An algorithm for symmetric generalized inverse eigenvalue problems

Transcription

1 Linear Algebra and its Applications 296 (999) 79±98 An algorithm for symmetric generalized inverse eigenvalue problems Hua Dai Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 26, People's Republic of hina Received 23 November 998; accepted 26 April 999 Submitted by P. Lancaster Abstract Using QR-like decomposition with column pivoting and least squares techniques, we propose a new and e cient algorithm for solving symmetric generalized inverse eigenvalue problems, and give its locally quadratic convergence analysis. We also present some numerical experiments which illustrate the behaviour of our algorithm. Ó 999 Elsevier Science Inc. All rights reserved. AMS classi cation: 65F5; 65H5 Keywords: Eigenvalue; Inverse problems; QR-like decomposition; Least squares; Gauss±Newton method. Introduction Let A c and B c be real n n twice continuously di erentiable symmetric matrix-valued functions depending on c ˆ c ;...;c n T 2R n, and B c be positive de nite whenever c 2 X, an open subset of R n. The generalized inverse eigenvalue problem(giep) under consideration is as follows. The research was supported by the National Natural Science Foundation of hina and the Jiangsu Province Natural Science Foundation /99/$ ± see front matter Ó 999 Elsevier Science Inc. All rights reserved. PII: S ( 9 9 ) 9-3

2 8 H. Dai / Linear Algebra and its Applications 296 (999) 79±98 GIEP. Given n real numbers k 6 k k n, nd c 2 R n such that the symmetric generalized eigenvalue problem A c x ˆ kb c x has the prescribed eigenvalues k ; k 2 ;... ; k n. The GIEP arises in various areas of applications, such as the discrete analogue of inverse Sturm±Liouville problems [2] and structural design [3]. A special case of the GIEP is obtained when A c and B c are de ned by A c ˆ A Xn c j A j ; B c ˆ I; jˆ where A ; A ;... ; A n are real symmetric n n matrices, and I is the identity matrix. This problem is known as the algebraic inverse eigenvalue problem. There is a large literature on various aspects of the existence theory as well as numerical methods for the problem. See, for example, [±4,,4,7,8,2,23± 25,27±3]. The well-known additive and multiplicative inverse eigenvalue problems [8,9] are also included in our formulation by taking A c ˆ A diag c ;... ; c n ; B c ˆ I and A c ˆ A; B c ˆ diag c ;... ; c n ; respectively, where A is an n n constant matrix, about which there has been considerable discussion. See, for example, [3] and the references contained therein. Recently, Dai and Lancaster [5] considered another special case of the GIEP, i.e., A c and B c are the following a ne families A c ˆ A Xn iˆ c i A i ; B c ˆ B Xn iˆ c i B i ; where A ; A ;... ; A n ; B ; B ;... ; B n are the real symmetric n n matrices, and B c is positive de nite whenever c 2 X. Let k c 6 k 2 c 6 6 k n c be the eigenvalues of the symmetric generalized eigenvalue problem A c x ˆ kb c x. Then there is a well-de ned map k : X 7! R n given by k c ˆ k c ;... ; k n c T. The numerical algorithm analysed in [5] is Newton's method for solving the following nonlinear system k c k ˆ ; 2 where k ˆ k ;... ; k n T. Each step in the numerical solution by Newton's method of the system (2), however, involves the complete solution of the generalized eigenvalue problem A c x ˆ kb c x. Based on determinant evaluations originating with Lancaster [5] and Biegler-Konig [2], an approach to the GIEP has been studied in [7] as well, but it is not computationally attractive for real symmetric matrices. When k ;... ; k n include multiple eigenvalues,

