The Interval Eigenvalue Problem

61

Short Communications ZAMM . 2. angew. Math.

Mech. 71 (1991) 1,61-64

Akademie-Verlag Berlin

DEW,A. S.

The Interval Eigenvalue Problem Characterization of the set of eigenvalues of a general interval matrix A' is introduced and criteria on the eigenpair (1,x) of Ax = Ax, A E A', are given. Upper and lower bounds of the eigenvalues are therefore found. In order to avoid cumbersome manipulations, the symmetric case is introduced first with the results generalized later on. I. I n t r o d u c t i o n

A great deal of work has been done in characterizing solutions of the linear interval equations

simulating a similar well known assumption in interval analysis [4], whereby setting A'x n b' 0 for the interval equations A'x = b' leads to the OETTLIand PRAGERresult in (3). The inequalities in (9) follow directly from (10) having that

+

A'x = [A'x - AA 1x1, ACx

Every eigenpair (4x) of AC + 6A, (6A(5 A A therefore satisfies the compatibility criterion (9). 111. B o u n d i n g r To facilitate our task, we study first the symmetric case (A' = (A')') with A simple. The inequality (8) provides a simple means for bounding r. Since each eigenpair ( 1 , ~ of ) (Ac 6 A ) x = Ax, 16AI 5 AA, satisfies

+

(1)

A'x = b'

in which A' and b' are respectively an interval matrix and an interval vector. OETTLIand PRAGER[9] have shown that x is a solution of (I), i.e. belonging to the set

+ AA 1x11

a maximum Max(1) 5

L and a minimum 1 of r are given respectively by

xTACx

+ lxlTAA 1x1 , X'X

X = { x : A x = b, A E A', b E b'}

+ Ab

(3)

where 1.1 denotes absolute values taken componentwise. AC and bC represent respectively the centre value of both A' and b', whereas AA and Ab are the matrix and vector of uncertainties, i.e. A' = [Ac - AA, AC + A A ] ,

b' = [bc - Ab, bC + Ab]

(4)

Methods have been established since then for obtaining upper and lower bounds for x. For a survey of these methods, the reader may consult the references [l, 3,6,8]. Some recent algorithms also compute the exact interval hull enclosing X; those of ROHN[ll] for instance. On the other hand, little is known regarding the algebraic eigenvalue problem (5)

A'x = I X

which possesses interesting properties and serves a wide range of applications in physics and engineering. In this paper, we show how to compute lower and upper bounds for 1,i = 1, ..., n. It will be assumed for simplicity that A is real, though the method applies to complex A, too.

XTX

The above two Rayleigh quotients yield respectiveIy for their maxima and minima the maximum and minimum eigenvalues of the whole spectrum [13]. To confine ourselves therefore to a particular Ai, i = 1, ..., n, we take it that the signs of the components of the vector xi are kept unchanged under all perturbations 6A. Denoting$ = diag(sgn (xi),..., sgn (xb)),xj 0, i.e. Sixi = [xi[> 0, it follows then from the Kuhn-Tucker theorem applied to the objective function

+

+ 1x1' AA 1x1)

F = Max {xTACx where XTX

= 1,

six > 0

(13)

that ACx + S'AAS'X - AX = 0

(14)

for a Lagrangian multiplier 1.Thus, we obtain

+

AA] is a real symmetric T h e o r e m 3.1: If A' = [Ac - AA, A' interval matrix, and S' = diag (sgn ( x f ) ,...,sgn (xi)), i = 1, ..., n, taken at A' is constant over A', then the eigenvalue li of A, A E A', ranges over the interval

Lf = [Ai(,'

- S' AA Si),Ii(AC

11. T h e s o l u t i o n set Example: For The basic problem is: Given a central matrix A' E R""",find for the interval matrix A' = { A , / A - ACI 5 AA} a description of the set of eigenvalues

r = {A E C, AX = AX, x

(A)zxTACx- lxlTAA 1x1

(2)

if and only if it satisfies ]ACx- bCI 5 AA 1x1

Min

0, A E A ' } .

