Properties and Computations of Matrix Pseudospectra Yuming Shen2,Jinxi Zhaol and Hongjun Fan3 1.State Key Laboratory for Novel Software Technology,Nanjing University,Nanjing 210093,P.R.China 2.Department of Mathematics,Guangxi Normal University,Guilin 541004.P.R.China 3.Department of Mathematics,Nanjing University,Nanjing 210008,P.R.China Abstract Pseudospectra were introduced as early as 1975 and became popular tool during the 1990s.In this paper,we give a new definition of pseudospectra by using QR decomposition. Some properties of pseduospectra are explored and an algorithm for the computation of pseduospectra is given. Key words and phrases:eigenvalues,pseudospectra,QR decomposition 1 Introduction Let A be an m x n matrix with m >n.An eigenvalue of the matrix A might be defined by the condition (A-λ)w=0, (1.1) for some nonzero n-vector v,where I denotes the m x n'identity'with 1 on the main diagonal and 0 elsewhere.If (A,v)satisfies (1.1),then we have v=0. A2 where A1 denotes the n x n upper part of A.Hence not only (A,v)must be an eigenpair of A1, but v must also be in the nullspace of A2.Obviously,if A is a square matrix,then we get the canonical definition of eigenvalue. Four equivalent definitions of pseudospectra of square matrix were introduced by [1,7,8,9]. Pseudospectra of rectangular matrix has been considered by Toh,Wright and Trefethen[6,11,12] Higham and Tisseur[2].Here we present these equivalent definitions of pseudospectra as follows 12], Definition 1.1 Let A E cmxn and e>0 be arbitrary.The e-pseudospectrum Ae(A)of A is the set of z∈C such that l(zi-A)‖≥e1, (1.2) where'denotes the pseudoinverse and I denotes the mxn identity with 1 on the main diagonal and 0 elsewhere,C denotes the complex plane. 1
Properties and Computations of Matrix Pseudospectra Y uming Shen1,2 , J inxi Zhao1 and Hongjun F an1,3 1.State Key Laboratory for Novel Software Technology, Nanjing University, Nanjing 210093, P.R.China 2.Department of Mathematics, Guangxi Normal University, Guilin 541004, P.R.China 3.Department of Mathematics, Nanjing University, Nanjing 210008, P.R.China Abstract Pseudospectra were introduced as early as 1975 and became popular tool during the 1990s.In this paper, we give a new definition of pseudospectra by using QR decomposition. Some properties of pseduospectra are explored and an algorithm for the computation of pseduospectra is given. Key words and phrases: eigenvalues, pseudospectra,QR decomposition 1 Introduction Let A be an m × n matrix with m ≥ n. An eigenvalue of the matrix A might be defined by the condition (A − λ˜I)ν = 0, (1.1) for some nonzero n-vector ν, where ˜I denotes the m × n ’identity’ with 1 on the main diagonal and 0 elsewhere. If (λ, ν) satisfies (1.1), then we have à A1 − λIn A2 ! ν = 0, where A1 denotes the n × n upper part of A. Hence not only (λ, ν) must be an eigenpair of A1, but ν must also be in the nullspace of A2. Obviously, if A is a square matrix, then we get the canonical definition of eigenvalue. Four equivalent definitions of pseudospectra of square matrix were introduced by [1, 7, 8, 9]. Pseudospectra of rectangular matrix has been considered by Toh,Wright and Trefethen[6, 11, 12] , Higham and Tisseur[2]. Here we present these equivalent definitions of pseudospectra as follows[12], Definition 1.1 Let A ∈ Cm×n and ² ≥ 0 be arbitrary. The ²−pseudospectrum Λ²(A) of A is the set of z ∈ C such that k(z ˜I − A) † k ≥ ² −1 , (1.2) where ’†’ denotes the pseudoinverse and ˜I denotes the m×n identity with 1 on the main diagonal and 0 elsewhere, C denotes the complex plane. 1