3 H. Dai / Linear Algebra and its Applications 296 (999) 79±98 8 however, the eigenvalues k c ;... ; k n c of the generalized eigenvalue problem A c x ˆ kb c x are not, in general, di erentiable [26] at a solution c. Furthermore, the eigenvectors are not unique, and they cannot generally be de ned to be continuous functions of c at c. The modi cation to the GIEP has been considered in [5], but the number of given eigenvalues and their multiplicities must satisfy a certain condition in the modi ed problem. In this paper, we propose a new, e cient and reliable algorithm for solving the GIEP, which consists of extension of ideas developed by [6,9,2]. The problem formulation and our algorithm do not need to be changed when multiple eigenvalues are present. The algorithm is suitable for both the distinct and multiple eigenvalue cases. Each step in the iterative process does not involve any solution of the generalized eigenvalue problem A c x ˆ kb c x. The paper is organized as follows. In Section 2 we recall some necessary di erentiability theory for QR-like decomposition of a matrix depending on several parameters. In Section 3 a new algorithm based on QR-like decomposition with column pivoting and least squares techniques is described, and its locally quadratic convergence analysis is given. In Section 4 some numerical examples are presented to illustrate the behaviour of our algorithm. For our consideration, we shall need the following notation. A solution to the GIEP will always be denoted by c. k k 2 denotes the Euclidean vector norm or induced spectral norm, and k k F the Frobenius matrix norm. For an n m matrix A ˆ a ;... ; a m Š, where a i is the ith column vector of A, we de ne a vector col A by col A ˆ a T ;... ; at m ŠT, and the norm kak w :ˆ max jˆ;...;m ka j k 2. The symbol denotes the Kronecker product of matrices. 2. QR-like decomposition and di erentiability Following Li [9], we rst de ne a QR-like decomposition of a real n n matrix A with index m 6 m 6 n to be a factorization A ˆ QR; R ˆ R R 2 ; 3 R 22 where Q 2 R nn is orthogonal, R is n m n m upper triangular, and R 22 is m m square. When m ˆ, this is just a classical QR decomposition of A. The existence of a QR-like decomposition is obvious. In fact, we need only construct a ``partial'' QR decomposition, see [], for example. In general, however, it is not unique as the following theorem shows. Theorem 2. (See [6,9]). Let A be an n n matrix whose first n m columns are linearly independent and let A ˆ QR be a QR-like decomposition with index m. Then A ˆ bqbr is also a QR-like decomposition with index m if and only if

4 82 H. Dai / Linear Algebra and its Applications 296 (999) 79±98 Q ˆ bqd; R ˆ D T br; 4 where D ˆ diag D ; D 22 ; D is an orthogonal diagonal matrix, and D 22 is an m m orthogonal matrix. The following is a theorem on perturbation of QR-like decomposition. Theorem 2.2 (See [6]). Let 2 R nn have its first n m columns linearly independent and let ˆ Q R be a QR-like decomposition with index m. Let 2 2 R nn be any matrix satisfying k 2 k 2 < e. Then, for sufficiently small e, 2 has a QR-like decomposition with index m, 2 ˆ Q 2 R 2, such that kq Q 2 k 2 6 j e; kr R 2 k 2 6 j 2 e 5 where j ; j 2 are constants independent on 2. Note that the linear independence hypothesis ensures that the R blocks of R and br are nonsingular. In order to make the submatrix R of QR-like decomposition nonsingular, we admit a permutation of the columns of matrix A. The QR-like decomposition with column pivoting of A 2 R nn may be expressed as AP ˆ QR; 6 where P is an n n permutation matrix, and R is similar to Eq. (3). If rank A ˆ n m, then the permutation matrix P can be chosen such that the rst n m columns of the matrix AP are linearly independent and the block upper triangular matrix R satis es je T Re j P je T 2 Re 2j P P je T n m Re n mj > ; R 22 ˆ : 7 Now let A c ˆ a ij c 2 R nn be a continuously di erentiable matrix-valued function of c 2 R n. Here, the di erentiability of A c with respect to c means, for any c 2 R n, we have A c ˆ A c Xn jˆ where c ˆ c ;... ; c n T ; c ˆ c ;... ; c n T, and oa c ˆ oa ij c j oc cˆc : j oa c c j c j o kc c k oc 2 ; 8 j Note that if A c is twice di erentiable, then o kc c k 2 in replaced by O kc c k 2 2. (8) may be

