ON THE GENERAL ALGEBRAIC INVERSE EIGENVALUE PROBLEMS. 1. Introduction

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1 Journal of Computatonal Mathematcs, Vol.22, No.4, 2004, ON THE GENERAL ALGEBRAIC INVERSE EIGENVALUE PROBLEMS Yu-ha Zhang (Department of Mathematcs, Shong Unversty, Jnan , Chna) (ICMSEC, Academy of Mathematcs System Scences, Chnese Academy of Scences, Beng , Chna) Abstract A number of new results on suffcent condtons for the solvabty numercal algorthms of the followng general algebrac nverse egenvalue problem are obtaned: Gven n+1 real n n matrces A =(a ), A =(a () )( =1, 2,...,n)n dstnct real numbers λ 1,λ 2,...,λ n, fnd n real numbers c 1,c 2,...,c n such that the matrx A(c) =A + n c A has egenvalues λ 1,λ 2,...,λ n. Mathematcs subject classfcaton: 15A18, 34A55. Key words: Lnear algebra, Egenvalue problem, Inverse problem. 1. Introducton We are nterested n solvng the followng nverse egenvalue problems: Problem A(Addtve nverse egenvalue problem). Gven an n n real matrx A =(a ), n dstnct real numbers λ 1,λ 2,...,λ n, fnd a real n n dagonal matrx D = dag(c 1,c 2,...,c n ) such that the matrx A + D has egenvalues λ 1,λ 2,...,λ n. Problem M(Multpcatve nverse egenvalue problem). Gven an n n real matrx A = (a ), n dstnct real numbers λ 1,λ 2,...,λ n, fnd a real n n dagonal matrx D = dag(c 1,c 2,...,c n ) such that the matrx DA has egenvalues λ 1,λ 2,...,λ n. Problem G(General nverse egenvalue problem). Gven n +1 real n n matrces A = (a ), A = (a () )( = 1, 2,...,n)n dstnct real numbers λ 1,λ 2,...,λ n, fnd n real numbers c 1,c 2,...,c n such that the matrx A(c) =A + n c A has egenvalues λ 1,λ 2,...,λ n. Evdently Problem A M are specal cases of Problem G. The solutons of Problem G are compcated. A number of results on suffcent condtons for the solvabty, stabty analyss of soluton numercal algorthms of Problem G wth real symmetrc matrces can be found n [1,3,11,12,14,16,19,20,21,22]. These results are all obtaned by studyng the followng nonnear system λ (A(c)) = λ, =1, 2,...,n (1) where λ (A(c)) s the th egenvalue of A(c),or det(a(c) λ I)=0, =1, 2,...,n. (2) Most numercal algorthms depend heavy on the fact that the egenvalues of real symmetrc matrx are real valued, hence, can be totally ordered [13]. But non-symmetrc matrces have not the fact. Less results on non-symmetrc problems can be found. In ths paper, we Receved Apr 17, 2002.

2 568 Y.H. ZHANG use another approach to nvestgate Problem G. The man dea s to treat Problem G as the followng equvalent problem. A(c)T = T Λ (3) where Λ = dag(λ 1,λ 2,...,λ n )T s a non-sngular matrx. We see that the columns of T are the egenvectors of A(c). (3) s equvalent to a polynomal system(see Secton 2). It s not necessary to consder orderng egenvalues to solve the polynomal system. In Secton 2 t s proved that problem G s equvalent to a polynomal system. In Secton 3 by studyng the system wth the help of Brouwer s fxed pont theorem we obtan some new suffcent condtons on the solvabty, whch mprove the results n[1,3,5,8,9]. In Secton 4, we propose a nearly convergent teratve algorthm a quadratcally convergent teratve algorthm. Several examples are gven n ths paper. Throughout ths paper we use the followng notaton. Let R n n be the set of all n n real matrces. R n = R n 1.Let h () = j=1, a (), h = h (), H =(h () ) R n n. Obvously, H s a nonnegatve matrx. Let ρ(h) be the spectral radus of H. For a permutaton π of the n tems {1,...,n}, let s = a + (λ π() a π(),π() )a (), l = s,,j =1, 2,...,n, j (4) l = j=1, l, =1, 2,...,n (5) 2. Equvalent Polynomal System Wthout loss of generaty we can suppose that[1,3,8,9] a =0( =1,...,n)nProblemA, a =1( =1,...,n)nProblemM, a () = δ (, =1,...,n)nProblemG. Theorem 1. Problem G has a soluton c 1,c 2,...,c n R f only f there exsts a permutaton π of the n tems {1,...,n} such that the followng polynomal system (λ π(j) a c )t =(a + n c a () )+ n (a + n c a () )t lj,,j =1,...,n, j,j λ π() a c = n (a + n c a () )t, =1,...,n (6) has a soluton c R, t R (, j =1,...,n, j). Proof. Suppose Problem G has a soluton c =(c 1,c 2,...,c n ) T R n. Snce the egenvalues λ 1,λ 2,...,λ n of A(c) are all dfferent, the Jordan canoncal form of A(c) s Λ = dag(λ 1,λ 2,...,λ n ), therefore there exsts a nonsngular matrx S =(s ) C n n such that A(c) =SΛS 1, that s A(c)S = SΛ. (7) Notng that A(c) s a real matrx only wth real egenvalues, then the smarty matrx S can be taen to be real. Notce that S R n n s nonsngular, hence dets 0, then there exsts a

