Rayleigh-Ritz and A. Weinstein Methods for Approximation of Eigenvalues. I. Operators in a Hilbert Space

Rayleigh-Ritz and A. Weinstein Methods for Approximation of Eigenvalues. I. Operators in aHilbert SpaceAuthor(s): N. AronszajnSource: Proceedings of the National Academy of Sciences of the United States of America,Vol. 34, No. 10 (Oct. 15, 1948), pp. 474-480Published by: National Academy of SciencesStable URL: http://www.jstor.org/stable/88381 .

Accessed: 07/05/2014 11:34

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

National Academy of Sciences is collaborating with JSTOR to digitize, preserve and extend access toProceedings of the National Academy of Sciences of the United States of America.

http://www.jstor.org

This content downloaded from 169.229.32.136 on Wed, 7 May 2014 11:34:09 AMAll use subject to JSTOR Terms and Conditions

http://www.jstor.org/action/showPublisher?publisherCode=nas

http://www.jstor.org/stable/88381?origin=JSTOR-pdf

http://www.jstor.org/page/info/about/policies/terms.jsp


474 MA THEMA TICS: N. ARONSZAJN PROC. N. A. S.

The sonic treatment alone did not cause aberrations. The increased yield is probably due to an increased movement of chromosomes and chromosome

fragments caused by, the sonic treatment, resulting in a decrease in the amount of restitution and an increase in the amount of detectable new reunions between the broken ends of chromosomes.

t A recent experimentl1 reports the production of chromosomal aberrations, mutations and other effects in Allium root tips, Helianthus stem apices, and Drosophila by treatment with ultrasonics alone. However, the treatment was quite different in frequency (400,000 cycles/second) and in other factors from the sonic exposures reported here.

* This work was performed under Contract No. W-7405-Eng-26. 1Bergmann, L., Ultrasonics, Their Scientific and Technical Applications, John Wiley

& Sons, New York, 1938, Chapter V. 2 Brumfield, R. T. PROC. NAT. ACAD. SCI. 29, 190-193 (1943). 3 Lea D. E., Actions of Radiation on Living Cells, Macmillan Co., New York, 1947. 4 Sax, K., Genetics, 32, 75-78 (1947). Sax, K., PROC. NAT. ACAD. SCI., 29, 18-21 (1943).

6 Sax, K., and Swanson, C. P., Am. J. Bot., 29, 52-59 (1941). 7 Shropshire, R. F., J. Bact., 53, 685-693 (1947). 8 Swanson, C. P., Genetics, 29, 61-68 (1944). 9 Swanson, C. P., and Hollaender, A., PROC. NAT. ACAD. SCI., 32, 295-302 (1946).

10 Wallace, R. H., Bushwell, R. J., and Newcomer, E. H., Science, 107, 577-578 (1948).

RA YLEIGII-RITZ AND A. WEINSTEIN METHIODS FOR APPROXI- MA TION OF EIGENVAL UES.* I. OPERA'TORS IN A HILBERT

SPA CE

BY N. ARONSZAJN

HARVARD GRADUATE, SCHOOL OF ENGINEERING

Communicated by Marston Morse, July 23, 1948

The well-known method of Rayleigh-Ritz' permits us to find upper bounds for the eigenvalues of a differential operator. In the late thirties, A. Weinstein introduced a new method, in some cases important for

applications, giving lower bounds for the eigenvalues.2 We were able to extend Weinstein's method, making it more precise and

simplifying it at the same time by the use of the Hilbert space theory. This method is seen then as a counterpart of the Rayleigh-Ritz method,

or more precisely, of a generalized Rayleigh-Ritz method for which similar

developments to Weinstein's method can be established. In the present paper we develop the theory for completely continuous,.

symmetric operators in a Hilbert space. In another paper we shall study the applications of this'theory to differential self-adjoint operators.

