., (AA’)“=s- limT,‘, n+m; the operators T., T.’ will have an exponential representation T,-exp[a,(A’A)], T,=exp[a,‘(AA’)], so that one can simultaneous1Y compute an approximation of the inverse operator provided that it exists. In the case of the factorization (2), the algorithms produc!-(siquences{u.(U)}, (w,(U)). where V is the unilateral shift operator, satisfying the conditions *(U)=s-llim.u;(U),‘ &(U)=s-limw.(U), w.(U)=exp[a,(U)]. conAs applications of our canonical factorizationalgorithms, two problems will be sidered: a) the numerical solution of the operator equation _ W))=Sa(r)U(r)dz, S(a(t))S'(a(t))=S(b(t)), 0 _

where S(b(t)) is a given operator and it is required to compute the operator s(a(t))and its inverse [S(a(t))l-‘ - such problems arise in applications in the field of time series analysis 131; b) the numerical solution of the operator equation [I+aU(to)lS(al(t))--S(ar(t)),

lal
where I is the identity operator, U(r) is the shift operator by T>O: U(T)*(~)--g(t-T), r,, a are fixed real numbers; the function a&) is assumed known and it is required to determine a, TO and the function at(t) - this is known in applications as the dereverberationproblem /4/. 1. Algorithm computing polar decanpositim. 1. The following lemma will play a major role in the design of algorithms computing the polar representation of a bounded operator defined in a Hilbert space H. “Zh.uychsit.Mat.mat.Pie.,3o,3,33g-354,1ggo

1

2 defined

.&em 1, Let A be a bounded operator operators

in x

uIl+r-exp

and IU,? the sequence of bound&

UFA,

nzd.

(1.1)

converges strongly as n-m to an operator P,, and {U.&*) Then the sequence IV.‘V.l converges strongly to an operator PI, sucbthat both P, and P, are projection operators in H. The sequence of operators bY.YJ.1 satisfies the equation Proof. U.Ull’)Urn at the same time, it is obvious that U.‘(J-U,U,‘)-(i-U.‘V~U~‘;

U~+,U.+,=-U.‘exp(Jhence

U:,,U.+,~exp~I-u*'u.)u,'u.. (1.2) U,*.U,,~ It follows directly from (1.2) that the sequence W.‘U3 of positive opertors and bounded above by the identity operator. beginning from ma& is monotone non-decreasing Thus the sequence has a strong limit p&Z such that P,=exp(Z-P,)P,, whence it follows that P, is a projection. Proceeding now to the sequence {U.U.*),we can write U,,,ET,,--expflU.U~'lU~U,', and therefore, using the same arguments, we conclude that W,U,‘l strongly as n-cm to a bounded operator P,, which is also a projection. It is obvious from (1.1) that the operator A can be written as “-1

A-mxp [f

(&J,Ui-nl)] L-e

V *WP [+( We will now show that the sequences

converges

(1.3)

VP

nx.

g ukw.-d)], *-*

W,,), tT.1, {T.‘}, nM,

where

converge strongly to bounded operators. Theorem 1. Let U, be the operators defined in Lemma 1, and T-and T.’ the operators defined by (1.4). Then the sequences (V,), (T-1, (T_‘) have bounded limits as n-+= in the strong operator topology of H. Proof.

Since

where

n--l

Tn’-exp ( +-[ g

VkVL-(n-iVii}

n)z,

,

it follows that in order to prove that the sequences {TJ and (T,,‘) are strongly convergent we need only show that the sequences {T.), tTs’l are strongly convergent. But by Lemma 1

are monotone non-decreasingsequences of positive operators,and therefore (TJ, (T,,‘) are monotone non-increasingsequences of positive operators; hence they have bounded strong limits. We will now show that the sequence {U.} has a strong limit. To that end we observe that U,-PtU.P,. Indeed, since n-l A’A-exp (z: U,‘U,-nl ) U.‘U.-V.W. h-0 and

P,-s-

limU.*V.,

it follows that I-‘

exp (“i

k”(

ViVb-nl)

