Low rank differential equations for Hamiltonian matrix nearness problems

Transcription

1 13 December 2013 Low rank differential equations for Hamiltonian matrix nearness problems Nicola Guglielmi Daniel Kressner Christian Lubich Abstract For a Hamiltonian matrix with purely imaginary eigenvalues, we aim to determine the nearest Hamiltonian matrix such that some or all eigenvalues leave the imaginary axis. Conversely, for a Hamiltonian matrix with all eigenvalues lying off the imaginary axis, we look for a nearest Hamiltonian matrix that has a pair of imaginary eigenvalues. The Hamiltonian matrices can be allowed to be complex or restricted to be real. Such Hamiltonian matrix nearness problems are motivated by applications such as the analysis of passive control systems. They are closely related to the problem of determining extremal points of Hamiltonian pseudospectra. We obtain a characterization of optimal perturbations, which turn out to be of low rank and are attractive stationary points of low-rank differential equations that we derive. We use a two-level approach, where in the inner level we determine extremal points of the Hamiltonian ε-pseudospectrum for a given ε by following the low-rank differential equations into a stationary point, and on the outer level we optimize for ε. This permits us to give fast algorithms- exhibiting quadratic convergence - for solving the considered Hamiltonian matrix nearness problems. Keywords Hamiltonian pseudospectrum, passivity radius, algebraic Riccati equations, low-rank dynamics, differential equations on Stiefel manifolds Nicola Guglielmi Dipartimento di Matematica Pura ed Applicata, Università degli Studi di L Aquila, Via Vetoio - Loc. Coppito, I L Aquila, Italy guglielm@univaq.it Daniel Kressner EPFL-SB-MATHICSE-ANCHP, Station 8, CH-1015 Lausanne, Switzerland daniel.kressner@epfl.ch Christian Lubich Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, D Tübingen, Germany lubich@na.uni-tuebingen.de

2 2 Nicola Guglielmi et al. Mathematics Subject Classification (2000) 15A18, 65K05, 93B36, 93B40, 49N35, 65F15, 93B52, 93C05 1 Introduction In this paper we propose and study algorithms for the following optimization problems: (A) Given a Hamiltonian matrix with no eigenvalues on the imaginary axis, find a nearest Hamiltonian matrix having some purely imaginary eigenvalue. (B) Given a Hamiltonian matrix with all eigenvalues on the imaginary axis, find a nearest Hamiltonian matrix such that arbitrarily close to that matrix there exist Hamiltonian matrices with eigenvalues off the imaginary axis. The notion of nearest depends on the choice of norm, for which we consider the matrix 2-norm. The Hamiltonian matrices can be allowed to be complex or restricted to be real. Such Hamiltonian matrix nearness problems arise in several important applications: in the solution of algebraic Riccati equations whose Hamiltonian matrix has eigenvalues on the imaginary axis (see, e.g., [And98, Hes09, PaVL81]), in passivation of linear time invariant control systems (see, e.g., [ABKMM11]), and in the stability of gyroscopic systems (see, e.g., [HKLP00]). Both problems (A) and (B) are closely related to the problem of finding extremal (locally leftmost or rightmost) points of the structured pseudospectrum (for this notion see, e.g., [BK04,WA05,Ru06,HP05,TE05,KKK10,Ka11]) of a given matrix A M with respect to perturbed matrices on a matrix manifold M C n n that are ε-close to A in a norm : Λ ε (A,M, ) = {λ C :λ is an eigenvalue of some B M with B A ε}. We will always consider the matrix 2-norm = 2 in this paper. In the Hamiltonian case considered here we take M as the real-linear space of Hamiltonian matrices, i.e., those satisfying (with even dimension n = 2d) ( ) 0 Id JA is complex hermitian (or real symmetric), where J =. I d 0 We recall that the eigenvalues of a Hamiltonian matrix lie symmetric to the imaginary axis. The paper is organized as follows. In Section 2 we characterize extremal complex Hamiltonian perturbations, which correspond to a locally leftmost or rightmost point in the ε-pseudospectrum. In Section 3 we deal with extremal real Hamiltonian perturbations. In both cases, in analogy to [GL11, GL11b, GL13] (see also [GO11]) we derive low-rank matrix differential equations that