5 H. Dai / Linear Algebra and its Applications 296 (999) 79±98 83 The next result which follows from Theorem 3.2 in [6] concerns the existence of a locally smooth QR-like decomposition of A c. Theorem 2.3. Let A c 2 R nn be twice continuously differentiable at c 2 R n and assume that rank A c P n m. Let P be a permutation matrix such that the first n m columns of A c P are linearly independent, and! A c P ˆ Q R ; R ˆ R R 2 9 R 22 be a QR-like decomposition of A c P with index m. Then there exists a neighbourhood N c of c in R n such that, for any c 2 N c, there is a QRlike decomposition of A c P with index m A c P ˆ Q c R c ; R c ˆ R c R 2 c R 22 c with the following properties:. Q c ˆ Q ; R c ˆ R. 2. All elements of Q c and R c are continuous in N c. 3. R 22 c and the diagonal elements r jj c ; j ˆ ;... ; n m, of R c are continuously differentiable at c. Moreover, if we write Q T oa c P ˆ Aj; A j;2 A j;2 A j;22 ; A j; 2 R n m n m ; j ˆ ;... ; n then R 22 c ˆ R 22 Xn jˆ A j;22 A j;2 R R 2 c j c j O kc c k 2 2 : 2 3. A new algorithm We rst present a new formulation of the GIEP, which is an extension of ideas developed in [6,9,2]. For convenience, we assume that only the rst eigenvalue is multiple, with multiplicity m, i.e., k ˆ ˆ k m < k m < < k n : 3 There is no di culty in generalizing all our results to an arbitrary set of given eigenvalues.

6 84 H. Dai / Linear Algebra and its Applications 296 (999) 79±98 ompute QR-like decomposition with column pivoting of A c k B c with index m A c k B c P c ˆ Q c R c ; R c ˆ R c R 2 c! ; 4 R 22 c where R c 2 R n m n m, and n m QR decompositions with column pivoting of A c k i B c i ˆ m ;... ; n A c k i B c P i c ˆ Q i c R i c ; R i c ˆ R i c R i 2 c! r nn i c ; 5 i ˆ m ;... ; n; where R i c 2 R n n. We assume permutation matrices P i c 2 R nn i ˆ ; m ;... ; n are constant matrices in a su ciently small neighbourhood of c for each i. If column pivoting is performed such that det R c 6ˆ ; jet R c e j P je T 2 R c e 2 j P P je T n m R c e n m j P kr 22 c k w 6 and je T R i c e j P je T 2 R i c e 2 j P P je T n R i c e n j ˆ jr i nn c j; i ˆ m ;... ; n 7 then the symmetric generalized eigenvalue problem A c x ˆ kb c x has the eigenvalues k ; k m ;... ; k n in which k is a multiple eigenvalue with multiplicity m if and only if R 22 c ˆ ; r nn i c ˆ ; i ˆ m ;... ; n: 8 Thus we consider solving the following least squares problem minimize F c with ( F c ˆ 2 kr 22 c k2 F Xn r i nn ): c 2 9 iˆm In fact, the GIEP has a solution c precisely when the function F c de ned by (9) has the minimal value zero at c. It is worth noting that F c may not be uniquely determined for any c because of the nonuniqueness of QR-like and QR decompositions. However, we