474 MA THEMA TICS: N. ARONSZAJN PROC. N. A. S.

The sonic treatment alone did not cause aberrations. The increased yield is probably due to an increased movement of chromosomes and chromosome

fragments caused by, the sonic treatment, resulting in a decrease in the amount of restitution and an increase in the amount of detectable new reunions between the broken ends of chromosomes.

t A recent experimentl1 reports the production of chromosomal aberrations, mutations and other effects in Allium root tips, Helianthus stem apices, and Drosophila by treatment with ultrasonics alone. However, the treatment was quite different in frequency (400,000 cycles/second) and in other factors from the sonic exposures reported here.

* This work was performed under Contract No. W-7405-Eng-26. 1Bergmann, L., Ultrasonics, Their Scientific and Technical Applications, John Wiley

& Sons, New York, 1938, Chapter V. 2 Brumfield, R. T. PROC. NAT. ACAD. SCI. 29, 190-193 (1943). 3 Lea D. E., Actions of Radiation on Living Cells, Macmillan Co., New York, 1947. 4 Sax, K., Genetics, 32, 75-78 (1947). Sax, K., PROC. NAT. ACAD. SCI., 29, 18-21 (1943).

6 Sax, K., and Swanson, C. P., Am. J. Bot., 29, 52-59 (1941). 7 Shropshire, R. F., J. Bact., 53, 685-693 (1947). 8 Swanson, C. P., Genetics, 29, 61-68 (1944). 9 Swanson, C. P., and Hollaender, A., PROC. NAT. ACAD. SCI., 32, 295-302 (1946).

10 Wallace, R. H., Bushwell, R. J., and Newcomer, E. H., Science, 107, 577-578 (1948).

RA YLEIGII-RITZ AND A. WEINSTEIN METHIODS FOR APPROXI- MA TION OF EIGENVAL UES.* I. OPERA'TORS IN A HILBERT

SPA CE

BY N. ARONSZAJN

HARVARD GRADUATE, SCHOOL OF ENGINEERING

Communicated by Marston Morse, July 23, 1948

The well-known method of Rayleigh-Ritz' permits us to find upper bounds for the eigenvalues of a differential operator. In the late thirties, A. Weinstein introduced a new method, in some cases important for

applications, giving lower bounds for the eigenvalues.2 We were able to extend Weinstein's method, making it more precise and

simplifying it at the same time by the use of the Hilbert space theory. This method is seen then as a counterpart of the Rayleigh-Ritz method,

or more precisely, of a generalized Rayleigh-Ritz method for which similar

developments to Weinstein's method can be established. In the present paper we develop the theory for completely continuous,.

symmetric operators in a Hilbert space. In another paper we shall study the applications of this'theory to differential self-adjoint operators.



VOL. 34, 1948 MATHEMATICS: N. ARONSZAJN 475

1. Consider in a Hilbert space x, with the scalar product (x, y), an

operator H, symmetric and completely continuous. For any linear closed subspace ? c Sa we denote by P the corresponding

projection. The operator L = PH, considered in the subspace ?, will be called

the part of II in ?. It is a symmetric and completely continuous operator (in ?). If H is positive, L is positive too.

The eigenvalues, eigenvectors, resolvent operator, etc., of L (in ?) will be called the eigenvalues, eigenvectors, resolvent operator, etc., of H in ?.

An eigenvector u and the corresponding eigenvalue X of H in ? satisfy the equation

Hu- u = p with p - . (1)

The resolvent operator Rx of H in ?, for anyf e ? satisfy the equation

-HRxf - XRxf = f + p for p - ?. (2)

2. The eigenvalues of H in ? can be defined by maximum-minimum

problems for the positive eigenvalues and by minimum-maximum problems for the negative eigenvalues.3 This is done in the following way. We define

(Htu, u) Q(u) = for u # 0 (3)

(u, u)

Q(0) = 0

and then for any set of n vectors (n = 0, 1, 2, . . .), 50p, 2, ... .n of aC we consider the maximum (or minimum) of Q(u) for all vectors u e ? satisfying the conditions (u, (k) = 0, k = 1, 2, .. , n. This maximum (or minimum) is attained by some vector and will be denoted by Xn{pjk} (or ,n{(Pk})- If we vary the n vectors ?k, the Xn { k} will attain its minimum X, which is the nth positive eigenvalue of H in ?. In the same way, the minimum n{ Ynk}, for varying Pk, will attain its Inaximum unB which is the nth negative eigenvalue of H in ?. In this way we get the positive and negative spectra of H in ?:

Xo X1 ... -* 0 positive spectrum, /o0 / ' ... -*> 0 negative spectrum.