3 On the General Algebrac Inverse Egenvalue Problems 569 permutaton π of the n tems {1,...,n} such that n s,π() 0. Wthout loss of generaty we can suppose that s,π() =1( =1, 2,...,n). Let =1 P =(p ) R n n where P s a permutaton matrx. Let p π(),j = δ,,j =1,...,n. T =(t )=SP, Λ π = dag(λ π(1),λ π(2),...,λ π(n) ). Clearly, t = s,π(j),t =1(, j =1, 2,...,n), Λ π = P T ΛP. Hence, A(c)T = T Λ π. (8) It s easy to show that (6) (8) are equvalent. Conversely, there exsts a permutaton π such that the system (6) has a soluton c R, t R,, j =1, 2,...,n, j. Lett =1Λ π = dag(λ π(1),λ π(2),...,λ π(n) ). Then t s easy to show that T =(t ) R n n,c=(c 1,c 2,...,c n ) T R n satsfy (8), that s, λ ( =1,...,n) are all the egenvalues of A(c). Hence c 1,c 2,...,c n s a soluton to Problem G. Remar 1. Let x = λ π() a c, =1, 2,...,n. (9) Then (4) can be wrtten as (λ π(j) λ π() )t,j s t lj = s x a () x t x n, j =1, 2,...,n, j a () t lj, (10) x + n x a () t = n s t, =1, 2,...,n. (11) Applyng Theorem 1 to the addtve multpcatve nverse egenvalue problems, we get the followng corollares. Corollary 1. Problem A has a soluton c 1,c 2,...,c n R f only f there exsts a permutaton π of {1, 2,...,n} such that the followng polynomal system (λ π() λ π(j) )t + a t j = a +( a t )t,,j =1,...,n, j,,j c = λ π() n (12) a t j,=1,...,n j=1, has a soluton c R, t R,, j =1, 2,...,n, j. Corollary 2. Problem M has a soluton c 1,c 2,...,c n R f only f there exsts a permutaton π of {1, 2,...,n} such that the followng polynomal system (λ π(j) c )t = c (a + n a t lj ),,j =1,...,n, j,j λ π() = c (1 + n (13) a t ), =1,...,n

4 570 Y.H. ZHANG has a soluton c R, t R,, j =1, 2,...,n, j. 3. Suffcent Condtons for the Exstence of Real Solutons Theorem 2. For Problem G, suppose that a () = δ,,, 2,...,n (14) there exst a constant K>0 a permutaton π of {1, 2,...,n} such that where σ,=1, 2,...,n satsfy ρ(h) < 1/K, (15) ] λ π() λ π(j) ( 1 K +1)σ +( 1 [l K 1) + σ a () (16) σ = Kl + K, j =1, 2,...,n, j σ h (), =1, 2,...,n (17) Then there exsts c =(c 1,c 2,...,c n ) T R n wth c (λ π() a ) σ, =1, 2,...,n (18) such that the egenvalues of A(c) are λ 1,λ 2,...,λ n. The proofs of Theorem 2 wl be based on the followng lemmas. Lemma 1. Under the condtons of Theorem 2 there exsts only one nonnegatve vector (σ 1,σ 2,...,σ n ) T R n satsfyng (17). Proof. Let σ =(σ 1,σ 2,...,σ n ) T R n, l =(l 1,l 2,...,l n ) T R n. Then (17) s equvalent to σ = Kl + KHσ that s (I KH)σ = Kl If ρ(h) < 1/K, ρ(kh) < 1, then I KH s nvertble (I KH) 1 = K n H n n=1 Hence, Proof of Theorem 2. Let σ = K n+1 H n l 0. n=1 t =(t 12,t 13,...,t 1n,t 21,t 23,...,t 2n,...,t n1,t n2,...,t n,n 1 ) T R n2 n, Defne c =(c 1,c 2,...,c n ) T R n, x =(x 1,x 2,...,x n ) T R n. Ω= { } t R n2 n : t K,, j =1, 2,...,n, j. (19)