3 Low rank ODEs for Hamiltonian matrix nearness problems 3 have the extremizers as attractive stationary points. The ranks are 2 in the complex case and 4 in the real case. Our approach to solving the Hamiltonian matrix nearness problems (A) and (B) consists of a two-level procedure, where on the inner level we determine an extremizer for a fixed perturbation size ε by following the differential equation into a stationary point, and then on the outer level we optimize over ε. While the matrix nearness problems treated here are of the same type as those studied in [ABKMM11], the algorithmic approach of the present paper is complementary. Whereas [ABKMM11] takes an inner approach by following eigenvalues on the imaginary axis and studying the sign characteristic in coalescent eigenvalues, we take an outer approach where we follow eigenvalues that are off the imaginary axis. We will demonstrate that the outer approach proposed here is very efficient in various example problems, but it is not the purpose of this paper to give an in-depth numerical comparison of the two fundamentally different approaches, each of which has its own merits. From an algorithmic point of view we are interested in robust variants of (A) and (B). We therefore introduce a thin vertical strip symmetric with respect to the imaginary axis and compute distances with respect to this strip instead of the imaginary axis itself, as considered also in [ABKMM11]. In Section 5 we present a fast algorithm for solving problem (A) and in Section 6 for problem (B). The first algorithm computes a Hamiltonian perturbation of minimal norm such that some pair of eigenvalues of a Hamiltonian matrix with purely imaginary spectrum is moved outside a vertical strip symmetric with respect to the imaginary axis, of a preassigned arbitrarily small width. The second algorithm computes a Hamiltonian perturbation of minimal norm such that a pair of eigenvalues of a Hamiltonian matrix A with spectrum bounded away from the imaginary axis is taken inside a given small strip that is symmetric with respect to the imaginary axis. In the examples we shall also deal with the following third problem, which is related to problem (B). (C) Given a Hamiltonian matrix with all eigenvalues on the imaginary axis, find a nearest Hamiltonian matrix such that arbitrarily close to that matrix there exist Hamiltonian matrices with all eigenvalues off the imaginary axis. This is associated to passivity measures. In Section 7 we present in some detail two interesting applications: passivation of linear control systems and gyroscopic stability. There are various extensions to the present work, which we do not report here: While we consider only the matrix 2-norm in this paper, a related but different theory and corresponding algorithms can be given also for the Frobenius norm. Analogous matrix nearness problems to this paper can be studied also for symplectic matrices, where the unit circle assumes the role of the imaginary axis for the Hamiltonian case. Another interesting extension is to Hamiltonian matrices with additional (block) structure, as they arise in control systems.

4 4 Nicola Guglielmi et al. We will make frequent use of the following standard perturbation result for eigenvalues; see, e.g., [Kat95, Section II.1.1]. Here and in the following, we denote = d/dt. Lemma 1.1 Consider the differentiable n n matrix valued function C(t) for t in a neighborhood of 0. Let λ(t) be an eigenvalue of C(t) converging to a simple eigenvalue λ 0 of C 0 = C(0) as t 0. Let x 0 and y 0 be left and right eigenvectors, respectively, of C 0 corresponding to λ 0, that is, (C 0 λ 0 I)y = 0 and x (C 0 λ 0 I) = 0. Then, x 0y 0 0 and λ(t) is differentiable near t = 0 with λ(0) = x 0Ċ(0)y 0 x 0 y. 0 2 Rank-2 dynamics to extremal points in complex Hamiltonian pseudospectra In this section we study Hamiltonian perturbations to the given Hamiltonian matrix A that correspond to extremal (locally leftmost or rightmost) points in the Hamiltonian ε-pseudospectrum. It is shown that to each extremal point the corresponding extremizer can be chosen of rank at most 2. We then proceed to derive and study a differential equation on the manifold of Hamiltonian rank-2 matrices that has the rank-2 extremizers as attractive stationary points. 2.1 Extremizers Theorem 2.1 For a Hamiltonian matrix A C n n and ε > 0, let λ / ir be a locally rightmost point in the Hamiltonian 2-norm ε-pseudospectrum, implying that there exists a Hamiltonian matrix E C n n of unit 2-norm such that λ is an eigenvalue of A+εE. If λ is a simple eigenvalue of A+εE and not an eigenvalue of A, then there exists a unique Hamiltonian matrix E of rank 2 such that λ is an eigenvalue of A+εE with the same left and right eigenvectors as for A+εE. Moreover, the nonzero eigenvalues of JE are +1 and 1. The proof of Theorem 2.1 shows that in addition to the unique extremizer E of rank 2, there exist also extremizers of any rank > 2. A characterization of the rank-2 extremizers is given in the following result. ( ) 1 0 Theorem 2.2 Assume that E = J T V V 0 1, where V C n 2 has orthonormal columns. Let λ / ir be a simple eigenvalue of A + εe, with left and right eigenvectors x and y, respectively, both of unit norm and with x y > 0. Let Y = (Jx,y) C n 2, and T C 2 2 be defined by ( ) T = RPR with R = V Y C and P =. 1 0 Then the following two statements are equivalent:

5 Low rank ODEs for Hamiltonian matrix nearness problems 5 1. Every differentiable path (E(t),λ(t)) (for small t 0) such that E(t) is Hamiltonian with E(t) 2 1 and λ(t) is an eigenvalue of A+εE(t), with E(0) = E and λ(0) = λ, has Re λ(0) ( 0. ) +τ Y and V have the same range, and T = with τ 0 τ 1,τ 2 > 0. 2 We remark that 1. is satisfied at points on the boundary of the Hamiltonian ε-pseudospectrum with a vertical tangent and the pseudospectrum lying locally to the left. This includes in particular locally rightmost points of the pseudospectrum, but includes also locally leftmost boundary points on the right boundary or inflection points with a vertical tangent. Proof (of Theorem 2.1) (a) Let V C n n be a unitary matrix obtained from a QR factorization of Y = (Jx,y): ( ) Y = V R with R C Let E(t), for small t 0, be a continuously differentiable path on the set of Hamiltonian matrices of 2-norm at most 1, with E(0) = E, which we write as with a hermitian matrix Ŝ(t) C n n = E(t) = J T VŜ(t) V ( ) S11 (t) S21(t), S S 21 (t) S 22 (t) 11 (t) C 2 2. We can choose the first 2 columns of the unitary matrix V such that ( ) σ1 0 S 11 (0) =. 0 σ 2 Clearly, σ 1,σ 2 are real and have absolute value at most 1. In the following, we will in fact show that σ 1 = σ 2 = 1 and that σ 1,σ 2 have opposite sign. (b) Let λ(t), for small t 0, be the continuously differentiable path of eigenvalues of A+εE(t) with λ(0) = λ. Since λ is locally rightmost, we have 0 Re λ(0) = ε x y Re(x Ė(0)y). Denoting by A,B = trace(a B) the standard matrix inner product and by A herm = 1 2 (A+A ) the hermitian part of a matrix, we observe 0 Re(x Ė(0)y) = xy,ė(0) = Jxy,JĖ(0) = (Jxy ) herm,jė(0) = 1 2 YPY, V Ŝ(0) V = 1 2 V YPY V, Ŝ(0) = 1 2 RPR,Ṡ11(0) = 1 2 T,Ṡ11(0). (2.1) This inequality holds for every path Ŝ(t) of unit norm, which in turn implies that it holds for every path S 11 (t) of norm at most one.