7 shall show that such nonuniqueness does not a ect the e ectiveness of our algorithm (see Theorem 3.). If m ˆ, i.e., the given eigenvalues k ; k 2 ;... ; k n are distinct, we may consider solving the nonlinear system r nn c r nn 2 c B A ˆ : 2 r nn n c The formulation (2) has been studied by Li [2] in the case of the algebraic inverse eigenvalue problem. If m ˆ and a solution of the GIEP exists, (9) and (2) are equivalent. Let c k be su ciently close to c. It follows from Theorem 2.3 that the matrix-valued function R 22 c and n m functions r i nn c i ˆ m ;... ; n are continuously di erentiable at c k, and R 22 c and r i nn c i ˆ m ;... ; n can be expressed as c k Xn where R 22 r i nn Q T Q T i c ˆ R 22 jˆ ˆ R 22 c k Xn c ˆ r i nn jˆ or 22 c k h O kc c k k 2 2 c k Xn jˆ ˆ r i nn c k Xn h oa c c h oa c c H. Dai / Linear Algebra and its Applications 296 (999) 79±98 85 jˆ T j;22 c k or nn i c k h t i j;22 c k c j c k j O kc c k k 2 2 i T j;2 c k R c k R 2 c k c j c k j t i j;2 O kc c k k 2 2 c R c k i k R i 2 c k i c j c k j 2 c j c k j O kc c k k 2 2 ; i ˆ m ;... ; n; 22 k ob c k i ob c i P c ˆ i P i c ˆ T j; c T j;2 c T j;2 c T j;22 c T i j; c t i j;2 c t i j;2 c t i j;22 c A; T j; c 2 R n m n m ; A; T i j; c 2 R n n : 23

8 86 H. Dai / Linear Algebra and its Applications 296 (999) 79±98 Let col R 22 c r nn m c f c ˆ B A : r nn n c 24 Then F c ˆ f T c f c : 25 2 We apply the Gauss±Newton method (see [2]) to solve the least squares problem (9). By use of (2) and (22), one step of Gauss±Newton method for the solution of (9) has the following form J T f c k J f c k c k c k ˆ J T f c k f c k ; 26 where J f c ˆ B col or 22 c or m nn c oc or n nn c oc oc col or 22 c oc n or m nn c oc n or n nn c oc n A 27 with or 22 c ˆ T j;22 c T j;2 c R c R 2 c ; or nn i c ˆ t i ;22 c t i j;2 c R i c R i 2 c ; j i ˆ m ;... ; n: 28 Thus the new method for solving the GIEP may be summarized as follows. Algorithm 3... hoose an initial approximation c to c, and for k ˆ ; ; ompute A c k k i B c k i ˆ ; m ;... ; n and oa c ob c k i ; i ˆ ; m ;... ; n; j ˆ ;... ; n : 3. ompute QR-like decomposition with column pivoting of A c k k B c k with index m:

9 A c k k B c k P c k ˆ Q c k R c k ; R c k ˆ R c k R 2 c k R 22 c k and n m QR decompositions with column pivoting of A c k k i B c k i ˆ m ;... ; n : A A c k k i B c k P i c k ˆ Q i c k R i c k ; R i c k ˆ H. Dai / Linear Algebra and its Applications 296 (999) 79±98 87! R i c k R i 2 c k : r nn i c k 4. If s kr 22 c k k 2 F Xn r nn c i k 2 iˆm is small enough stop; Otherwise. 5. Form f c k and J f c k using (24) and (27). 6. Find c k by solving linear system (26). 7. Go to 2. We rst compare the computational requirements of Algorithm 3. and algorithms in [5]. Since in all the algorithms for solving GIEP, Step 2 and 6 are indispensable, our comparison does not include the computational requirements for Step 2 and 6. It is well-known that the QR decomposition with column pivoting for each A c k k i B c k requires 2=3n 3 ops, while the QR-like decomposition with column pivoting of A c k k B c k with index m requires about 2=3n 2 n m ops (see, for example, []). Therefore Step 3 requires approximately 2=3 n 3 n 2 n m ops. It is easy to verify that Step 5 requires approximately n 4 ops. Thus Algorithm 3. requires approximately n 4 2=3 n 3 n 2 n m ops per iteration, while algorithms in [5] require only about n 4 8n 3 ops per iteration. However, our numerical experiments showed that Algorithm 3. took generally less iterations than algorithms in [5]. In the next section we will comment on the numerical behaviour of the algorithms. Now we prove that the iterates c k generated by Algorithm 3. do not vary with di erent QR-like decompositions of A c k k B c k P c k and different QR decompositions of A c k k i B c k P i c k i ˆ m ;... ; n.