3 and consequently (I-I

1 (A'A)"=s-limexp [ 2

aL-e

)IP,-

Vh’Vh-nl

(1.5)

Similarly, (AA')":-a-limexp [f(Z

L-0

P*-

UJX-d)]

0.6)

“-1

and therefore (A*A)“==P~(A’A)“P, and (AA’)“‘-P,(AA’)“P,. Hence, in turn, using the polar and PJ-P,(AA’)“V==A, decomposition, we can write the relations API- V(A’A)“P,-A from which it follows that V,,==Pd.Pd

We will now use the obvious equality "+n-, Un+,=exp[ +-( ml-

x WY;)] *-n

V.

and show that whence it follows that the sequence &!LJ.j has a strong limit. Since V,=V,P,,

we can write

VII)+"-,

WL+n--L V.P,( I-exp[

f (ml-

E V;Vk)]}9 k-n

and so

(1+1)-t Il~V~-V.+.~~ll~~llV~li~~~P~(~-~x~[~(~~-

But

IIVnlla<& n,l,

VW&

&L(l,~)])rpJ~. I-0

and so it will suffice to show that the sequence of operators @,I, where -+"-a ~~=P,{z-exP[f(mz-r,v~v*)]},

n+o~, Vm>O

I-n

converges strongly to the zero operator. But by Lemma 1 Ilk+"-, s-lim so we

v
can write “+-A

d-limP,(I-exP[+(

d-E

P,( I-exp[f m (I-P,)

I-*

II-0

Vh*s)]}-

4

a-limE,,-O,n+=, proving the theorem. whence it follows that of the sequences (V.), Thus, in view of (1.3) and Theorem 1, the strong limits V,T,T’ We shall now show that this yield a factorizationof the operator A:A=VT-T’V. (T.}, (T.‘) is the polar decomposition of A. be the sequences defined in Theorem 1. Then A has Corollary 1. Let W.1, lT.1, CT.‘) the polar decomposition V-s- limU., T-s- limT., T’-=s- limT,,‘. A=VT=T’V, (1.7) for all n>O; but by Theorem 1 the It follows from (1.3) that A=V.T,-T,‘V, Proof. have strong limits, so that the representation (1.7) holds. It sequences (V,),(T.),(T,,‘) will therefore be sufficient to show that (A’A)“=T, (AA’)“-T’. But it is obvious from (1.5) and (AA’)‘h=PJ”=T’P1. But since Urn-V,,P,, and (1.6) that (A-A) ‘h-TP,=P,T o--L

T,=exp

[ -21 (~vh%-nz)]b-0

esP[&wk8Z)P,1exp[+(P,-l)l; L-0 at the same time

s-limexp[‘/,n(P,-Z)]-P,,

and so

..~. V;V,-nZ

-PST-TPfi.

It can be shown in the same way that T’-P,T’-T”P,. 2. Hitherto we have been considering arbitrary bounded operators A. If A is a bounded positive selfadjoint operator, we can write v.+,=exp(z-v~)u?l, II,-A, n>O,

(1.8a)

I-,

T,-exp(x U.-d), L-0

A-s-lim T.,

(1.8bl

and if at the same time A has a left inverse with dense domain, then “-1

(1.9)

A-l-s-limerp(nZ-XLI,). L-0 Under these conditions we can also define the continuous at zero:

semigroup of fractional powers of A, which is

(I--L

Ar==s-limerp[E(z uh-nZ)]. 1-O

E.30,

(1.10)

as well as the operator-valuedfunction 1nA and fractional powers of the inverse A-‘:

lnd-e-lirn(&J+~Z),

A-8-a-limerp[&(sZ-~ oh)].