Definition 1.2 Ae(A)is the set of zEc such that zEA(A+E)for some EE cmxn with IE≤e. Definition1.3Ae(A)is the set of z∈C such that‖(zi-A)vl≤e for some v∈Cn with wl=1. Definition 1.4 (assuming that the norm is.2)A(A)is the set of zEC such that omin(zI-A)≤e, (1.3) where omin denotes the smallest singular value. In section 2 we give a new definition of pseudospectra.In section 3 we consider some fun- damental properties of this new definition.In section4 we present some numerical examples to examine our conclusions.For simplicity,our norm.will always be the vector 2-norm. 2 A new definition of matrices pseudospectra Let B =2-A =[b1,62,...,on].It is shown that a system of vectors (o1,b2,...,on}is de- pendence if and only if G[b1,b2,...,n]=0,where G[61,b2,...,bn]is Gram determinant,i.e., G[b1,b2,...,n]=det(B*B).We can see that if 2A(A)is an eigenvalue of A then we must have deti(B*B)=0.Based on this consideration we give another definition of pseudospectra. On the other hand,let A be an m x n matrix with m>n,we write A as follows, A=[a1,a2,...,an]. (2.1) A system of vectors {a,a2,..,1kn is e-linear dependence,if Ga,a2,..ak for any given e>0[3].Obviously,if a system of vectorsfa1,a2,...,ak his e-linear dependence then a system of vectorsfai,a2,...,ar}with r>k is also c-linear dependence.And we can have the following result[4]. Suppose (61,b2,...,b&}is an orthogonal system and lbill =llaill,i 1,2,...,k then G[a1,a2,,a]≤G[b1,b2,…,bk (2.2) The equality is satisfied if and only if fal,a2,..,ak}is also an orthogonal system. Based on this consideration we give a new definition of pseudospectra. Definition 2.1 Let AE Cmxn and e>0 be arbitrary.The e-pseudospectrum Ac(A)of A is the set of z∈C such that c(A)={z∈C:G2(zi-A)=G[b1,b2,,bnl≤e} (2.3) As we will show,A(A)depends continuously on A(for e>0)and is nonempty for sufficiently large e. 2
Definition 1.2 Λ²(A) is the set of z ∈ C such that z ∈ Λ(A + E) for some E ∈ Cm×n with kEk ≤ ². Definition 1.3 Λ²(A) is the set of z ∈ C such that k(z ˜I − A)νk ≤ ² for some ν ∈ Cn with kνk = 1. Definition 1.4 (assuming that the norm is k.k2) Λ²(A) is the set of z ∈ C such that σmin(z ˜I − A) ≤ ², (1.3) where σmin denotes the smallest singular value. In section 2 we give a new definition of pseudospectra. In section 3 we consider some fundamental properties of this new definition.In section4 we present some numerical examples to examine our conclusions. For simplicity, our norm k.k will always be the vector 2-norm. 2 A new definition of matrices pseudospectra Let B = z ˜I − A = [b1, b2, ..., bn]. It is shown that a system of vectors {b1, b2, ..., bn} is dependence if and only if G[b1, b2, ..., bn] = 0, where G[b1, b2, ..., bn] is Gram determinant, i.e., G[b1, b2, ..., bn] ≡ det(B∗B). We can see that if z ∈ Λ(A) is an eigenvalue of A then we must