5 On the General Algebrac Inverse Egenvalue Problems 571 Obvously, Ω s a nonempty convex closed set n R n2 n.nowletf be the map wth wth f :Ω R n2 n f(t) = (F 12 (t),f 13 (t),...,f 1n (t),f 21 (t),f 23 (t),...,f 2n (t),...,f n1 (t),f n2 (t),...,f n,n 1 (t)) T (λ π(j) λ π() + x )F where x ( =1, 2,...,n)satsfy x + x n,j (s x a (), j =1, 2,...,n, j a () t = )F lj = s x a (), (20) s t, =1, 2,...,n. (21) We show that f(ω) Ω contnuous. Let t Ω, that s, t K,, j =1, 2,...,n, j. By (15) (16), we have x = n s t + x a () t K l + K x h (). (22) Then we have where x Kl + KH x (23) x =( x 1, x 2,..., x n ) T. Notng that (I KH) 1 > 0, hence x K(I KH) 1 l = σ. Suppose that p, q satsfy Then we have λ π(q) λ π(p) F pq = s pq l pq + l pq + t pq = t =max j t. x a () pq x p F pq + l=1, p,q x a () pq + x p F pq + σ a () pq + σ p F pq + (s pl l=1, p,q l=1, p,q x a () (l pl + (l pl + pl )F lq x a () pl ) F lq σ a () pl ) F pq.

6 572 Y.H. ZHANG Hence by (17) (16) we can get F pq = K. l pq + n λ π(q) λ π(p) σ p l pq + n σ a () pq l=1, p,q (l pl + n σ a () pq λ π(q) λ π(p) (1 + 1/K)σ p +(l pq + n σ a () pl ) (24) σ a () pq ) Let t Ω. By (15), (21) the mpct functon theorem, the vector-valued functon x :Ω Γ={x R n : x σ} s analytc. By (20) (16) the mpct functon theorem, the vector-valued functon f :Ω Γ Ω s analytc. By the chan rule, f :Ω Ωsanalytc. Then we have f(ω) Ω contnuous. By Brouwer s fxed pont theorem, f has a fxed pont n Ω. Hence, by Remar 1 Theorem 1, we can get Theorem 2. Applyng Theorem 2 to the addtve multpcatve nverse egenvalue problems, we get the followng corollares. Corollary 3. For Problem A, suppose that (25) a =0, =1, 2,...,n (26) there exst a constant K>0 a permutaton π of {1, 2,...,n} such that λ π() λ π(j) (K +1) Then there exsts D = dag(c 1,c 2,...,c n ) R n n wth c λ π() K a +( 1 K 1) a,,j =1, 2,...,n, j. (27) j=1, such that the egenvalues of A + D are λ 1,λ 2,...,λ n. Corollary 4. For Problem M, suppose that a, =1, 2,...,n (28) a =1, =1, 2,...,n (29) there exst a constant K>0 a permutaton π of {1, 2,...,n} such that g = a < 1, =1, 2,...,n (30) K λ π() λ π(j) λ π() 1 Kg j=1, (K +1) Then there exsts D = dag(c 1,c 2,...,c n ) R n n wth a +( 1 K 1) a,,j =1, 2,...,n, j (31) c λ π() K λ π() g 1 Kg, =1, 2,...,n (32)