I-0

L-e

(i.ii)

Relations (1.8)-(1.11)yield constructive algorithms for computing functions of bounded positive selfadjoint operators, which are commonly encountered in applications. 2. Algorithms for factorizing operator-valued functions of the unilateral shift operator 1. In our construction of an algorithm computing the canonical factorization (2), the role of the completely non-unitary contraction will be played by the unilateral shift operator IIdefined on the real H-space E, of square-summablesequences k&)0-,x&*--J:

u: ho,. . ,a,,. . .)-to, a,,. . . ,a,-,, . . .}* Let

u(U) - z ahUh, l>O

u(V)eH,“;

together with IIwe shall consider its unitarydilatationu, defined in the H-space 1, of square-summabledoubly infinite sequences (a,L,-, and the corresponding function u(U). Defining a sequence (u,,(u))of operator-valuedfunctions by

5

%+1(U)'exe [$[I-2E.(u)l}u.,

~o(t7)==W),

(2.ia)

~.(II)+ii,'(17)-u"(8)u,'(B),

(2.lb)

where

we shall show that the sequences 6a,(U)~,(wn(U))rwhere n-i

(2.2)

lo.(U)-exp(f[2r,f.(U)-n~]}, L-0

have strong limits in the space I,. First, however, we will establish the following lemma. Lemma 2. The operator sequence {E.(U)) converges to the operator 112 in the strong operator topology of t,.

Proof. Since

is strongly convergent to a projection operator P(U). the sequence (s,'(U)u,,(U)) time,

At the same

u'(U)u(U)-exp[r:ukYU)u+U)-+JU)u.(U). L-, and consequently

” . u'(U)u(U)-8-limexp [&h'(UMn-nZ]P(U),

I-0

But the operator u(U)=&' in tr has a dense range, therefore P(B)u’(B)u(~)-u’(~)u(~). In turn, it follows from the condition s-limu.'(U)u,(U)=I thats-limii.(U)= and so P(U)-I. I/2. Now, using the condition s-lim ii.(U)-I/2, we shall show that the sequences bz.(U)),bn(U)) have strong limits. Theorem

2.

The sequences (u,(U)), h,(U)} have strong limits s-limu.(U)=u,(U), 8-lim is a unitary operator. u(U) -a*(U)u.(U), where u,(U)

w,(U)-u.(U), and

Proof. We can write u(U)-u,(U)la.(U) for any n>O. In turn, to prove that the seare strongly convergent, we need only show that the operator sequences Cr;.(~~jlCw.(U)) u.+,(U)), (w,(U)-w,+,(U)}, where m>O is arbitrary, converge strongly to quences u, the zero operator in 1%. On the other hand,

;;ittherefore, as the sequences

Jlu,(U)llr,,~~w,, (U)llr, are uniformly

bounded,

we need only

show

“+111-l (z-sXp(f[ml-2~ii~(U)]))+0,

n+o,

VnGQ

h-n

in the strong operator topology of r,. But this follows immediately from the fact thau, by Lemma 2,

6 Ill+?&--L

Vm>O.

E&(o)-nr112

s-limz

awn

of the sequences {u,,(ti)), (to,(g)}, and Thus there exist strong limits u,(U) and 40) moreover u‘(U) is a unitary operator, since a'(ti)~(U)~~~(17)u*(U)-~ Using this theorem, it is now not difficult to establish the following result for factorizationof the operator-valuedfunction u(U)eHVg in I,. CorolZary 2. Let k(U)), (w.(U)} be sequences of the type (2.1) and (2.21, with u (w.(U)} Then the sequences k,(zI)~, replaced by its isometric restriction U to the subspace la. have strong limits: s-lim~(U)=u,(U), and moreover a(~)-~(~)u,(U),

where

e(U)

s-limw,(U)===u*(U), is an isometry.

are the projections of the operators m_(B). Ppoof. The operators k(U) and w&V la.@) onto E,, and hence there exist strong limits u*(U) and u.(U) of the sequences k(U)A (W"(U)}, which are projections of the operators u@), u,(r);and the projection of the unitary operator u,(g) onto Et is an isometry. For a constructive method producing the operator sequences u,(U),w,(U), it suffices to observe that if

then it follows directly from (2.1)and (2.2)that

and

where the symbol * denotes convolution of sequences: bgua&- El h-n%, ma3

k>O;

exp'(k) is the exponential function of a sequence {a*) with multiplication understood in the sense of convolution: exp'(dk) = 6, +