A' =

[

[4,81 [4,121 [4, 121 [- 10, -21

]

+ Si A A S')],

i = 1,

...,n .

'

(6)

To do this, we consider the perturbed problem (Ac + 6 A ) x = AX

(7)

in which 16AJ=< AA. For each such a matrix 6A, there exists an eigenpair (A, x ) obtained by solving (7) which satisfies by (3) ~A'x

- AX^ 5 AA 1x1.

(8)

Whereas, by isolating real and imaginary parts of (7) denoted by r and y, respectively, the following inequalities are obtained:

They can also be reached upon setting A'x n Ix 9

0

AC has eigenvalues I = 10, -10 with eigenvectors x1 = (2, xz = (1, -2)T, thus S' = diag(1, l), Sz = diag(1, -I), and A C + S ' A A S 1 = [ 12 -212],

L,=16,

(15)

62

ZAMM . Z. angew. Math. Mech. 71 (1991) 1

Hence

1:

=

A:

[5.062, 161,

=

as well as normal matrices [lo]. It combines advantages of both the Rayleigh-quotient for approximating an eigenvalue and the inverseiteration method for updating an eigenvector. It runs as follows: Given a real symmetric matrix B, pick a unit vector xo, then for k = 0, 1,2, ...

[-16.892, -3.4031.

Net we conclude Corollary: For a symmetric interval matrix A', ifthe components 1,have equal signs over A', then

i)

of xi pertaining to some

1; = [Add),A i ( 4 1

bychoosinga,, = -4,a,, = l , L = 1 _+ 2j,j = l/-l.Still,A1has an eigenvalue with maximum real part situated at the point 5 + j 0 of the matrix in which a l z = aZ1 = -4. This does not violate theorem 3.1, since xi becomes complex for some A E A'; the eigenvectors therefore change sign. An important q u e s t i o n follows: What guarantees that the sign of X I remains invariant for all 6 A E [ - AA, AA]? The following theorem provides a sufficient bound for A A . It is proved for any real xi with A' not necessarily symmetric.

=

xkT~xk

ii) solve ( B - A,I) y = xk for y

(16)

where A = AC - AA, 2 = AC + A A as the lower and upper bounds of A'. A result which follows directly from theorem 3.1 upon taking s' = I . The reader should note that for a symmetric A', 1 could be complex, but this is precluded from the above theorem, like in the case

A,

iii) set xk+' = y/liyli

.

Repeat from step (i), end.

The sequences (A,, xk) tend to an eigenpair (L, x) of B as k + co. The proof of convergence relies upon the monotonicity of the residual [lo]. To implement the algorithm on our problem is direct. Set B = AC S' AA S' and take as xo the ith-eigenvectorof A', then perform two Rayleigh sequences. Example: For

of which AC has an eigenvalue 1, = 10 and a corresponding eigenvector x 1 = (2, -1, -2)T we seek to maximize Al over all possible SA E [ - A A , AA]. For this, we search for the eigenvalue of the matrix Ac+SIAAS1=[l; 9

; 11.

-3

-6

Theorem 3.2: If AA is such that The following results were reached after two iterations only: then AC and AC + &A,for all 16AI 5 AA, have eigenvectors with same signs for the components. T is the modal matrix of A', i.e. which brings i t to the diagonal form containing the eigenvalues under a similarity transformation. 8 represents the separation between the eigenvalues of A', that is

step

I

X

1 2

16.33333 16.34847

0.6738944 0.6738873

-0.3028739 -0.3029054

-0.6738944 -0.6738874

IV. C a s e of a general A' The above approach can be equally adapted to a general interval matrix A'. By forming inner products with the eigenrow of A in (7), we have

and cp is a bound on the shijis in the eigenvalues of A' under the perturbation 6A, it is taken as

/I IT-ll AA IT1 1 1 .

4J 5

(19)

P r o of: From the eigenvalue problem written for all eigenvalues and eigenvectors (Ac

+ 6A)(T+ 6T) = (T+ 6T)(A + 6A)

(20)

we get AST

+ 6AT + GAST=

T6A

+ 6TA + ST6A.