have det 1 2 (B∗B) = 0. Based on this consideration we give another definition of pseudospectra. On the other hand , let A be an m × n matrix with m ≥ n, we write A as follows, A = [a1, a2, ..., an]. (2.1) A system of vectors {a1, a2, ..., ak}, 1 ≤ k ≤ n is ²−linear dependence, if G 1 2 [a1, a2, ..., ak] ≤ ² for any given ² ≥ 0[3]. Obviously, if a system of vectors{a1, a2, ..., ak}is ²−linear dependence then a system of vectors{a1, a2, ..., ar} with r > k is also ²−linear dependence. And we can have the following result[4]. Suppose {b1, b2, ..., bk} is an orthogonal system and kbik = kaik, i = 1, 2, ..., k then G[a1, a2, ..., ar] ≤ G[b1, b2, ..., bk] (2.2) . The equality is satisfied if and only if {a1, a2, ..., ak} is also an orthogonal system. Based on this consideration we give a new definition of pseudospectra. Definition 2.1 Let A ∈ Cm×n and ² ≥ 0 be arbitrary.The ²−pseudospectrum Λ²(A) of A is the set of z ∈ C such that Λ²(A) = {z ∈ C : G 1 2 (z ˜I − A) = G 1 2 [b1, b2, ..., bn] ≤ ²} (2.3) As we will show, Λ²(A) depends continuously on A(for ² > 0) and is nonempty for sufficiently large ². 2
3 Basic properties Theorem 3.1 Let A be an m x n matrix,B =zI-A=QR.Then (i)A(A)CA(A),where A(A)denotes the set of eigenvalues of A. (ii)Nea-1(A)=aAc(A)for any a>0. ()e(A6,1:)Cps++a(A(,1:k十1)，1≤k<n, where the monotonicity result is expressed in 'MATLAB notation,QR denotes QR decomposition and pkk is the main diagonal elements of matrir R. Proof.(i)For any 2A(A),we obtain that a system of vectors {b1,..,bn}is linear dependence, i.e.,G61,62...,on]=0.which yields,z Ae(A). (ii)This result follows immediately from the definition of A(A). (iii)The idea is to factor the matrix B as B=2-A=QR,where p11 p22 0 with P1≥lp22l≥·≥IPan land Q is an m×n unitary matrix. This is trivial for 2 A(A),since 2I-A is singular.If A(A),i.e.,rank(B)=n then Ipnn0.Consider that Go,b2,..]=det(BB)=det(RiR)=pi.p22...k:where Rk is a k x k upper triangular matrix of R,B=[61,b2,...,bk]. This formula yields Gb1,b2,,bk+1]=G[b1,b2,…,b]p2+1k+1, (3.1) which implies 不c(A,1:)Cep+1k+1(A(,1:k+1), (3.2) If +1 then we get A(A(:,1:K))CAc(A(:,1:k+1)). ▣ Theorem 3.2 (Pseudospectra of Similarity Transformation)Let m n,S is an nonsingular matrix and C=S-1AS.Then 工(A)=(C) (3.3) Proof.Let C=S-1AS=[c1,c2,..,cn]then we have G[c1,...cn.]=det(CTC)=det(ATA). which implies G[b1;b2;...;bn]=G[c1,c2,....cn], (3.4) i.e., A(A)=A.(C) ◇ The result demonstrates that pseudospectra are invariant under similarity transformation. Consider Definition2.1,we know that A(A)CA(5)(C).((S)=S)The results follows from the inequality e1<(-A)<S]S-(I-C)-.This means an ill-conditioned similarity transformation can alter pseudospectra. 3
3 Basic properties Theorem 3.1 Let A be an m × n matrix,B = z ˜I − A = QR.Then (i) Λ(A) ⊆ Λ²(A) ,where Λ(A) denotes the set of eigenvalues of A. (ii) Λ²α−1 (A) = αΛ²(A) for any α > 0. (iii) Λ²(A(:, 1 : k)) ⊆ Λ²|ρk+1,k+1| (A(:, 1 : k + 1)), 1 ≤ k < n, where the monotonicity result is expressed in ’MATLAB notation’,QR denotes QR decomposition and ρkk is the main diagonal elements of matrix R. Proof. (i) For any z ∈ Λ(A), we obtain that a system of vectors {b1, ..., bn} is linear dependence, i.e.,G[b1, b2..., bn] = 0. which yields, z ∈ Λ²(A). (ii) This result follows immediately from the definition of Λ²(A). (iii)The idea is to factor the matrix B as B = z ˜I − A = QR, where R = Ã R˜ 0 ! , R˜ = ρ11 ρ22 ∗ . . . ρnn , with |ρ11| ≥ |ρ22| ≥ · · · ≥ |ρnn|and Q is an m × n unitary matrix. This is trivial for z ∈ Λ(A), since z ˜I − A is singular. If z 6∈ Λ(A), i.e., rank(B) = n then |ρnn| 6= 0. Consider that G[b1, b2, ..., bk] = det(B∗ kBk) = det(R˜∗ kR˜ k) = ρ 2 11.