6 6 Nicola Guglielmi et al. Suppose that one of the diagonal entries of S 11 (0), say σ 2, has absolute value less than one. For the moment, let us additionally assume that σ 1 σ 2. Then we consider the following three different paths of norm at most one, locally for small t 0: S (1) 11 (t) = ( σ1 0 0 σ 2 ±t ) ( ) ( ), S (2) a(t) ±t 11 (t) =, S (3) a(t) it ±t b(t) 11 (t) =, ±it b(t) where in the latter two cases a(t),b(t) are chosen such that the matrices have the eigenvalues σ 1 and σ 2 : a(t) = σ 1 +σ 2 (σ1 σ 2 ) + 2 t 2 4 2, b(t) = σ 1 +σ 2 a(t). In particular, ȧ(0) = ḃ(0) = 0. We then obtain from (2.1) that T,Ṡ(1) 11 (0) = ±t 22 = t 22 = 0, T,Ṡ(2) 11 (0) = ±(t 21 +t 21 ) = Re(t 21 ) = 0, T,Ṡ(3) 11 (0) = ±(t 21 t 21 ) = Im(t 21 ) = 0. In particular, this means that T = RPR is singular and hence R itself is singular. By the definition of R, this would imply that the eigenvectors Jx and y arelinearlydependent.hence,jxwouldbearighteigenvectorofh = A+εE to the eigenvalue λ. On the other hand, since H is Hamiltonian, Jx is a right eigenvalue of H to the eigenvalue λ, because HJx = JH x = J λ x = λ Jx. We would thus have λ = λ, so that λ is on the imaginary axis, in contradiction to our assumption. Therefore, σ 1 = σ 2 = 1. The same conclusion holds when σ 1 = σ 2, using appropriately modified paths. (c) Now suppose that σ 1 and σ 2 have the same sign, so that S 11 (0) = ±I 2. We then choose S 11 (t) = ±e tm with a hermitian positive definite matrix M, so that Ṡ11(0) = M. The inequality 0 T,Ṡ11(0) = T,M would then hold for every positive definite hermitian M, which would imply that T is positive semi-definite (negative semi-definite). This is not possible, as P and therefore also T = RPR is an indefinite nonsingular matrix. In view of this contradiction, S 11 (0) must have two eigenvalues of different sign, +1 and 1. (d) Finally, σ 1 = σ 2 = 1 implies that S 21 (0) = 0, since otherwise one of the first two columns of Ŝ(0) has norm larger than one, in contradiction to Ŝ(0) 2 = 1. This shows that A+εE, with ( ) E = J T S11 (0) 0 ˆV ˆV, 0 0

7 Low rank ODEs for Hamiltonian matrix nearness problems 7 satisfies (A+εE )y = (A+εE)y = λ y and x (A+εE ) = x (A+εE) = λ x. The rank-2 matrix E is unique because the block structure of Ŝ(0) and the orthogonality of the columns of V show that for every other rank-2 reduction of E(t) we have Ey = 0 or EJx = 0, which would imply that λ is an eigenvalue of A, contrary to the assumption. Proof (of Theorem 2.2) 1. implies 2.: We know from the proof of Theorem 2.1 (that proof uses precisely condition 1. rather than the slightly stronger property of λ being locally rightmost) that Y and V have the same range, and we write Y = VR with an invertible 2 2 matrix R. We note that T = RPR is the matrix of which we want to show that it is diagonal with entries of different sign. Let us denote ( ) 1 0 D =. 0 1 We consider the path of matrices of unit 2-norm JE(t) = Ve tz De tz V with a skew-hermitian matrix Z C 2 2 and the corresponding path of eigenvalues λ(t) of A+εE(t) with λ(0) = λ. By 1., we have 0 Re(x Ė(0)y) = YPY,Ė(0) = T,ZD DZ, which holds for every skew-hermitian matrix Z. The matrix ZD DZ is hermitian with zero diagonal. It thus follows that T is diagonal. Since T = RPR is hermitian, indefinite and nonsingular, its diagonal entries must be real and have opposite sign. Using the paths E(t) = Ve tm De tm V with a diagonal positive semi-definite matrix M we then conclude from 0 Re(x Ė(0)y) = T,MD +DM that the(1,1) entryoft is positive andthe(2,2) entryis negative. This proves implies 1.: Let E(t), for small t 0, be a continuously differentiable path of Hamiltonian matrices of 2-norm at most one, with E(0) = J T VDV. We extend V C n 2 to a unitary matrix V C n n and write V = (V,V ). We define the hermitian matrix Ŝ(t) by writing E(t) as E(t) = J T VŜ(t) V. ( ) With the matrix T = RPR = V YPY τ1 0 V satisfying T = with 0 τ 2 τ 1,τ 2 > 0, and with S(t) denoting the upper left 2 2 block of Ŝ(t), we have