7 On the General Algebrac Inverse Egenvalue Problems 573 such that the egenvalues of DA are λ 1,λ 2,...,λ n. Remar 2. In fact, K s the bound of the normazed egenvectors n Theorem 2, Corollary 3 Corollary 4. We can get many suffcent condtons on the solvabty by choosng dfferent values of K. Especally, lettng K = 1, we ( can obtan ) the results ( n [1,3,5,8]. ) ( ) Example 1. For λ 1 =4,λ 2 = 8, A=,A =,A =, consder Problem G. It can be verfed that f π(1) = 1, π(2) = 2, K = 0.8 thenσ 1 = , σ 2 = Applyng Theorem 2, we now that Problem G n ths example s solvable. In fact c 1 = , c 2 = We can t nfer the solvabty of Example 1 from the results n [1,3,5]. 4. Numercal Methods 4.1. A Lnearly Convergent Iteratve Algorthm Algorthm L. 1)Chooseastartngvaluet (0) =0foral, j =1, 2,...,n, j. Form =1, 2, 3,..., ) compute c (m), =1, 2,...,n by solvng the near system x (m) + n x (m) a () t (m 1) = n s t (m 1), =1, 2,...,n. (33) )compute t (m),,j=1, 2,...,n, j by solvng n near systems (λ π(j) λ π() + x (m) )t (m),j (s x (m) a () )t (m) lj for j=1,2,...,n. 2) Compute c = λ π() a x (m),=1, 2,...,n. The followng theorem s the man result of ths secton. Theorem 3. For Problem G, suppose that = s x (m) a (), =1, 2,...,n, j (34) a () = δ,,, 2,...,n (35) there exst a constant K>0 a permutaton π of {1, 2,...,n} such that [ λ π() λ π(j) max {( 1K +1)σ +( 1K 1) l + σ (l + where σ,τ =1, 2,...,n satsfy σ = Kl + K ρ(h) < 1/K, (36) σ a () )+(K +1)τ +(K 1), j =1, 2,...,n, j σ a () ], τ a () } (37) σ h (), =1, 2,...,n, (38)

8 574 Y.H. ZHANG τ = σ /K + K τ h (), =1, 2,...,n, (39) Then () there exsts c =(c 1,c 2,...,c n ) T R n wth c (λ π() a ) σ, =1, 2,...,n (40) such that the egenvalues of A(c) are λ 1,λ 2,...,λ n, () c 1,c 2,...,c n can be obtaned wth Algorthm L. The terates {t (m) } j generated by Algorthm L converge nearly to the unque soluton {t } j of the equton (6) wth max max t t (m) j t(m+1) t (m) ϱm j 1 ϱ 1 ϱ max j t(+1) t (), (41) m (42) where ϱ =max j (K +1)τ +(K 1) n τ a () σ /K λ π() λ π(j) (1 + 1/K)σ (l + n σ a ) (43) Proof. We use the notatons of the proof of Theorem 2. From the proof of Theorem 2, we now that F (Ω) Ω. It s suffcent to show that F s a contracton operator mappng Ω nto tself. Let t (1),t (2) Ω. We have (λ π(j) λ π() + x (1) )F (t (1) ) where x (1) ( =1, 2,...,n)satsfy x (1) + (λ π(j) λ π() + x (2) x (1) )F (t (2) ) where x (2) ( =1, 2,...,n)satsfy x (2) + x (2),j (s x (1) a(), j =1, 2,...,n, j a () t (1) =,j (s )F lj (t (1) )=s x (1) a() (44) s t (1), =1, 2,...,n (45) x (2) a(), j =1, 2,...,n, j a () t (2) = )F lj (t (2) )=s x (2) a() (46) s t (2), =1, 2,...,n. (47)

9 On the General Algebrac Inverse Egenvalue Problems 575 Subtractng (47) from (45), we can get = x (1) + (x (1) n ) n s (t (1) t (2) ) x (1) =1, 2,...,n. a () t (2) a () (t (1) t (2) ), (48) Then that s, we have K = K K (49) s equvalent to n where =( 1 1 that s, h() h() a () t (2) + n + t (1) t (2) ( s + ( s + a() ) t (1) t (2) σ a () ) + 1 K σ t (1) t (2), =1, 2,...,n, h() 1 K σ t (1) t (2), =1, 2,...,n. (49) (I KH) 1 K t(1) t (2) σ (50), x(1) 2 2,, x(1) n n ) T. Hence we have 1 K (I KH) 1 t (1) t (2) σ = t (1) t (2) τ, (51) Subtractng (44) from (46), we can get t (1) t (2) τ, =1, 2,,n. (52) (λ π(j) λ π() + x (1) )(F (t (1) ) F (t (2) )) (s x (1) a() )(F lj (t (1) ) F lj (t (2) )),j = (x (1) )F (t (2) ),j (x (1) for all, j =1, 2,,n,, j. Suppose that p, q satsfy F (1) pq F (2) pq =max j )a() F lj (t (2) ) F (1) F (2) = F (1) F (2). (x (1) )a() (53)