ρ2 22...ρ2 kk, where R˜ k is a k × k upper triangular matrix of R, B ˜ k = [b1, b2, ..., bk]. This formula yields G[b1, b2, ..., bk+1] = G[b1, b2, ..., bk]ρ 2 k+1,k+1, (3.1) which implies Λ²(A(:, 1 : k)) ⊆ Λ²ρk+1,k+1 (A(:, 1 : k + 1)), (3.2) If |ρk+1,k+1| ≤ 1 then we get Λ²(A(:, 1 : k)) ⊆ Λ²(A(:, 1 : k + 1)). ut Theorem 3.2 (Pseudospectra of Similarity Transformation) Let m = n, S is an nonsingular matrix and C = S −1AS. Then Λ²(A) = Λ²(C). (3.3) Proof. Let C = S −1AS = [c1, c2, ..., cn] then we have G[c1, ..., cn] = det(C T C) = det(AT A). which implies G[b1, b2, ..., bn] = G[c1, c2, ..., cn], (3.4) i.e., Λ²(A) = Λ²(C) ut The result demonstrates that pseudospectra are invariant under similarity transformation. Consider Definition2.1, we know that Λ²(A) ⊆ Λκ(S)² (C).(κ(S) = kSkkS −1k) The results follows from the inequality ² −1 ≤ k(z ˜I −A) −1k ≤ kSkkS −1kk(z ˜I −C) −1k. This means an ill-conditioned similarity transformation can alter pseudospectra. 3
Theorem 3.3 Suppose A is a normal matrix then (i)A(A)=A(A),where A is a diagonal matrix with eigenvalues of A on the main diagonal. (i)for any z∈工e(A),there erists入s∈A(A)such that z-入sl≤e六whee入s is an eigenvalue of A that minimum z-入k for1≤k≤n. Proof.(i)This result follows from Theorem3.2. (i)）Let B-:1-A (3.5) then det(BTB)=Iz-A1l2lz-212...J-An2. Hence we get G[b1,b2,bn=z-1lz-2lz-入n≥z-入sm (3.6) Ifz∈c(A)then we have-Asl≤e. ◇ Theorem 3.4 Let A,B are square matrices then Ae(AB)=Ae(BA). Proof.Notice that A(AB)=A(BA).Then we have det(AI-BA)det(XI-AB). Let C1=AI-BA=[c1...,cn],C2=AI-AB=[d,...,ch]then Glc1,...,cn]det(CfC1)=det(CC2)=Gld,. (3.7) which yields K(AB)=K(BA). 0 The same proof shows that if A is an m x n matrix and B is an n x m matrix,then AB and BA have the same psedoeigenvalues except that the product which is of higher order has m-n extra zero eigenvalues. The following theorem gives relationship between two definitions of pseudospectra. Theorem3.5 For any given c≥0，A(A)Sc(A)∈Ah(A). Proof.From Theorem3.1 we have that G b2on]=Ipiillp221...IPnnl, where Pnn is the element of matrix R,with IPu≥lp22l≥·≥IPnnl From the definition of the minimum singular value of a matrix B, omin(A)=doin(B) Since B =QR and the unitary invariance of the 2-norm,let x =en we have that omin(zi-A)≤IPnnl 4