10 88 H. Dai / Linear Algebra and its Applications 296 (999) 79±98 Theorem 3.. In Algorithm 3., for any fixed k, suppose A c k k B c k P c k ˆ Q c k R c k ˆ eq c k er c k ;! er er c k ˆ c k er 2 c k er 22 c k 29 are two (different) QR-like decompositions with column pivoting of A c k k B c k with index m, and A c k k i B c k P i c k ˆ Q i c k R i c k ˆ eq i c k er i c k ; er i c k i ˆ er c k er i 2 c k ; i ˆ m ;... ; n er i nn c k 3 are two (different) QR decompositions with column pivoting of A c k k i B c k i ˆ m ;... ; n, J f c k ; f c k and ej f c k ; e f c k are obtained in Step 5 of Algorithm 3. corresponding to two different decompositions of (29) and (3). Then J T f c k J f c k ˆ ej T f c k ej f c k ; 3 J T f c k f c k ˆ ej T f c k e f c k : 32 Proof. It follows from Theorem 2. that there exist a partitioned orthogonal matrix D ˆ diag D ; D 22 where D is an orthogonal diagonal matrix, D 22 is an m m orthogonal matrix, and n m orthogonal diagonal matrices D i ˆ diag d i ;... ; d i n where d i j ˆ i ˆ m ;... ; n such that Q c k ˆ eq c k D ; R c k ˆ D T e R c k ; Q i c k ˆ eq i c k D i ; R i c k ˆ D i er i c k ; i ˆ m ;... ; n: 33 By use of (23), (28) and (33), we have R 22 c k ˆ D T 22 e R 22 c k ; r i nn c k ˆ d i n er i nn c k ; or 22 c k ˆ D T o ~R 22 c k 22 ; j ˆ ;... ; n; or i nn c k ˆ d i n o~r i nn c k ; i ˆ m ;... ; n: 34 From (34) and the properties of the Kronecker product (see [6]), we obtain col R 22 c k ˆ I D T 22 col er 22 c k ; col or 22 c k ˆ I D T 22 col o ~R 22 c k ; j ˆ ;... ; n: 35

11 H. Dai / Linear Algebra and its Applications 296 (999) 79±98 89 From (24), (27), (34) and (35), the matrices J f c k, ej f c k and the vectors f c k, e f c k obtained in Algorithm 3. satisfy J f c k ˆ diag I D T 22 ; d m n ;... ; d n n ej f c k ; 36 f c k ˆ diag I D T 22 ; d m n ;... ; d n n e f c k : 37 Hence (3) and (32) hold. Although QR-like and QR decompositions are not unique, Theorem 3. shows that the iterates c k generated by Algorithm 3. do not depend on the used decompositions. Theorem 3.2. Suppose that the GIEP has a solution c, and that in Algorithm 3. P i c k ˆ P i c i ˆ ; m ;... ; n are independent on k when kc c k k 2 is sufficiently small. Assume also that J f c 2 R m2 n m n corresponding to a QRlike decomposition of A c k B c P c with index m and to QR decompositions of A c k i B c P i c i ˆ m ;... ; n is of full rank. Then Algorithm 3. is locally quadratically convergent. Proof. First form the QR-like decomposition of A c k B c P c with index m and n m QR decompositions of A c k i B c P i c i ˆ m ;... ; n A c k B c P c ˆ Q c R c ; A c k i B c P i c ˆ Q i c R i c ; i ˆ m ;... ; n: 38 Note that the matrix J f c corresponding to the decompositions (38), by assumption, has full rank, and that J T f c J f c is invertible. Assuming that kc c k k 2 is su ciently small, we can form a QR-like decomposition of A c k k B c k P c with index m and n m QR decompositions of A c k k i B c k P i c i ˆ m ;... ; n A c k k B c k P c ˆ eq c k er c k ; A c k k i B c k P i c ˆ eq i c k er i c k i ˆ m ;... ; n : 39 It follows from Theorem 2.2 that kq i c eq i c k k 2 6 j i e; kr i c er i c k k 2 6 j i 2 e; i ˆ ; m ;... ; n; 4