10 576 Y.H. ZHANG By (53) (52), we have λ π(q) λ π(p) Hence (Kτ p + K F (1) F (2) max j = max j l=1, p,q l=1, p,q l pl σ p l=1, p,q σ a () F pq (1) F (2) pl pq n τ a () pl + τ a () pq ) t(1) t (2). (54) Kτ + K n,j λ π(j) λ π() n,j τ a () + n l σ (K +1)τ +(K 1) n n τ a (),j τ a () σ /K λ π(j) λ π() (1 + 1/K)σ +(l + n = ϱ t (1) t (2) σ a () t (1) t (2) σ a () ) t (1) t (2) From (37) (43), we now that ϱ<1, t follows that F s a contracton wth contracton number ϱ. Now the statements of the theorem can be deduced from the Banach fxed pont theorem Newton s Method Algorthm N. 1) Choose a startng value x (0) x (m),t (m),,j=1, 2,,n, j. Form =1, 2,,M compute,,j=1, 2,,n, j by solvng the near system,t (0) (λ π(j) λ π() + x (m 1) )t (m) n (s n,j + n x (m) (a () + n a () t (m 1) lj ),j = s + x (m 1) t (m 1) + n + n = n x (m) x (m 1) x (m) a () t (m 1) x (m 1),j x (m 1) a () t (m 1) lj, a () t (m 1) n (s n a () )t (m) lj x (m 1), j =1, 2,,n j. + x (m) t (m 1) a () )t (m) 2) Compute c = λ π() a x (M), =1, 2,,n. Remar 3. By a stard argument (see [18]), t follows that the terates {c (m) } generated by Algorthm N converge quadratcally to the soluton {c } when a startng value {t(0) } j s suffcently close to the soluton of (6) Numercal Examples We have tested Algorthms descrbed n ths paper wth Matlab 5.3.

11 On the General Algebrac Inverse Egenvalue Problems 577 Examples 2 [22]. Ths s a general nverse egenvalue problem wth symmetrc matrces A = , A 1 = , A 2 = A 4 = , A 3 =, λ = 30, 10, 10, Let π() =, = 1, 2,, n. We have calculated ths example wth Algorthm L Algorthm N. We choose the startng pont wth zeros. After 10 teratons wth Algorthm L or after 4 teratons wth Algorthm N we obtan the numercal results as follows c = , , , λ (A(c)) = , , , After 5 3 teratons wth Algorthm L we obtan the numercal results as follows, respectvely. c = , , , c = , , , Wth Algorthm N, after 3 2 teratons we obtan the numercal results as follows, respectvely. c = , , , λ (A(c)) = , , , c = , , , λ (A(c)) = , , , Example 3. Ths s a general nverse egenvalue problem wth nonsymmetrc matrces A = , A 1 = , A 2 = A 4 = , A 3 = , λ = 30, 10, 10, 30.,,

12 578 Y.H. ZHANG Let π() =, = 1, 2,, n. We have calculated ths example wth Algorthm L Algorthm N. We choose the startng pont wth zeros. Wth Algorthm L, after 50 teratons we obtan the numercal results as follows c = , , , λ (A(c)) = , , , after teratons we obtan the numercal results as follows, respectvely. c = , , , λ (A(c)) = , , , c = , , , λ (A(c)) = , , , Wth Algorthm N, after 4,3 2 teratons we obtan the numercal results as follows, respectvely. c = , , , , λ (A(c)) = , , , , c = , , , , λ (A(c)) = , , , , c = , , , , λ (A(c)) = , , , Example 4 [11]. Consder Problem A wth symmetrc matrces. Let A = 2 0 4, λ =6, 3, It s easy to verfy that c =2, 1, 1 s an exact soluton of ths problem. Applyng Algorthm N to ths problem wth the startng pont wth x (0) =( 7.8, 3.7, 4.2) T t (0) beng the (, j) element of the egenmatrx wth dagonal elements beng 1 of A + dag(1.8, 0, 7, 1.2) we fnd M c (M) c 2 λ λ (M)