Theorem 3.3 Suppose A is a normal matrix then (i) Λ²(A) = Λ²(Λ), where Λ is a diagonal matrix with eigenvalues of A on the main diagonal. (ii) for any z ∈ Λ²(A), there exists λs ∈ λ(A)such that |z − λs| ≤ ² 1 n where λs is an eigenvalue of A that minimum |z − λk| for 1 ≤ k ≤ n. Proof. (i) This result follows from Theorem3.2. (ii) Let B = z ˜I − Λ = z − λ1 . . . z − λn , (3.5) then det(B T B) = |z − λ1| 2 |z − λ2| 2 ...|z − λn| 2 . Hence we get G 1 2 [b1, b2..., bn] = |z − λ1||z − λ2|...|z − λn| ≥ |z − λs| n . (3.6) If z ∈ Λ²(A) then we have |z − λs| ≤ ² 1 n . ut Theorem 3.4 Let A,B are square matrices then Λ²(AB) = Λ²(BA). Proof. Notice that Λ(AB) = Λ(BA). Then we have det(λI − BA) = det(λI − AB). Let C1 = λI − BA = [c1, ..., cn], C2 = λI − AB = [c 0 1 , ..., c0 n ] then G[c1, ..., cn] = det(C T 1 C1) = det(C T 2 C2) = G[c 0 1 , ..., c0 n ], (3.7) which yields Λ²(AB) = Λ²(BA). ut The same proof shows that if A is an m × n matrix and B is an n × m matrix, then AB and BA have the same psedoeigenvalues except that the product which is of higher order has |m − n| extra zero eigenvalues. The following theorem gives relationship between two definitions of pseudospectra. Theorem 3.5 For any given ² ≥ 0 ,Λ(A) ⊆ Λ²(A) ⊆ Λ ² 1 n (A). Proof. From Theorem3.1 we have that G 1 2 [b1, b2, ..., bn] = |ρ11||ρ22|...|ρnn|, where ρnn is the element of matrix R˜,with |ρ11| ≥ |ρ22| ≥ · · · ≥ |ρnn|. From the definition of the minimum singular value of a matrix B, σmin(z ˜I − A) = σmin(B) = min kxk2=1 kBxk2. Since B = QR and the unitary invariance of the 2-norm, let x = en we have that σmin(z ˜I − A) ≤ |ρnn|. 4
This formula implies omin(B)0. 4 Numerical experiments Now let us calculate pseudospectra properly.The place to begin is with the column pivoting QR decomposition.In numerical experiments,we observe that,if pk>>1,k=1,2,...,n then G,b21.In order to avoid this situation,we modify our formula G2 e as Gb1,62,..,onl/pil<e/p1 Hence,the algorithm is to compute column pivoting QR de- composition of zI-A for values of z on a grid in the plane and then generate a contour plot from this data.At last,we also notice that if Gb,b2,...,on]e then we get omin(B)e, but the converse may not be true (see Lawson and Hanson[5]p31). Algorithm5.1 (1)For each ze grid computing B=2I-A=[b1;...o]; (2)Computing the column pivoting QR decomposition of B; (3)IfG[b1,b2,,bnl/lp1l≤e/IPul then z∈c(A) else goto step (1). Now we present some numerical examples to examine our conclusions. Examplel We denote the matrix A=rand(5,5), 0.19340.69790.49660.6602 0.7271 0.68220.37840.8998 0.34200.3093 A= 0.30280.86000.82160.28970.8385 0.54170.85370.64490.34120.5681 0.15090.59360.81800.53410.3704 Figure 1 shows the pseudospectra of A,which the eigenvalue drawn as dots.Note that the sets are nested as indicated in Theorem3.1.In figure2,we see the 5 x 4 matrix of A(:,1:4).The inclusion properties of Theorem3.1(iii)can be clearly seen that the pseudospectra of the square matrix A are bigger than those of A(:,1:4). 6
This formula implies σ n min(B) ≤ |ρ11||ρ22|...|ρnn|, i.e., Λ²(A) ⊆ Λ ² 1 n (A). ut Remark. Since the singular values and ρnn are continuous functions of the matrix entries, hence if ρnn −→ 0 then σmin(B) −→ 0. The converse is also true. It also can be seen that Λ²(A) and Λ²(A) change continuously with ² > 0. 