$-

,

dp=;dz+....d,. 7

2. The results obtained above can be generalized immediately by considering, instead of the unilateral shift operator U, a semigroup U(T),r>O, of unilateral shift operators actdefined on the half-line &[O,m): ing in the space of functions W)

and instead of the operator-valuedfunctions u(U),operators of the form

It is readily seen that S(a(t)) is a continuous analogue of u(U), since the set {U"), may be considered as a discrete semigroup of shift operators. If we now use ,..., 0, this analogy, considering U(T) in the H-space &(O, =) of square-integrablefunctions, then U(r) is a strongly continuous semigroup of isometric operators, and if aft) is in the class L,(O,=) of functions integrable on the half-line, then S(o(t)) is a bounded operator. n-O,i

7

o(r),TE(--oo, m), acting in the H-space L,(-00,-) Next, if we define a one-parameter group t=(--, m), which is a unitary dilatation of of square-integrablefunctions on the real line the semigroup U(r), t=[O,-), we can define the following operators in L,(-00,Q)): _ S(s(t))-Sa(7)~(r)dr, 0 where if a(t)~L,(0, -)llL,(O, m), then the operator S(a(t))has range dense in L,(-00,-). This last assertion follows from the isomorphism between L,(O,=) and the Hardy class Hz of functions in the half-plane, by virtue of which the Wiener-Paley condition /5/ holds:

(2.3) where C?(O) is the Fourier transform of a(t)&,(O,m). Indeed, it is obvious from (2.3) that (i(o)can vanish only on a set of measure zero, and this now implies directly that has range dense in L,(-oo,m) (see /6, p.570, Theorem 21.2.5/). Thus, the situS@(l)) ation in the case of the continuous group C(T),2=(-m. -), is analogous to that already established for the discrete group {8"),n=O, f%...,fm.Hence, using similar arguments, we can prove the following assertion: let S(a(t))be the restriction of S(a(t)), to the subspace G(O,oo) S(b,(r)) have strong of L*(-=% -), with a(t)~&(O,a)G(O, 00).Then the sequences S(a.(t)), limits: s-limS(a,(t))=S(a("(t)), s-limS(b,(t))=S(a'"(t)), where the sequences

(a,(t)), (b.(t)}are defined by

c5l+&)=exp{+[ 6(i)-w])*a.w,

aaW=a(t),

“-1

W)=exp*(f[ 2z a,(t)-w)]}; L-0

here 6(t) is the Dirac delta-function,

and the symbol

l

denotes the operator

d,(~)+d,W- j d,(t-r)d,(z)dz,

eo,

and moreover the strong limits S(a(')(t)), S(&)(t)) of the sequences (S(a,(t))), U(b,(t))}define factorization of the operator S@(t)) -S(a")(t))S(a("(t)). To conclude this section we note that we have been considering the factorization of operator-valued functions u(U) and S(a(t)) acting in the spaces of infinite sequences and functions defined on the half-line ro,00). However, it is also of interest to study the factorization of operator-valued functions u(0) and S(a(t)) acting in the spaces of finite sequences (u~)~H and functions defined in a finite interval t=[O,T): u: (u,....,u,,...,u,)-+(O, ao,...,ai-l,...,U”-,), t40, T). ~(a(t))~(t)-ja(T)rp(t-l)dr. This problem can be reduced to the previous one by usingdilatationsof u(D) and .9@(t))acting respectively in &. and L*(O,m). But since the dilatationsin this case are not unique, one encounters the question of whether the factorizationsof u(o) and S(a(t)) are unique. On the other hand, it is possible to make direct use of (2.1) and (2.2) to define operatorvalued functions u(o) and S(u(t)) which, provided the sequences (u"$7Y;)o&(u)A s(&(t)), in the s@,(t)) are convergent, determine unique factorisationsof u(U) and spaces of finite sequences and functions in a finite interval. 3. Apptications. Rmmptes of mmertht Gnptementation. The above canonical factorization algorithms, using the exponential representation, enable one to compute other factorisations,which yield solutions of operator equations of importance in applications. We shall consider two problems of numerical factorization: 1) decomposition of the form S(b(t))=S(u(t))S(n(--t)) for an operator ,!J(b(t)), where

8 is a positive-definitefunction and a(t) is defined on the half-line t=[O,co); 2) decomposition of an operator S(a(t)) intoafaytors 8(&(t)) and S(a,(t)),where 1 a singular generalized function. al(t) is a regular function on the half-line and Factorizationsof the first type are connected with time series analysis /3/, and those of the second with dereverberationproblems /4/. its dilatation in 1. Let S(a(t)) be an operator acting in L,(O,m) and S(a(t)) Consider the equation M-m, -).