In this definition we do not exclude the vector u = 0. This fact makes it necessary to accept the following convention. If there are only a finite number of positive (or negative) eigenvalues, then the positive (or negative) spectrum has to be completed by an infinite number of zeros so that the sequences (4) will always have an infinite number of terms.4 The eigenvectors corresponding to the eigenvalues (4) will be denoted by



47C AL4 THEMA TICS: N. ARONSZAJN PROC. N. A. S.

u, u -, ... (5)

Uo, Ul ....

When, following our convention, we continue the positive (or negative) spectrum by zeros, the corresponding uZ or u will be equal to zero. All the other eigenvectors will be supposed normalized (11 u+ II = -I u, II - 1). The operator H may admit in ? the eigenvalue 0. A corresponding complete system of orthonormal eigenvectors will be denoted by

0 0 0 u1, U2, u, ..., (5')

3. The following two theorems are well known.5 THEOREM A. If ? C ?' then for the corresponding eigenvalues we have

Xk Xk, /,k ? u, k = 0, , 2, ... THEOREM B. If ? C ?' and ?" 0 ? is n-dimensional, then

Xk > x x .+n. /k 2 < a ,-+ k = 0, , ....

The following theorem seems new even in the case of an n-dimensional

space, n < co

THEOREM I. If ?' C ?, ?" = ? 0 ?', we have the inequalities:

Xi+j +- o 0 A X. + i, /ji

+ Xo > ,/ + 1, for i, j = 0, , , ....

An interesting special case of this theorem is COROLLARY I'. Under the assumptions of Theorem I, if H is positive

definite we have X,+j < X + x'. A sequence of subspaces ?(n) is said to converge to a subspace ? if for

every u e C, p(n) u -> pu. We then have the theorem THEOREM II. If ?(n) -- ? then X) - Xk, 2(n) --> Ek. If ?(n) is an increasing (or decreasing) sequence of subspaces, then it

converges to the smallest closed subspace ? containing all of them (or the intersection ? of all of them). It follows from theorems A and II that

COROLLARY II'. If (n) / ? (or ?( \ ?), X(' / Xk, () X Y (or

kX X,, k 7 / k). 4. Consider two subspaces, ?' c ? with ? 0 ?' n-dimensional.

Consider further, a system of n vectors pi, P2, . . , Pn generating the sub-

space ? 0 ?', the resolvent operator Rx of H in ? and

Un(X) = Rxpn, m = 1, 2, ..., n, X any complex number. (6)

We will denote by W(1) (Weinstein's determinant) the nth order determi-

nant

W(s) = det. {()Um(), Pk)} (7)

and by P - rP{p}, the Gramm's determinant

r{p } = det. {(p,, pk)}. (8)



VOL. 34, 1948 MATHEMATICS: N. ARONSZAJN 477

We have then the following fundamental theorem.

THEOREM III. The function m(D) = ,W(} is dependent only on the

operator H, and the subspaces ? and ?'. '(I) is meromorphic in the whole plane v with the exception of v = 0 (where it may have an essential singularity), is regular at =- oo and representable by the product

I n1 =

k n n 4) (9) v k=O o(D - ) k=O(0 - /)

where Xk, /k, Xk, /r are the eigenvalues of H in ,? and ?', respectively. Consider now the projection P' on ?' and the resolvent operator Rx of

H in ?'. We write

wk(X) = -RxP'IHp. (10)

Denoting by )D() the determinant of nth order

D(v) = det. {(Iwk + IHpk - P, pi), (11)

We have

THEOREM IV. The function () = r{- is given by

1() O (r - k=) -O (' - X- )

5. The proof of theorems III and IV is based on a few lemmas. We use first

LEMMA 1. The function I(~) of (9) does not depend on the choice of vectors

{pk} generating ? e ?'. This lemma allows us to limit ourselves to the case of orthonormal

vectors pk. In writing

_(q) - (13)

we can state

LEMMA 2. If ?' C ?" C ?, then ' F, ? P?,, ? - 1= 2?p. This lemma is proved firstly in the case when ? O 2? is one dimensional

and the general case follows immediately.