4 Numerical experiments Now let us calculate pseudospectra properly. The place to begin is with the column pivoting QR decomposition. In numerical experiments,we observe that, if |ρkk| >> 1, k = 1, 2, ..., n then G 1 2 [b1, b2, ..., bn] >> 1. In order to avoid this situation, we modify our formula G 1 2 [b1, b2, ..., bn] ≤ ² as G 1 2 [b1, b2, ..., bn]/|ρ11| ≤ ²/|ρ11| Hence, the algorithm is to compute column pivoting QR decomposition of z ˜I − A for values of z on a grid in the plane and then generate a contour plot from this data. At last, we also notice that if G 1 2 [b1, b2, ..., bn] ≤ ² n then we get σmin(B) ≤ ², but the converse may not be true (see Lawson and Hanson[5]p31). Algorithm5.1 (1)For each z ∈ grid computing B = z ˜I − A = [b1, ..., bn]; (2) Computing the column pivoting QR decomposition of B; (3) If G 1 2 [b1, b2, ..., bn]/|ρ11| ≤ ²/|ρ11| then z ∈ Λ²(A) else goto step (1). Now we present some numerical examples to examine our conclusions. Example1 We denote the matrix A=rand(5,5), A = 0.1934 0.6979 0.4966 0.6602 0.7271 0.6822 0.3784 0.8998 0.3420 0.3093 0.3028 0.8600 0.8216 0.2897 0.8385 0.5417 0.8537 0.6449 0.3412 0.5681 0.1509 0.5936 0.8180 0.5341 0.3704 . Figure 1 shows the pseudospectra of A,which the eigenvalue drawn as dots. Note that the sets are nested as indicated in Theorem3.1.In figure2, we see the 5 × 4 matrix of A(:,1:4). The inclusion properties of Theorem3.1(iii) can be clearly seen that the pseudospectra of the square matrix A are bigger than those of A(:,1:4). 5
23 Figure 1:c-pseudospectra (e=0.5,0.4,0.3,0.2,0.1)for the matrix A.The grid points are selected with v=80. 3 2 Figure 2:e-pseudospectra(e=0.5,0.4,0.3,0.2,0.1)for the matrix A(:,1:4).The grid points are selected with v=80. Example2 This test matrix is constructed by L.N.Trefethen(see[10]p255).Let A=AN=20 and grid points are selected with v=80,the numerical result is shown in Figure 3. 6
−3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 fig.1 Figure 1:²−pseudospectra (² = 0.5, 0.4, 0.3, 0.2, 0.1) for the matrix A. The grid points are selected with v=80. −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 fig.2 Figure 2:²−pseudospectra (² = 0.5, 0.4, 0.3, 0.2, 0.1) for the matrix A(:,1:4). The grid points are selected with v=80. Example2 This test matrix is constructed by L.N.Trefethen (see[10]p255). Let A = AN=20 and grid points are selected with v=80, the numerical result is shown in Figure 3. 6
300 400 -300 500 Figure 3 e-pseudospectra (e=10-1,10-2,10-3,10-4,10-5,10-6,10-7)for the matrix A. The grid points are selected with v=80. 5 Summary We present a new definition of pseudospectra of matrices by using QR decomposition.Based on this definition we get some basic properties.An algorithm for the computation of pseudospec- tra is given. References [1]H.J.Landau,On Szego's eigenvalue distribution theory and non-Hermitian ker- nels.J.Analyse Math.,28(1975),pp.335-357 [2]N.J.Higham and F.Tisseur,More on pseudospectra for polynomial eigenvalue problems and applications in control theory.Tech.Reort 372(2001),Maths Dept.,University of Manchester [3]X.He,Theory of numerical linear dependence and its applications.Numer.Math.J.Chinese Univ.,1(1979),pp.11-19 [4]X.He;The fundamental properties of pseudoinverse and its applications.Shanghai Scientific Technology Pulishing House,1985 [5]C.L.Lawson and R.J.Hanson,Solving Least Squares Problems.Prentice-Hall,Englewood Cliffs,NJ.Reprinted with a detailed "new developments'appendix in 1996 by SIAM Pub- lications,Philadelphia,PA [6]K.-C.Toh and L.N.Trefethen,Calculation of pseudospectra by the Arnoldi iteraion .SIAM J.Sci.Comput.,17(1996),pp.1-15