8 8 Nicola Guglielmi et al. as in (2.1) Re(x Ė(0)y) = YPY, V Ŝ(0) V ( ) R = P(R,0), Ŝ(0) = RPR,Ṡ(0) = T,Ṡ(0) 0 = τ 1 ṡ 11 (0) τ 2 ṡ 22 (0) 0, since s 11 (0) = 1 and s 22 (0) = 1 imply ṡ 11 (0) 0 and ṡ 22 (0) 0. This yields statement Rank-2 dynamics for the complex case Transfering the approach of [GL11, GL13] to the Hamiltonian case, we search for a differential equation for the perturbation E(t) in A+εE(t) such that the extremizers corresponding to locally rightmost points in the Hamiltonian 2- norm pseudospectrum are attractive stationary points of the differential equation. Since Theorem 2.1 yields the existence of extremizers E of rank 2 with the property that the two nonzero eigenvalues of JE are ±1, we search for a differential equation on the manifold M = {E C n n : E is Hamiltonian of rank 2 and the nonzero eigenvalues of JE are equal to +1 and 1}. We represent matrices in M in a non-unique way as JE = VQV, (2.2) where V C n 2 has orthonormal columns and Q C 2 2 is a hermitian unitary matrix with eigenvalues +1 and 1. Such a Q can be written as a complex Householder matrix Q = I 2 2uu, u C 2, u u = 1. (2.3) For a given choice of V and Q, tangent matrices Ė T E M can then be uniquely written as (cf. [KL07]) JĖ = VQV +V QV +VQ V with V V = 0, u u = 0. (2.4) We have given up a unique representation of E for the benefit of the very useful orthogonality relations for V and u. Our objective is to find differential equations for u(t) and V(t) along which an eigenvalue (most interestingly, the rightmost eigenvalue) λ(t) of A + εe(t) increases monotonically with t and which have attractive stationary points where the extremality condition of Theorem 2.2 is satisfied. Assuming that λ(t) is simple for all t under consideration, we denote the left and right eigenvectors by x(t) and y(t), respectively, both of unit norm and with x(t) y(t) > 0. We let Y(t) = (Jx(t),y(t)). We omit the argument t in the following when its presence is clear from the context.

9 Low rank ODEs for Hamiltonian matrix nearness problems 9 Motivated by the orthogonality relations in (2.4) and the objective that Y and V have the same range when V = 0, we make the ansatz, with as yet unknown f C 2 and G C 2 2, u = (I 2 uu )f V = (I n VV )YG. (2.5) We call a tangent matrix Ė T EM admissible if Ė is of the form (2.4) with (2.5) and satisfies the normalizing constraint that Ė has Frobenius norm at most one. The set of admissible tangent matrices at E M is denoted by A E = {Z T E M : Z is admissible}. Given E(t) M at some t, we want to move in an admissible direction Ė(t) T E(t) M such that Re λ(t) is as large as possible. We thus maximize Re λ = εre(x Ėy)/x y (see Lemma 1.1) over the admissible set A E. Since Ė = 0 is admissible, the maximum value will be nonnegative. In this context we have the following result. (Note that in this lemma the dots are not time derivatives, but just suggestive notational symbols.) Lemma 2.1 Let A be a Hamiltonian matrix, let E M with the factorization (2.2) (2.3), and ε > 0. Let λ be a simple eigenvalue of A+εE with left and right eigenvectors x and y, respectively, both of unit norm and with x y > 0. Then, among all admissible tangent matrices Z A E, Re(x Zy) is maximal for Z = J 1 ( VQV +V QV +VQ V ), where Q = 2( uu +u u ), and u and V are of the form (2.5) with f = 1 4µ Tu, G = 1 2µ PR Q. (2.6) ( ) 0 1 Here P =, T = RPR 1 0 C 2 2 with R = V Y for Y = (Jx,y) C n 2. If u or V are different from zero for any scaling factor µ > 0 then µ is chosen such that Z has unit Frobenius norm. Otherwise, µ can be chosen arbitrarily. Proof Inserting (2.4), we obtain for any Ė A E that Re(x Ėy) = (Jxy ) herm,jė = 1 2 YPY,JĖ = 1 2 YPY, VQV +V QV +VQ V = 1 2 YPR Q, V T, Q QRPY, V = 1 2 T, Q +Re YPR Q, V. We further note that, with the hermitian positive semi-definite 2 2 matrices L = I 2 uu, M = Y (I n VV )Y,