Following these two lemmas, the proof of Theorem I is reduced to the case when 2? ?' is one dimensional. In this case we have only one

vector, pi, 11 P ]1 =I 1, and using the spectral representation of the resolvent

operator, we obtain

OD I o t+12 o | |2 1 CO

4(r) = +E I Z- E +

laki2, (14) k=o k - S k/ =o - Sk=i

where a+ = (P, a = (), a ), a (pi u2).



478 -MATHEMATICS: N. ARONSZAJN PROC. N. A. S.

We use then the following lemma on analytic functions. LEMMA 3. Any meromqrphic function of the form

ao .+ co

(v) - + + E - (15) v k+ =o ko -" k=-o -

where y, > 0, y > O, X > O, y + ZE(Ty + - y) = 0, C > C > ... ->

wo < wo < ... -- O, is representable in the form

II - H _Co- Co(16) +r ?r- ? k o r

-

where oAk > rk > wk+i, C < r0 < wC+,. Inversely, every function of the

form (16) is representable in theform (15). Both representations are unique, the r+ and rk are the zeros of 4 and we have the formulas:

n k I, + -t

Tk

+ H + 41 4 Cok CO k C

+ + + + C1 kfl c ~

k1 - w k

oI col -~ CWk kFl CW - Cok

This lemma gives us the representation (16), sinilar to (9) for the case

of n = 1. In order to obtain the exact formula (9) from (16) (for n = 1) we use the following lemma.

LEMMA 4. In the case of a one-dimensional ? 0 ?', for any positive (or negative) number ?', the difference of multiplicity of D' in {X}j and {X} (or

{/ }and {f/k }) is equal to -1, 0 or 1, depending on r' being a pole, an ordinary

point or a zero of 4(r). The term "multiplicity of `'" in a monotonic sequence {P2} means the

number of times D' appears in this sequence. After having proved Theorem III, we pass to Theorem IV. It is

sufficient to prove here that T = 1/4. This is achieved by comparing the expressions of the determinants W(() and D(v).

6. The problem in which we are interested is the calculation of the

eigenvalues of the operator H in a given subspace ?. On the basis of the

preceding theorems, we will develop two methods for the computation of

these eigenvalues. A. Weinstein's Method.-Suppose that for some subspace ?(0) D ?

we know the eigenvalues and eigenvectors of Hin ?(O), and suppose further,

that for a sequence of vectors, Pi, P2, ... of the subspace ?() 0e ?, we

know the vectors vk(X) = R()pk. We consider then the subspace ?(n)



VOL. 34, 1948 MATHEMATiCS: N. ARONSZAJN 479

(0') O (p(n) where (P(n) is the n-dimensional subspace generated by the n first vectors pk.

Our assumptions allow us to form explicitly the function Bn(g) =

?p(n) ?(0)(?), and by Theorem III, knowing the eigenvalues {X') and

{/()} we can find the eigenvalues IX('} and {lJ()} By Theorem A, we have

Xn ) h" ') >Xk /(n) > /2-n 1) ,

If the sequence {pk} is complete in the subspace 2() 0 ?, the subspaces f?() X ? and by Corollary II', X() Xk, and /() / Ak.

Remark: We can give here a practical rule for establishing the sequence {Xn} (for the /k) there is a similar rule). Consider the sequence of

positive poles wol > o2 > ... and of positive zeros ri > r2 > ... of Bn(~).

{cok is a subsequence of the sequence tX?) }, i.e., it is of the form t()'j, I 1, 2, Consider then the complementary terms X(?) for all k' 7 ki and order all these terms together with the zeros rk in one non-increasing

sequence. The obtained sequence will be X(n}. B. Rayleigh-Ritz Method.-Suppose now that for some subspace

2() c ? we know completely the eigenvalues, eigenvectors, resolvent

operator and the part of H in ?(0).