(21)

Premultiplying by T - ' and setting T - ' 6 T = C, we have AC - C A = - T - ' G A T -

T-'6A7C

+ 6.4 + C6A

(22)

IAC - CAI 2 ICI 0

(23)

we obtain

ICI

+ c p l + ICI 4~

(24)

(27)

+ tr ( A A [Re xy*l),

Re y*A'x

Min (nl) 2 Re y*ACx - tr ( A A [Re xy*l). Likewise Max (Ay)6 Im y*ACx

+ tr ( A A IIm xy*l),

Min (Ay) 2 Im y*A'x - tr ( A A IIm xy*l).

(29)

To obtain therefore Max (A'), we seek to maximize the function

+ y:ACx, + tr ( A A IXJ; + xyy:l)

subject to

+ y;x,

Shj(Y4rj

=

1,

y:x, - y;x, = 0 , > 0, k,.i = 1, ..., n ,

YykXyj)

(30)

sLj being the sign of the above bracket evaluated at A', that is

or that

+

+

_I 1T-'I A A IT1 IT-'/ AA IT1 IC[ 91. (25) After putting IT1 ICI 2 16TI as well as imposing 16TI 5 ITI, we reach finally the above result in its final form. Despite the fact that theorem 3.1 determines exact eigenvalue bounds, one has to solve two eigenvalue problems for each ii This . does not often appeal to workers accustomed to writing simple computer programs for their problems. For them, we recommend the Rayleigh Quotient algorithm valid for real symmetric matrices

(8- 9)ICI

Max (Ar)

y:x,

+ IT-'GATI

+ y*GAx

where x and y are respectively the eigenvector and reciprocal eigenvector of A' + 6 A and normalized such that y*x = 1. Thus

F = y:ACx,

and having that

8 ICI 5 IT-'6AT

Ay*x = y*ACx

s i j = sgn (yYf&

+

ykkxij). This results in the equations (Ac + AA 0 S') X : = Arx:

-

(Ac + A A 0 S') X :

+ 1{x:,

=

Xxf

(31)

Ayx;,

(ACT+ AAT 0 SiT)y: = Alyf + Lyy: , (ACT+ AAT SiT)yi = 1;~:- L y ~ f ,

(32)

Short Communications

63

where A, and 1, are Lagrange multipliers associated with the equality constraints in (30). The symbol denotes componentwise multiplication, i.e. the entries of C = A Bare ckj = akjbkj.We therefore have 0

0

+

Theorem 4.1: If A' = [Ac - AA, AC AA] is a real interval matrix and s t j in (31) evaluated at AC is constant over A', then the real part of the eigenvalue liof A E A' ranges over the interval

Afl

=

A;(AC + AA 0 S')],

[Ai(Ac - AA S'), 0

i = 1, ..., n . (33)

Max (2')can be obtained likewise by maximizing the function F = yTACxy- yiACx,

+ tr (AA Ixyy:

-

x,y:l)

(34)

V. A p e r t u r b a t i o n a p p r o a c h

In this section, we study the interval eigenvalue problem using a different approach. If A'exhibits a slight variation SA in its elements, then, by perturbation techniques, the eigenvalue ,Ii of AC will consequently exhibit a shift 6Ai given by

6Ai

= (y',

6Ax')

(37)

if diis simple, and where xi and y' are respectively the eigenvector and reciprocal eigenvector of ACcorresponding to Aiand normalized such that y'*x' = 1 [2]. To carry out therefore a sensitivity analysis of the eigenvalue .Ziwith respect to the elements of AC relies upon watching the signs of a;ii/aa:j, whereby a positive change 6akj will increase the value of 62' for positive 81i/aakj.The latter is obtained from (37) and

together with a set of similar constraints. One obtains (Ac

+ AA

0

S') xi = @xi

+ Xxf,

where Iy and 1' are Lagrange multipliers associated with (30). We therefore have

+

Theorem 4.2: If A' = [Ac - AA, A' AA] is a real interval matrix, with s t j in (35) evaluated at AC is constant over A', then the imaginary part of the eigenvalue liof A E A' ranges over the interval

ny' = [n;(AC- AA

S'), n;(AC + AA S')] ,

0

0

to, 01

6Ai =

C yfxj6akj

i = 1, ..., n .