12 9 H. Dai / Linear Algebra and its Applications 296 (999) 79±98 where e ˆ max iˆ;m ;...;n fk A c A c k k i B c B c k ŠP i c k 2 g: orresponding to the decompositions (39), we obtain a matrix ej f c k 2 R m2 n m n. From the de nition of J f c and (4) we know that kj f c ej f c k k 2 is su ciently small, and so is kjf T c J f c ej f T c k ej f c k k 2 when c k is su ciently close to c. Therefore, ej f T c k ej f c k is invertible, ej f c k has full rank, and kej f c k k 2 is bounded. The QR-like decomposition and n m QR decompositions obtained in Algorithm 3. at c k are not necessarily (39). Write them 8 < A c k k B c k P c ˆ Q c k R c k ; 4 : A c k k i B c k P i c ˆ Q i c k R i c k i ˆ m ;... ; n : It follows from Theorem 4. and (36) that J f c k corresponding to the decompositions (4) also has full rank, kjf T c J f c Jf T c k J f c k k 2 is su ciently small, and kj f c k k 2 is bounded if kc c k k 2 is small enough. Using the perturbation theory for the inversion of a matrix (see, for example, [22]), we have h h J T f c k J f c i k 2 6 J T f c J f c i 2 x e 42 for the su ciently small kc c k k 2, where x e P is a continuous function of e and x ˆ. Now we use Theorem 2.3 to extend smoothly the decompositions (4) to a neighbourhood of c k which may be assumed to include c. Then, abbreviating (2) and (22) R 22 c ˆ R 22 c k Xn r i nn c ˆ r i nn c k Xn jˆ jˆ or 22 c k c j or nn i c k c j c k j O kc c k k 2 2 ; c k j O kc c k k 2 2 : But R 22 c ˆ, r nn i c ˆ i ˆ m ;... ; n, and we have f c k J f c k c c k ˆ O kc c k k 2 2 : Since kj f c k k 2 is bounded, then J T f c k J f c k c c k ˆ J T f c k f c k O kc c k k 2 2 : 43 omparing (43) with Eq. (26) de ning c k, we have J T f c k J f c k c c k ˆ O kc c k k 2 2 :

13 H. Dai / Linear Algebra and its Applications 296 (999) 79±98 9 It follows from this and (42) that kc c k k 2 ˆ O kc c k k 2 2 : as required. 4. Numerical experiments Now we present some of our numerical experiments with Algorithm 3., and also give a numerical comparison between Algorithm 3. and algorithms in [5] for our examples. The following tests were made on a SUN workstation. Double precision arithmetic was used throughout. The starting points were chosen close to the solution, so that few iterations were required for convergence. We were interested in verifying that locally quadratic convergence takes place in practice, in both the distinct and multiple eigenvalue cases. The iterations were stopped when the norm kf c k k 2 was less than 9. For convenience, all vectors will be written as row-vectors.when specifying a symmetric matrix we will only write its upper triangular part. Example 4.. This is a generalized inverse eigenvalue problem with distinct eigenvalues. Let n ˆ 5, A ˆ diag 9; ; ; 8; 4 ; B ˆ diag ; 3; 5; ; ; A ˆ B ˆ I; 2 A 2 ˆ ; B 2 ˆ B B A ; A A 3 ˆ B 3 2 A ; B 3 ˆ B A ;