W

S(a(t))S’(a(t))=S(b(t)),

(3.1)

a known function and it is required to compute the operator S(a(t)) where b(t) and its (possiblyunbounded) inverse S-'(a(t)), which has dense range. The problem has a solution if -)flL(--, -),S(b(t)) a(t)EL,(O, ~)nL(O, =). Indeed, if that is the case then b(t)+(--m, is a bounded positive selfadjoint operator, which has a possibly unbounded inverse with dense range; hence by (1.8), this last operator may be written as is

n-1

limexpW(d,(O)l, d,(t)==x

S(b(t))--s-

b,(t)-%3(t),

(3.2)

k-0

where b,(t)-b(t). b.+,(t)=exp'I6(t)-b.(t)l*b,(t),

Under these conditions the sequence (fl(a,(t)))

(3.3)

has a strong limit:

~(a.(t))=exp[~(~(~)d.(t))l=S(exp'[~(t)~(t)I) and thus S(b(t))=[s-limS(a,(t))][s-limS'(a,(t))]= [s-lim9(a,(t))][s-limS(a.(--t))]. If attention is confined to cperators S(a(t)) for which the interior factor is the identity operator, the solution of Eq.(3.1) is unique apart from sign and has the form S(a(t))=s-limS(exp'[H(t)d.(t)]), and at the same time [S(a(t))]-'-s-limS(exp'[-H(t)d,(t)]). onto the If we now take the projections S(a(t)) and S-'(a(t))of S(a(t)) and S-‘(a(t)) space L,(O,m), we obtain a solution of the problem stated above for Eq.(3.1). Thus, the solution reduces to the computation of a sequence of function Id.(t)1 by the formulae b,+r(t)-exp.Ig(t)-b,(t)l~b,(t),

b&)==b(q,

“-1 d.(+~b,(+nd(t). I-0

To estimate the efficiency of the above scheme for solving Eq.(3.1) in a numerical experiment, we approximated (3.1) by a finite-differenceequation

The numbers (ah) were computed by a formula which is a finite-differenceapproximation of (3.3):

n--l ah'uLexp'

where the elements

{br”)

were

computed using the iterative scheme bF’==exp’(&-b:“‘)

lbr’,

b (0) k - BL,

which is a finite-differenceanalogue of (3.3). Fig.la shows the results of our numerical computation for the case in which the sequence tar) is a finite-differenceapproximationof a triangular pulse: curve I corresponds to the exact solution {a}, curve 2 to the quantities {PA),and the dots denote the computed values of (a-i"'),n-25. To compute (b$“)} we took 5 terms in the expansion of the exponential function:

9

to compute

64”)1

we used the approximation

and the number of points where the solution was sought was N=10'. Fig.lb shows the computed values of I$)),n-25, corresponding to the inverse operator: -

In order to determine the degree of approximation of the numerical to the exact solution, we computed the sequence (A:"'): A:"'-$' *a,, whose deviation from the sequence (i&),&=l,&=O, k>O. can serve as a measure of the accuracy of the solution {$')). The table lists values of (A(:)),n-25.As the numerical experiments show, our numerical factorization scheme is fairly efficient: it converges quickly and is not too sensitive to round-off errors. QK 2200