(36)

I-3, - 11

having

(39)

k,j

=

+

[(ytkxfj y i k x i j ) f k,j

Example: For [O, 21

where aA/aakjis an n x n matrix, the elements of which are all zero except the element (k ,j ) substituted by with unity. Thus (38) finally reads

1/-1(yikxij - yikxtj)]6akj

(40)

for 6akj real. Applying therefore the above technique to the interval matrix A'reveals that for small enough AA, choosing hakj = dakjcontributes in increasing the real part of 612, whenever the sign of the quantity yikxtj + $,,xij is found positive. Obviously, the same is also true if the latter was found negative, and 6akj = -Aakj becomes the right choice. To obtain therefore an upper bound for l;(AC + 6A), for all 1SAI 5 AA, we make the following choice for 6akj, &akj= Aakj sgn (YfkXij

+ &Xtj)

141)

in virtue of (40), and coinciding with theorem 4.1. Likewise, the upper bound for nl(AC 6A), for all 16AI 5 AA can also be reached by letting

+

the first matrix has the eigenvalues 2:: = 0,n; = 2 + j , 2; = 2 - j . Suppose we wish to bound &(AC + 6A), for all 16AI 5 AA, then the choice of &akjaccording to (31) and based upon xf = (1 - j , 2 1 - 1 . - 1 2 .T ' -j, and y c lo^, m - &, 3 gives 2

0

-2

AC-AAoS'=

1

having & = 4.7346 and get

-2

by virtue of (40), and coinciding also with theorem 4.2.

Acknowledgement: The author wishes to thank Dr. A. NEUfor helpfull comments, as well as the referee for constructive criticism.

-1

= 0.2874.

(42)

C o n cl u s i o n : Most researchers on eigenvalue bounds treat the case of an "exact" eigenvalue problem and once the matrix contains intervals, the enclosures they get tend to overestimate the true eigenvalue ranges; and in particular when the intervals are wide. Our approach is new in this respect since it handles the interval situation directly and compute exact eigenvalue ranges.

-3

[I 1-11 -1

6akj = A a k j .sgn (ytkxij - & X f j )

MAIER

While if based upon (35), we References 1 ALEFELD, G.; HERZBERGER, J.: Introduction to interval computations. Academic Press, New York 1983. 2 DEIF,A.: Advanced matrix theory for scientists and engineers. Halsted Press div., Wiley, New York 1982, pp. 19 and Sec. 6.3. 3 DEIF, A,: Sensitivity analysis in linear systems. Springer-Verlag, Berlin and Heidelberg 1986.

having

1;

1%= 2.1753 and 1% = 0.0. Thus

= [0.2874,4.7346] = j[O, 2.17531

4 HANSEN, E.: On linear algebraic equations with interval coefficients. In HANSEN,E. (ed): Topics in interval analysis. Clarendon Press, Oxford 1969. 5 JAHN, K. : Eine Theorie der Gleichungssysteme mit Intervall-Koefizienten. ZAMM 54 (1974), 405 -412. 6 MOORE, R.: Interval analysis. Prentice-Hall, New Jersey 1966. 7 NICKEL, K.: Bounding eigenvectors of a symmetric matrix. In NICKEL,K. (ed.): Interval mathematics. Academic Press 1980. 8 NEUMAIER, A,: Linear interval equations. In NICKEL, K. (ed.): Interval mathematics. Springer-Verlag 1985.