10 10 Nicola Guglielmi et al. we have by (2.5) 1 2 T, Q = T, uu +u u = 2Re Tu, u = 2Re LTu,f Re YPR Q, V = Re MPR Q,G. The normalizing constraint becomes, on inserting (2.4), 1 JĖ 2 F = VQV 2 F + V QV 2 F + VQ V 2 F = Q 2 F +2 V 2 F. Because of (2.5) we obtain Q 2 F = 4 u u + uu 2 F = 8 u 2 2 = 8f Lf V 2 F = trace(g MG), where we used L 2 = L and (I n VV ) 2 = (I n VV ). Summing up, we want to find f C 2 and G C 2 2 that maximize under the quadratic constraint 2Re LTu,f +Re MPR Q,G 1 8f Lf +2trace(G MG). We thus have an optimization problem for z = vec(f,g) of the type Re(c z) max, z Kz 1 (2.7) with a symmetric positive semi-definite matrix K. Note that c is in the range of K, that is, there is z with c = K z. For c 0, the solution of (2.7) is thus given by z = z/µ with µ = z K z > 0. For c = 0, we can still choose z = z/µ = 0 with an arbitrary µ. In our situation, c = ( 2LTu MPR Q ) which yields the result. z = 1 2 ( 1 2 Tu PR Q ) ( ) f = 1 ( 1 G 2µ 2 Tu PR Q Lemma 2.1 leads us to the following system of differential equations for u(t) and V(t) where we rescale time by choosing µ = 1/2 in (2.6) (a different factor µ > 0 just leads to a different scaling of time, dt = µdt, which is irrelevant in the following): ), u = 1 2 (I 2 uu )Tu V = (I n VV )YPY VQ. (2.8) Here Y(t) = (Jx(t),y(t)) contains the left and right eigenvectors x(t) and y(t), of unit norm and with x(t) y(t) > 0, corresponding to a simple eigenvalue λ(t) of A + εe(t) with JE(t) = V(t)Q(t)V(t) ( and ) Q(t) = I 2 2u(t)u(t) with 0 1 u(t) C 2 of unit norm. Moreover, P = and T(t) = (V 1 0 YPY V)(t). We then have the following monotonicity result.

11 Low rank ODEs for Hamiltonian matrix nearness problems 11 Theorem 2.3 Let E(t) = J T V(t)(I 2u(t)u(t) )V(t) with u(t),v(t) satisfying the differential equations (2.8) and with initial values such that u(0) C 2 has unit Euclidean norm and V(0) C n 2 has orthonormal columns. If the eigenvalue λ(t) of A+εE(t) is simple, then Re λ(t) 0 with equality only if u = 0 and V = 0. Proof From the proof of Lemma 2.1 and the choice of the scaling factor µ = 1 2 we have the identity (x y)re λ = Re(x Ėy) = 1 2 T, Q +Re YPR Q, V = 2Re LTu,f +Re MPR Q,G = LTu 2 + MPR Q 2 F = 4 u 2 + V 2 F. Since x y > 0, this yields the stated result. Since Re λ(t) is bounded for all t (by the ε-pseudospectral abscissa) and Re λ(t) 0, we must have Re λ(t) 0 as t. Since 0 < x y 1, the above formula implies that u(t) and V(t) must tend to zero as t. In the following result we make the generic assumption that Y V C 2 2 is invertible. We note that if this were not satisfied, then Jx and y would span an invariant subspace of the unperturbed matrix A (in addition to A+εE). As the following theorem shows in comparison with Theorem 2.2, stationary points of (2.8) are extremizers. Let u be a vector of unit norm orthogonal to u, and let T = (u,u) T(u,u). Theorem 2.4 Suppose that Y V is invertible. If λ / ir, then (u,v) is a stationary point of (2.8) if and only if Y and V have the same range and T is diagonal. ( ) 1 0 We note that the transformation Ṽ = V(u,u) yields JE = Ṽ Ṽ 0 1, and therefore T corresponds to T of Theorem 2.2. T has the same eigenvalues as the hermitian matrix T = RPR, which are of different sign. Proof From (2.8) it is immediate that (V,u) is a stationary point if and only if the range of Y is contained in the range of V and (u ) Tu = 0. Since Y has rank 2 for an eigenvalue λ / ir, this yields the result.

12 12 Nicola Guglielmi et al. 2.3 An illustrative example Consider the Hamiltonian matrix (2.9) Its eigenvalues are given by ±0.4060i, ± i, ± i. In Figure 2.1 we show the boundary of the Hamiltonian ε-pseudospectrum, computed by Structured Eigtool [KKK10], and the path of rightmost eigenvalues of A+εE(t) for ε = 0.1, where E(t) = J T V(t)(I 2u(t)u(t) )V(t) is computed by solving (2.8) x Fig. 2.1 Hamiltonian pseudospectrum of the matrix A in Example (2.9) for ε = 0.1 and trajectory of the rightmost eigenvalue λ(t) (in blue) associated to the solution of (2.8). Right picture: zoom close to the rightmost point. We note in the right picture of Figure 2.1 that the trajectory approaches the rightmost point of the Hamiltonian ε-pseudospectrum tangentially, as it happens for the unstructured ε-pseudospectrum case. This can be explained by analogous arguments.