−500 −400 −300 −200 −100 0 100 200 300 400 500 −500 −400 −300 −200 −100 0 100 200 300 400 500 fig.3 Figure 3 :²−pseudospectra (² = 10−1 , 10−2 , 10−3 , 10−4 , 10−5 , 10−6 , 10−7 ) for the matrix A. The grid points are selected with v=80. 5 Summary We present a new definition of pseudospectra of matrices by using QR decomposition. Based on this definition we get some basic properties. An algorithm for the computation of pseudospectra is given. References [1] H.J.Landau, On Szeg¨o’s eigenvalue distribution theory and non-Hermitian kernels.J.Analyse Math., 28(1975),pp.335-357 [2] N.J.Higham and F.Tisseur, More on pseudospectra for polynomial eigenvalue problems and applications in control theory.Tech. Reort 372(2001),Maths Dept.,University of Manchester [3] X.He, Theory of numerical linear dependence and its applications.Numer.Math.J.Chinese Univ.,1(1979),pp.11-19 [4] X.He , The fundamental properties of pseudoinverse and its applications.Shanghai Scientific Technology Pulishing House,1985 [5] C.L.Lawson and R.J.Hanson, Solving Least Squares Problems.Prentice-Hall,Englewood Cliffs,NJ.Reprinted with a detailed ”new developments’appendix in 1996 by SIAM Publications,Philadelphia,PA [6] K.-C.Toh and L.N.Trefethen, Calculation of pseudospectra by the Arnoldi iteraion .SIAM J.Sci.Comput.,17(1996),pp.1-15 7
[7]L.N.Trefethen,Pseudospectra of matrices,in Numerical Analysis 1991,D.F.Griffiths and G.A.Watson,eds.,Longman Scientific and Technical,Harlow,UK,1992,pp.234-266. [8]L.N.Trefethen,Pseudospectra of linear operators,SIAM Rev.,39(1997),pp.383-406. [9]L.N.Trefethen,Spectra and pseudospectra:The behavior of non-normal matrices and opera- tors,in The Graduate Student's Guide to Numerical Analysis,M.Ainsworth,J.Levesley,and M.Marletta,eds.,Springer-Verlag,Berlin,1999,pp.217-250. [10]L.N.Trefethen,Computation of pesudospectra.In Acta numerical 1999,pp.247-295,Cambridge University Press,Cambridge [11]T.G.Wright and L.N.Trefethen,Computation of pseudospectra using ARPACK and eigs.SIAM J.Sci.Comput.,23(2001),pp.591-605 [12]T.G.Wright and L.N.Trefethen,Eigenvalues and pseudospectra of rectangular matrices Tech.report 01/13,Oxford University Computing Laboratory Numerical Analysis Group 8
[7] L.N.Trefethen,Pseudospectra of matrices,in Numerical Analysis 1991,D.F.Griffiths and G.A.Watson,eds.,Longman Scientific and Technical,Harlow,UK,1992,pp.234-266. [8] L.N.Trefethen,Pseudospectra of linear operators,SIAM Rev.,39(1997),pp.383-406. [9] L.N.Trefethen,Spectra and pseudospectra:The behavior of non-normal matrices and operators,in The Graduate Student’s Guide to Numerical Analysis,M.Ainsworth,J.Levesley, and M.Marletta,eds.,Springer-Verlag,Berlin,1999,pp.217-250. [10] L.N.Trefethen,Computation of pesudospectra.In Acta numerical 1999,pp.247-295,Cambridge University Press,Cambridge [11] T.G.Wright and L.N.Trefethen,Computation of pseudospectra using ARPACK and eigs.SIAM J.Sci.Comput., 23(2001),pp.591-605 [12] T.G.Wright and L.N.Trefethen,Eigenvalues and pseudospectra of rectangular matrices ,Tech. report 01/13, Oxford University Computing Laboratory Numerical Analysis Group 8