64

ZAMM . Z. angew. Math. Mech. 71 (1991) 1

9 OETTLI,W.; PRAGER,W.: Compatibility of approximate solutions of linear equations with given error bounds for coeficients and right-hand sides. Numer. Math. 6 (1964), 405 - 409. 10 PARLETT,B.: The symmetric eigenvalue problem. Prentice-Hall, New Jersey 1980. 11 ROHN,J.: An algorithm for solving interval linear systems and inverting interval matrices. In NICKEL,K. (ed.):Freiburger Intervall-Berichte, 82/5, Universitat Freiburg 1982. 12 SMITH,B.; BOYLE,J.; IKBBE,Y.; KLEMA,V.; MOLER,C.: Matrix eigensystem routines EISPACK -guide. Springer-Verlaa Berlin 1970. . 13 STOER,J.; BULIRSCH,R.: Introduction to numerical analysis. Springer-Verlag, New York 1980, pp. 395ff. 14 SYMM, H.; WILKINSON, J.: Realistic error bounds for a simple eigenvalue and its associated eigenvector. Numer. Math. 35 (1980), 113 - 126.

Received May 31, 1988, revised March 9, 1989

Address: Prof. Dr. ASSEMDEIF, Dept. of Engineering Mathematics, Faculty of Engineering, Cairo University, Giza, Egypt and Faculty of Science, King Saud University, Riyadh 11451, Saudi Arabia

BOOK REVIEWS

White, N. (ed.), T h e o r y o f M a t r o i d s . Cambridge etc., Cambridge University Press 1986. XVII, 316 pp. ISBN 0-521-30937-9 (Encyclopedia of Mathematics and Its Application 26)

Halin, R., G r a p hen t heo r ie. Berlin, Akademie-Verlag 1989. 322 S., DM 79, -. ISBN 3-05-500405-1 (Mathematische Lehrbucher und Monographien, I. Abt. 40)

Given a set E and a family &? of subsets of E, the pair (E, &?)is called a matroid (on E ) if 1$. $, &? is an antichain in E , and for every X , Y 2 E, X c I:and B,, Bz E with X c B , and B , 5 X there exists B , E &? with X 5 B , & Y The concept of a matroid is one of the most important tools for giving insight into the common foundation of different mathematical areas. “It is a rare event that mathematicians, cozily ensconced in the world of established theories, should extract, by dint of pioneering work, some new gem that later generations will spend decades polishing and refining. The hard-won theory of matroids is one such instance. Rich in connections with mathematics, pure and applied, deeply rooted in the utmost reaches of combinatorial thinking, strongly motivated by the toughest combinatorial problems of our day, this theory has emerged as the proving ground of the idea that combinatorics, too, can yield to the power of systematic thinking. . . . Matroid theory is unique in mathematics in the number and variety of its equivalent axiom systems; this accounts in part for the versatility and applicability of the subject.” (From the Foreword by the Series Editor G.-C. ROTA.)The present book is the first volume of a three part comprehensive compendium on the topic (besides: Combinatorial Geometries (ibid. 1987), Advances in Matroid Theory (in preparation)) unifying contributions of several experts in the field. The volume in question presents the basic of the many facets of the theory in a exacting manner, it will give pleasure and profit to every reader who has some knowledge on linear algebra, projective geometry, graph theory and related areas and it is qualified for serving as a standard textbook on matroids. Contributions: Ch. 1: Examples and basic concepts (H. CRAPO), p. 1; Ch. 2: Axiom systems (G. NICOLETTI, N. WHITE)p. 29; Ch. 3: Lattices (U. FAIGLE),p. 45; Ch. 4: Basis-exchange properties (J. P. S. KUNG), p. 62; Ch. 5: Orthogonality (H. CRAPO),p. 76; Ch. 6: Graphs and series-parallel networks (J. OXLEY),p. 97; Ch. 7: Contributions (T. BRYLAWSKI), p. 127; c h . 8: Strong maps (J. P. S. KUNG),p. 224; Ch. 9: Weak maps (J. P. S. KUNG,H. Q. NGUYEN), p. 254; Ch. 10: Semimodular functions (H. Q. NGUYEN),p. 272; Appendix of matroid cryptomorphisms (T. BRYLAWSKI), p. 298. The chapters are related to each other and use a “common language”. Most of them contain exercises and references. A common index concludes the valuable volume. Berlin W. WESSEL