13 On the General Algebrac Inverse Egenvalue Problems 579 Here λ (M) denotes the vector of the egenvalues of A(c (M) ). Observe that the speed of convergence s sghtly faster than Algorthm n [11] t s the same as Algothm n [11]. In Alorthm n [11] all the egenvecors of A(c (M) ) have to be computed per step, whch s very tme consumng. In Algorthm N n ths paper n Algorthm n [11] only some near systems have to be solved per step, whch s less tme consumng. From these examples we fnd that the convergent speed of Algorthm L s much slower than Algorthm N, but t requres less operatons n each teraton. Acnowledgments. The author would e to express hs grattude to Professor Zhu Benren Professor Zhang Guanquan for helpng to arrange hs vsts to Insttute of Computatonal Mathematcs Scentfc/Engneerng Computng, Academy of Mathematcs System Scences, Chnese Academy of Scences for ther consstent encouragement support. References [1] F.W. Begler-Köng,Suffcent condtons for the solubty of nverse egenvalue problems, Lnear Algebra Appl., 40 (1981), [2] S. Fredl, Inverse egenvalue problems, Lnear Algebra Appl., 17 (1977), [3] K.P. Hadeler, Exstenz-und Endeutgetssätze für nverse Egenwertaufgaben mt Hfe des topologschen. Abbdungsgrades, Arch. Ratonal Mech. Anal., 42 (1971), [4] R. Horn C. Johnson, Matrx Analyss, Cambrdge U. P., New Yor, [5] Sun Jguang, On the suffcent condtons for the solubty of algebrac nverse egenvalue problems, Math. Numer. Snca, 9 (1987), [6] Sun Jguang Ye Qang, The unsolvabty of nverse algebrac egenvalue problems almost everywhere, J. Comput. Math., 4 (1986), [7] S. Fredl, J. Nocedal M.L. Overton, The formulaton analyss of numercal methods for nverse egenvalue problems, SIAM J. Numer. Anal., 24 (1987), [8] Zhang Yuha Zhu Benren, On the suffcent condtons for the solubty of general algebrac nverse egenvalue problems, Chnese J. of Num. Math. Appl., 18 (1996), [9] Zhang Yuha, On the suffcent condtons for the solubty of multpcatve nverse egenvalue problems, Math. Numer. Snca, 19 (1997), [10] Xngzh J, On matrx nverse egenvalue problems, Inverse Problems, 14 (1998), [11] Shufang Xu, An Introducton to Inverse Algebrac Egenvalue Problems, Peng Unversty Press, [12] Zhou Shuquan Da Hua, The Algebrac Inverse Egenvalue Problems, Henan Scence Tecnology Press, Zhengzhou, Chna, [13] Chu M.T., Inverse egenvalue problems, SIAM Revew, 40 (1998), [14] Sun J-guang, The stabty analyss of the solutons of nverse egenvalue problems, J. Comp. Math., 4 (1986), [15] Zhang Yuha, On the addtve nverse egenvalue problems, Chnese J. Num. Math. & Appl., 23:4 (2001), [16] Shufang Xu, The stabty analyss of the solutons of general algebrac nverse egenvalue problems, Math. Numer. Snca, 14 (1992), [17] J.M. Varah, A lower-bound for the smallest sngular value of a matrx, Ln. Alg. Appl., 11 (1975), 3-5. [18] Ortega J.M., Rheeboldt W.C., Iteratve soluton of nonear equton n several varables, New Yor, Academc Press, [19] Q. Ye, A class of teratve algorthm for solvng nverse egenvalue problems, Math.Numer.Snca, 9 (1987),

14 580 Y.H. ZHANG [20] Begler-Kong, F.W., A Newton teraton process for nverse egenvalue problems, Numer. Math., 37 (1981), [21] S. Fredl, J. Nocedal M.L. Overton, The formulaton analyss of numercal methods for nverse egenvalue problems, SIAM J. Numer. Anal. 24 (1987), [22] S.F. Xu, A homotopy algorthm for solvng nverse egenvalue problems for complex symmetrc matrces, J. Comput. Math., 11 (1993), [23] Stewart, G.W., Error perturbaton bounds for subspaces assocated wth certan egenvalue problems, SIAM Rev., 15 (1973), [24] Hu Jagan, The teratve methods for the soluton of near algebrac equatons, Academc Press, Beng, 1991.