1800

600

YK

b Ah

-15

1 0

I

I

20

40

II

I

I\

I

I

60

80

100

Fig.1

2. Before going on to a numerical scheme for factorization into regular and singular parts, we will consider an analytical example which should clarify the entire problem. Let have the form a,(t) “(4n)-gt-“exp(-t-‘/4),t>O; then, as the reader will know, the ofi is a member of the strongly continuous semigroup of operators operator 8(&(t)) SW;

t))“exP

wS(bltt))l,

eao,

s(a,(t)k-S(a(t-l;

t)l,

where &(t)=[r(-‘/,)]-‘~,t>O is a singular generalized function, and r(z) is the Gamma function. Moreover, let s(%(t))-[ZtaU(7o)lS(al(t)), lal
r,J=ya.8(,_,), “>O

we can write

a,(t)-(a(t)+aa(t-o)l+a,(t). In turn, since do(t)

vanishes at

t-0, while

b,(t)

is

regular in a neighbourhood of the

10 set

where

~H-n~p,n--l,2,...,-l, it follows that

X”(t)

is

the characteristicfunction of the set

(H)

in the generalized sense:

(xH(t)d,(t), rp(t))--lim(&(d, dd+-ndM)), s>O, s+=, q(t) being a test function equal to unity in Thus, in order to determine from k(t) the values of a,rO and a neighbourhoodof t-0. the function al(t), it is sufficient to factorize s*(t) into regular and singular parts, and for this factorization in turn it suffices to express 40) as a sum of two components, one of which is regular at DO, the other singular. We will now consider the operator equation lz+aU(ro)lS(s*(t))=S(a*(t)), lalO, where the function a,(t)&(O, =)llL.,(O, m) is assumed known a, 70 and the function a,(t)EL,(O, *)f&(O, m). In order to ence and uniqueness of the solution to this problem, we factorizationof the operator S(a,(t))-S(o,(“(t))S(a,‘~)(t)),

(3.4) and the problem is to determine establish the conditions of existwill consider the canonical where S(a,")(t))-s-limexp[S(d,,.(t))].

. . defines an unbounded operator as In the general case, S@,,.(r)) n-m, while a,.,,(l), n+m, Analogous formulae can be derived for determines a singular generalized function a,(t). S(a*(G), and we have s(a,“‘(t))-s(a,(“(t)), If we now require that the function sidered above, we can write

s(al”‘(t))~exp[S((l,(t)+$(t))J.

a%(t) be regular for

S(~(t))-S(a:“(t))exp[S(i

G-0,

then, as in Example 2 con-

@))lexp[S(d:'a (t))],

with

where a:,:(t), b;:(t)

are the regular and singular (at 00)

components in the resolution

aP'(t)-a:;; +a:'d(t,. Thus, regularity of d,(')(t) for t>O is a sufficient condition for the existence and uniqueness of a solution to our problem for the operator Eq.(3.4). In the numerical implementationof factorization into singular and regular parts, it is natural to start with a finite-differenceapproximation u(U) of the operator S(a,(t)): u(u)-

ahrV, z I>0

where (CQ) is a function on a uniform grid which approximates a,(t). factorizationof u(U), we can express ain the form

Using the canonical

Now, dividing the sum under the exponential sign into a regular component (a$)

and an

irregular component (dtk),

we can write

ai;’-exp' where (a$,),n+m

(2dj,?),

is a finite-differenceapproximation to the regular Part

o,(t) and (a$$~

to the singular part of (6(t)-c&(&To)). To estimate the numerical efficiency of the factorizationalgorithm, we carried out some numerical experiments. Fig.2 shows the results of our numerical Canonical faCtOrization in the case in which (Q was replaced by a grid function k-0,1,. . . . at,,-((6r+0.96r-r,)*k'exp(-0.05k)sin0.5k,

11 and the ex-

Curve 2, a correspondsto &x&, curves2, b, c to the interiorfactor and curve 2, d to the grid function teriorfactor {~a), respectively,

(3.5)

a a.04 .

b a.02 -

A

0.00

v

A

Fig.3

Fig.3 il.lustrates the numericalfactorization of 14 parts:curves 3, a, b represent the regularcomponent rdp;E tdimkl

of

into regularand irregular and the irregularcomponent

the grid function13.5f,and curves 3, c,d the iegufarpart (ai$]

and irregular

corpart {ai)zjof {a$$; the circlesin Fig.3,care the valuesof at,-~exp(-~~5k)sin#.5k. respondingto the exact valuesof the regularpart of {a&. The exponentialfunctionwas computedas in Example1; the numberof pointsat which the solutionwas computedwas N--l@ and the initialsequencewas normalizedto