Das Buch ist eine uberarbeitete Neuauflage der zweibandigen Ausgabe von 1980/81. Bis auf wenige Erganzungen stimmen beide Ausgaben uberein. In den ersten drei Kapiteln werden grundlegende Begriffe und Konzepte eingefuhrt, verschiedene Darstellungsformen von Graphen angegeben und Resultate fur Baume diskutiert. Das vierte Kapitel ist einem Basissatz der Graphentheorie, dem Satz von Menger, gewidmet. Implikationen und Varianten werden behandelt. Kapitel 5 diskutiert sogenannte Reichhaltigkeitssatze, d. h. Bedingungen, unter denen gewisse Substrukturen in Graphen zu finden sind. Die beiden folgenden Kapitel befassen sich mit zwei Teilgebieten, die schon seit langer Zeit das Interesse vieler Forscher gefunden haben und fur die sehr schone Resultate existieren: Farbungsprobleme und planare Graphen. Die wichtigen Beziehungen zwischen Graphen und Matroiden sind in Kapitel 8 zu finden. Kapitel 9 beschaftigt sich mit Gruppen und Graphen. Die beiden abschlieBenden Kapitel diskutieren Zerlegungen von Graphen und Zusammenhangsprobleme, Fragestellungen, die auch fur die Entwicklung von Algorithmen von groBer Bedeutung sind. Diese kurze Inhaltsangabe belegt, daB die wichtigsten Teilgebiete der Graphentheorie behandelt werden. Dies geschieht im vorliegenden Buch sehr exakt und ausfuhrlich, was auch durch die umfangreiche Liste berucksichtigter Literatur belegt wird. Insbesondere ist anzumerken, da5 unendlichen Graphen ein breiter Raum gewidmet wird, was in den meisten anderen Biichern zu diesem Gebiet nicht der Fall ist. Zusammen mit einer umfangreichen Kollektion an Obungsaufgaben wird das notwendige Basiswissen zur erfolgreichen Beschaftigung mit der Graphentheorie bereitgestellt. Negativ mochte der Rezensent anmerken, daB die im Buch venvendete Notation fur ihn recht ungewohnt ist und vielleicht auch anderen Lesern das Studium des Buches etwas erschweren mag. Wie der Autor im Vorwort selbst schreibt, spiegelt das Buch im wesentlichen den Stand der Graphentheorie von etwa 1979 wider. Dem Nichtexperten wird dadurch nicht klar, auf welchen Gebieten mittlenveile wesentliche neue Resultate erzielt wurden und welche Schwerpunkte die moderne Graphentheorie setzt. Wie der Autor zu recht bemerkt, wurde es naturlich den Rahmen des Buches sprengen, die Literatur der letzten Jahre detailliert aufzuarbeiten. Vielleicht ware es aber fur eine zukunftige Neuauflage eine gute Idee, einen kurzen AbriB uber die Entwicklung der Graphentheorie in der jungsten Vergangenheit aus der Sicht des Autors zu geben oder eine kommentierte Bibliographie wichtiger neuerer Arbeiten anzufiigen. Zusammenfassend ist zu bemerken, daB das vorliegende Buch eine enonne Stoffvielfalt und -fiille bietet. Es ist abgesehen von den kleinen obigen Anmerkungen sowohl als Einfuhrung in die Graphentheorie als auch als Nachschlagewerk fur grundlegende Resultate in einer Vielzahl von Teilgebieten zu empfehlen. G. REINELT Augsburg

Siemons, J. (ed.), Surveys i n C o m b i n a t o r i c s , 1989. Cambridge etc., Cambridge University Press 1989. VIII, 217 pp.. & 17.50 P/b. ISBN 0-521-37823-0 (London Mathematical Society. Lecture Note Series 141) This year the twelfth British Combinatorial Conference is being held in Norwich. This volume contains the contributions of the principal speakers. They were invited to prepare a survey paper for this book and to deliver a lecture in an area of their expertise. (From the Preface)