Consider then a sequence of vectors {Pk} contained in 2? 0 ?(0). We introduce the increasing sequence of subspaces ?(n) = ?(0) G (P(n), where

(p(n) is the n-dimensional subspace generated by the n first pk. By our

assumptions we can calculate the function Tn(~) =?0 ?o(n(~), and know-

ing the eigenvalues of H in 2(), we obtain, by Theorem IV, the eigenvalues of H in (n). By Theorem A, we have

X(n) <

(n+l)X < (rn) > ,,(n+l) %

If the pk form a complete system in ? 0O 20), we get Xln' / X,, and /2n) \

Uk. This method will be called the generalized Rayleigh-Ritz method.

The ordinary Rayleigh-Ritz method corresponds to the case when we take ?(0) = (0), i.e., the subspace containing only the zero vector. It is clear

that in this case all our assumptions will be fulfilled so that the ordinary

Rayleigh-Ritz method may always be applied. 7. It is clear now that the two methods are complementary in character.

The Rayleigh-Ritz method gives lower bounds for Xk, upper bounds for

/k2, whereas the Weinstein method gives upper bounds for Xk, lower bounds for ,1u. The combination of the two (even by the use of a small number of pk) may lead to quite good approximations of the eigenvalues of H in

?. A method may be developed for evaluating the errors made in taking Xk' instead of Xk. In the case of a definite positive operator we can use, for instance, Corollary I' of ? 3 and obtain inequalities of the type I Xn -

XSI q Xn , where Xn1} are the eigenvalues of H in the subspace 2(0) O ?



480 MAITHEMATICS: N. ARONSZAJN PROC . N. A. S.

for the Weinstein method and ? O ?(0) for the Rayleigh-Ritz method,

respectively. 8. Lemma 3 of ? 5 allows the complete solution of the following problem.

Given the eigenvalues {Xk} and {/k} of H in ?, find the characteristic

properties of the eigenvalues X,} and {i} of H in a subspace ?' c ?,

We will mention here only the solution of this problem in the case of a

positive definite H. We have then only the sequence Xo >Xi > X2 > . . ->

0. For Xo > X' > X2 > ... -- 0 to be the sequence of the eigenvalues of

Hin a subspace ?' c ?, it is necessary and sufficient that Xk < Xk for every k and that for some h = 1, 2, ..., the following condition be fulfilled

co t

Xo Xn,h Xn X'

, o, (17,)

n= 0 Xn

where X,1 =h max. (X', X,+h). It should be remarked that for two sequences { Xk} and {X } 0, satisfy-

ing the condition X, < X., the conditions (17k), for any values of h, are

always equivalent. From (17,) follows the condition

nH - 0. (18) 0 Xm

We can get more precise information about X' if we require them to be

the eigenvalues of H in a subspace ?' c ? such that ? )O ?' be n-dimen-

sional, n < co. Then the necessary and sufficient conditions are X, > X >D X ?+,, together with the condition (18).

* The results of the present and the following papers were obtained by the author

in 1943 as far as the positive definite operators are concerned. A resume of these results

was circulated in a few copies among a number of mathematicians, but were never

published. The extension to indefinite operators is much more recent. 1 Ritz, W., J. Reine Angew. Math., 135, 1 (1908); Courant, R. Bull. Am. Math. Soc.

49, 1 (1943). 2 Weinstein, A., Memorial des Sciences Math., 88 (1937). 3 Courant-Hilbert, Methoden der lrath. Physik, Vol. I, 2nd ed., Berlin, pp. 26-29,

112-113, 351. 4 This kind of conventiont was first introduced by H. Weyl, Math. Ann., 71, 443

(footnote), 1912. 5 These theorems are known especially for differential operators, cf. Courant-Hilbert,

loc. cit., p. 353. These operators in usual cases are the inverses of completely continuous

integral operators and consequently the inequalities in the theorems have to be inverted.



Documents

Rayleigh-Ritz and A. Weinstein Methods for Approximation of Eigenvalues. I. Operators in a Hilbert Space