14 92 H. Dai / Linear Algebra and its Applications 296 (999) 79±98 2 A 4 ˆ B A ; B 4 ˆ B A ; A 5 ˆ B A ˆ B 5; A c ˆ A Xn iˆ c i A i ; B c ˆ B Xn iˆ c i B i : The eigenvalues are prescribed to be k ˆ : ; : ; : ; : ; : With the starting point c ˆ :25; :5; :5; :95; :85, Algorithm 3. converges to a solution c ˆ ; ; ; ; and the results are displayed in Table. Example 4.2. This is a generalized inverse eigenvalue problem with multiple eigenvalues. Let n ˆ 6, Table Iteration Algorithm 3. Algorithm 2. in [5] kc c k k 2 kf c k k 2 kc c k k 2 kk k c k k 2 :34E :83E :34E :34E :4E :5E :97E :97E 3 2 :5E 2 :49E 3 :2E 2 :2E 4 3 :4E 6 :8E 7 :23E 6 :39E 8 4 :4E 2 :68E 4 :42E 3 :2E 4

15 26 889: :4 4:6 858:2 66:3 483:75 82:39 43:2 598:29 24:375 2:425 3:5 54:325 A ˆ 58: : :2 ; B 29:35 54:57 A 36:63 B ˆ I B ˆ diag 43: ; 43: ; 43: ; 43: ; 43: ; 43: u ˆ B H. Dai / Linear Algebra and its Applications 296 (999) 79± :5 :4 :2 : ; u 2 ˆ B A 2 :5 :2 u 5 ˆ B ; u 6 ˆ ; B 2 A A 2 ; u 3 ˆ B A 2 :5 :4 ; u 4 ˆ B A : : : : : : : : : V ˆ : : : ; B : : A : :5 ; A A k ˆ u k u T k ; B k ˆ X6 jˆk v k ;j e k e T j e je T k ; k ˆ ;... ; 6; A c ˆ A X6 iˆ c i A i ; B c ˆ B X6 iˆ c i B i : The eigenvalues are prescribed by k ˆ :5; :5; :5; 3: ; 9: ; 33: :

16 94 H. Dai / Linear Algebra and its Applications 296 (999) 79±98 With the starting point c ˆ 3; 4; 3; 4; ; 8, Algorithm 3. and Algorithm 2. in [5] converge the locally unique solution c ˆ 3: ; 4: ; 2: ; 3: ; : ; 7: : hoosing the target eigenvalues k ˆ :5; :5; :5 and using the same starting point c, Algorithm 5. in [5] also converges the same solution c. Table 2 displays the residuals. Example 4.3. This is also an example with multiple eigenvalues. Let n ˆ 8, A ˆ, B ˆ diag 99; 3; ; 5; ; 7; ; 8 ; B ˆ I; : : A ˆ ; :2 B 99 2:8 A 48: B ˆ : 9 B 22 A Now we de ne the matrices fa k g, fb k g, A c and B c from A and B as follows: A k ˆ Xk a kj e k e T j e je T k a kk e k e T k ; k ˆ ;... ; 8; jˆ B k ˆ X8 jˆk b k ;j e k e T j e je T k ; k ˆ 2;... ; 8;

17 H. Dai / Linear Algebra and its Applications 296 (999) 79±98 95 Table 2 Iteration Algorithm 3. Algorithm 2. in [5] Algorithm 5. in [5] kc c k k kc c k k kc c k k kf c k k 2 kk k c k k 2 kk k c k k 2 :E :62E 3 :E :46E :E :43E :E :26E 2 :79E :54E :34E :2E 2 :97E 3 :74E :26E :9E :34E :E 3 :62E 6 :42E 5 :3E :4E 2 :47E 3 :4E 3 4 :94E :86E :48E 5 :4E 5 :82E 7 :25E 7 5 :88E :7E :64E :8E 4 Table 3 Iteration Algorithm 3. Algorithm 2. in [5] Algorithm 5. in [5] kc c k k kc c k k kc c k k kf c k k 2 kk k c k k 2 kk k c k k 2 :28E :E 4 :28E :8E :28E :34E :36E 2 :39E :64E :49E :2E :76E 2 :87E 4 :8E :29E 2 :42E :23E 3 :E 2 3 :E 8 :5E 6 :24E 4 :23E 3 :5E 7 :28E 6 4 :93E 4 :3E :2E 8 :8E 7 :22E 3 :2E 2 5 :8E 3 :47E 2