FinaLly,we remarkthat all these graphswere constructedusing linearinterpolation of the grid functions. RWERENCES 1. ANON N. and SCHWARTZJ.T., Linear Operators,Part TX: Spectral Theory. Seffadjoint Operatorsin HilbertSpace, Interscience, Wew York, 3.963. 2. SZ.-NAGYR. and FOIAS C., Analyseharmoniquedes operateursde l'espacede Hilbert.Massie et Cie. Paris,Akadeniiai Kiado,Budapest,1967.

12 3. BOX G.E.P. and JENKINS G.M., Time Series Analysis. Forecasting and Control. Holden-Day, San Francisco, 1970. 4. CHILDERS D.G., SKINNER D.P. and KEMERAIT R-C., The Cepstrum: a guide to processing. Proc. IEEE, 65, 10, 1428-1443, 1977. 5. HOFFMAN K., Banach Spaces of Analytic Functions. Prentice-Hall,Englewood Cliffs, N.J., 1962. 6. HILLE E. and PHILLIPS R.S., Functional Analysis and Semigroups, American Mathematical Society,Providence, R.I., 1957.

Translated by D.L.

U.S.S.R. Comput.Maths.Math.Phys.,Vo1.30,No.2,pp.12-19,1990 Printed in Great Britain

0041-5553/90 $10.00+0.00 01991 Pergamon Press plc

APPROXIMATELY OPTIMAL ALGORITHMS FOR DETERMINING EXTREMA

IN A CERTAIN CLASS OF FUNCTIONS* A.G. KOROTCHENKO

The problem of designing approximately optimal algorithms to find the extrema of functions in a certain class, which is closed under various natural operations and contains concave, convex and Lipschitz functions, is considered. Introduction. The efficiency of a number of methods for maximizing functions of several variables depends on the rapid solution of certain auxiliary problems in one dimension. It seems natural, therefore, to tackle the solution of such problems by using optimal algorithms. However, such algorithms are as a rule rather complicated and difficult to use in practice (see, e.g., /l-3/). It is thus important to design easy-to-use, approximately optimal algorithms which preserve some of the properties of the optimal ones. In this paper we consider the design of this kind of aglorithm for a class of multimodal functions which contain concave, convex and Lipschitz functions. The class is closed under several natural operations (the basic convolution operations of a vector criterion). We shall construct successive single-stage and block algorithms (where by a block algorithm we mean one in which the function is computed in the search phase at a whole series of points). Necessary bounds will be derived. 1. Definition and properties of the ctaes of functions. Let @, denote the class of all functions p(x) satisfying the following conditions. Condition

1.

Condition

2.

The function

qw-~d x1--a

q(z)

is defined and continuous in an interval [a,A].

cP(sr)-K, x,-a

$4x,)--K, b-xl

cp(d--K b-x,

’

where $-x1,x1,~=(a, b),K is a real constant. This class was first defined in /3/. It follows from Condition 2 that cp(z)>K if cp(s)EQ)I. We list a few properties of functions belonging to class &. Some questions relating to the properties of IQ,
Property 1. Q)K contains all the concave functions bounded below by [a,b] and continuous at x-a, b. Property

if

2.

A convex function

K

which are defined

q(x) which is differentiableat a and b belongs to cDx

K-min[cp(b)-cp’(b)(b-a),cp(a)fq’(a)(b-&)I*

Note that the differentiabilityof cp(z) at a and b is not absolutely essential. The definition of the constant K requires no more than a lower bound for the right derivative at a and an upper bound for the left derivative at b. in

Property 3. A function rp(x)which satisfies a Lipschitz condition with constant L [a,b] belongs to the class Qn if

*Zh.uychisZ.Mat.mat.Fiz.,30,3,355-365,199O