13 Low rank ODEs for Hamiltonian matrix nearness problems 13 3 Rank-4 dynamics to extremal points in real Hamiltonian pseudospectra 3.1 Extremizers We have the following real analog of Theorem 2.1. Theorem 3.1 For a Hamiltonian matrix A R n n and ε > 0, let λ / R ir be a locally rightmost point in the real Hamiltonian 2-norm ε-pseudospectrum, implying that there exists a real Hamiltonian matrix E R n n of unit 2-norm such that λ is an eigenvalue of A+εE. If λ is a simple eigenvalue of A+εE and not an eigenvalue of A, then there exists a unique real Hamiltonian matrix E of rank 4 such that λ is an eigenvalue of A+εE with the same left and right eigenvectors as for A + εe. Moreover, the nonzero eigenvalues of JE are +1 and 1, each of multiplicity 2. For λ R we have the rank-2 matrix E of Theorem 2.1, which is real for real λ. The following characterization of the rank-4 real Hamiltonian extremizers is given in the following simultaneous analog of Theorem 2.2 and of [GL13, Theorem 3.2] on extremizers for unstructured real perturbations. We denote by x R and x I the real part and the imaginary part of x, respectively, and similarly by y R and y I the real part and the imaginary part of y. Theorem 3.2 Assume that E = J T Vdiag(1,1, 1, 1)V T, where V R n 4 has orthonormal columns. Let λ / R ir be a simple eigenvalue of A+εE, with left and right eigenvectors x and y, respectively, both of unit norm and with x y > 0. Let Y = (Jx R,y R,Jx I,y I ) R n 4 and T R 4 4 be defined by ( ) ( ) T = RPR T with R = V T Y R 4 4 P and P = with P 0 P 2 = Then the following two statements are equivalent: 1. Every differentiable path (E(t),λ(t)) (for small t 0) such that E(t) is real Hamiltonian with E(t) 2 1 and λ(t) is an eigenvalue of A+εE(t), with E(0) = E and λ(0) = λ, has Re λ(0) ( 0. ) +T Y and V have the same range, and T = with symmetric 0 T 2 positive definite blocks T 1,T 2 R 2 2. Proof (of Theorem 3.1) Let V R n n be an orthogonal matrix obtained from a QR factorization of Y = (Jx R,y R,Jx I,y I ): ( ) Y = V R with R R Let E(t), for small t > 0, be a continuously differentiable path on the set of real Hamiltonian matrices of at most unit 2-norm, with E(0) = E, which we write as E(t) = J T VŜ(t) V T

14 14 Nicola Guglielmi et al. with a symmetric matrix Ŝ(t) R n n = ( ) S11 (t) S21(t) T, S S 21 (t) S 22 (t) 11 (t) R 4 4, We can choose the first 4 columns of the orthogonal matrix V such that S 11 (0) is diagonal. The diagonal entries σ j are real and have absolute value at most 1. Using the same arguments as in the proof of Theorem 2.1 it is shown that σ j = 1 for j = 1,2,3,4 and that two of the σ j are positive and two are negative. This implies that S 21 (0) = 0, which shows that A + εe, with the rank-4 matrix ( ) E = J T S11 (0) 0 ˆV ˆV T, 0 0 satisfies (A+εE )y = (A+εE)y = λ y and x (A+εE ) = x (A+εE) = λ x. By the same argument as in the proof of Theorem 2.1, E is the unique rank-4 matrix with this property. Proof (of Theorem 3.2) 1. implies 2.: We know from the proof of Theorem 3.1 that Y and V have the same range, and we write Y = VR with an invertible 4 4 real matrix R. We denote D = diag(1,1, 1, 1). We consider the path of matrices of unit 2-norm JE(t) = Ve tz De tz V T with a skew-symmetric matrix Z R 4 4 and the corresponding path of eigenvalues λ(t) of A+εE(t) with λ(0) = λ. By 1., we have 0 Re(x T Ė(0)y) = YPY,Ė(0) = T,ZD DZ, which holds for every skew-symmetric matrix Z. The matrix ZD DZ is symmetric and its upper and lower diagonal 2 2 blocks are zero. It thus follows that T is block-diagonal with 2 2 blocks. Moreover, T = RPR T is symmetric and nonsingular. Using the paths E(t) = Ve tm De tm V T with positive semi-definite matrices M that are block diagonal with 2 2 blocks, we then conclude from 0 Re(x Ė(0)y) = T,MD +DM that the upper diagonal block of T is positive definite and the lower block is negative definite; compare [GL13, Section 3] for an analogous situation in the characterization of extremizers among general real ε-perturbations. This proves implies 1.: This is proven by a direct generalization of the corresponding argument in the proof of Theorem 2.2.