18 96 H. Dai / Linear Algebra and its Applications 296 (999) 79±98 A c ˆ A Xn c i A i ; iˆ B c ˆ B Xn iˆ c i B i : The given eigenvalues are k ˆ 2; 2; 2; 3: ; : ; 8: ; 36: ; 723: : With the starting point c ˆ :99; :99; :99; :99; :; :; :; :, Algorithm 3. and Algorithm 2. in [5] converge to the exact solution c ˆ ; ; ; ; ; ; ; : Specifying one eigenvalue of multiplicity 3 and 2 distinct eigenvalues, i.e., k ˆ 2; 2; 2; 3: ; : and using the same starting point c, the solution found by Algorithm 5. in [5] is precisely c. The results are displayed in Table 3. Example 4.4. In this example we consider the usefulness of the GIEP in structural design. Fig. shows a -bar truss structure where Young's modulus E ˆ 6:95 N=m 2, weight density P ˆ 265 kg=m 3, l ˆ m, acceleration of gravity g ˆ 9:8 m=s 2, nonstructural mass at all nodes m ˆ 425 kg. The design parameters are the areas of cross section of the bars. Since the number of design parameters exceeds the order of the global sti ness and mass matrices by two, the areas of cross section of the 7th and 8th bars are xed with the values :865m 2 and :65m 2, respectively. The global sti ness and mass matrices of the structure can be expressed, respectively as Fig.. -bar Truss structure.

19 H. Dai / Linear Algebra and its Applications 296 (999) 79±98 97 Table 4 Iteration Algorithm 3. Algorithm 2. in [5] kc c k k 2 kf c k k 2 kc c k k 2 kk k c k k 2 :5E 4 :72E 4 :5E 4 :2E 4 :2E 4 :4E 3 :2E 4 :3E 2 2 :22E 6 :3E :77E 6 :8E 3 :2E 9 :23E 2 :26E 8 :46E 2 4 :2E 5 :87E 9 :33E 3 :56E 7 5 :7E 5 :74E A c ˆ A X8 c i A i ; iˆ B c ˆ B X8 iˆ c i B i where A ; B ; A i and B i i ˆ ;... ; 8 are 8 8 symmetric matrices, c ;... ; c 8 are the areas of cross section of the bars -6, 9,, respectively. The frequencies of the structure are prescribed to be x j ˆ 5j j ˆ ;... ; 8, i.e., the given eigenvalues are k j ˆ 2px j 2 j ˆ ;... ; 8. With the starting point c ˆ 3 :7; :4; :7; :6;, :7; :; :4; :, Algorithm 3. and Algorithm 2. in [5] converge to a solution c ˆ 3 :724; :423; :6994; :6; :756; :64; :439; :92 : The results are displayed in Table 4. These examples and our other numerical experiments with Algorithm 3. indicate that quadratic convergence indeed occurs in practice. We observed in most of our tests that Algorithm 3. took less iterations than algorithms in [5]. Acknowledgements The author would like to thank Professor Peter Lancaster for his careful reading of a preliminary version of this paper. The author is also indebted to the referees for their comments and suggestions. References [] F.W. Biegler-Konig, Su cient conditions for the solubility of inverse eigenvalue problems, Linear Algebra Appl. 4 (98) 89±. [2] F.W. Biegler-Konig, A Newton iteration process for inverse eigenvalue problems, Numer. Math. 37 (98) 349±354. [3] Z. Bohte, Numerical solution of the inverse algebraic eigenvalue problem, omput. J. (968) 385±388.

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