15 Low rank ODEs for Hamiltonian matrix nearness problems Rank-4 dynamics for the real case In view of the preceding theorem, we search for a differential equation for matrices of the form JE = VQV T, where V R n 4 has orthonormal columns and Q R 4 4 is a symmetric orthogonal matrix with double eigenvalues +1 and 1, to be determined in the form of a double Householder transformation Q = I 4 2UU T, U R 4 2, U T U = I 2. For a given choice of V and Q, the tangent matrices Ė T EM can then be uniquely written as (cf. [KL07]) JĖ = VQV T +V QV T +VQ V T with V T V = 0, U T U = 0. (3.1) In the same way as in Section 2.2, we derive the following differential equations for V and U: U = 1 2 (I 4 UU T )TU V = (I n VV T )YPY T VQ. (3.2) Let us recall that Y = (Jx R,y R,Jx I,y I ) R n 4 contains the real and imaginary parts of the left and right eigenvectors x and y, of unit norm and with x y > 0, corresponding to an eigenvalue λ of A+εE with JE = VQV T and Q = I 2UU T with U having two orthonormal columns. Further, P R 4 4 and T R 4 4 are defined as in Theorem 3.2. We obtain again the desired monotonicity result. Theorem 3.3 Let E(t) = J T V(t)(I 2U(t)U(t) T )V(t) T with U(t),V(t) satisfying the differential equations (3.2) and with initial values such that U(0) R 4 2 and V(0) R n 4 have orthonormal columns. If λ(t) is a simple eigenvalue of A+εE(t), then Re λ(t) 0, with equality only if U = 0 and V = 0. Let Ũ R4 4 be an orthogonal matrix such that its last two columns are those of U, and let T = ŨTTŨ. The following characterization of the stationary points is then obtained by direct inspection of (3.2), and similar to the proof of Theorem 2.4. Theorem 3.4 Suppose that Y T V is invertible. If λ / R ir, then (U,V) is a stationary point of (3.2) if and only if Y and V have the same range and T is block-diagonal with 2 2 blocks. Note that the transformation Ṽ = VŨ yields JE = Ṽdiag(1,1, 1, 1)Ṽ T, and therefore T corresponds to T of Theorem 3.2. Since T = RPR T, we know that T always has two positive and two negative eigenvalues.

16 16 Nicola Guglielmi et al. 4 An algorithm to compute locally rightmost points in the Hamiltonian pseudospectrum and their extremizers We discretize the differential equation (2.8) or (3.2) by the explicit Euler method with an adaptively chosen stepsize. At each discretization step, we require according to the monotonicity property of Theorem 2.3 that the real part of the rightmost eigenvalue λ of A + εe is increased, for a given ε > 0. In this way the method determines a sequence (λ n,e n ) such that Reλ n > Reλ n 1, until E n approaches a stationary point. We consider here the case where the matrix A and the perturbations are complex. Given E n = J T V n Q n V n of rank 2 and unit 2-norm, such that Re(λ n ) 0, whereλ n istherightmosteigenvalueofa+εe n,wherev n C n 2 withv nv n = I 2, and u n C 2 is such that u n = 1 and hence Q n = I 2 2u n u n C 2 2 is unitary, Algorithm 1 determines an approximation at time t n+1 = t n +h n. In Algorithm 1, γ > 1 denotes a given scaling factor for the stepsize control and Orth(B) denotes the unitary matrix obtained by orthogonalizing the columns of a given (rectangular) matrix B. Algorithm 1: adaptive Euler step Data: E n,v n,q n,x n,y n,λ n and ρ n (step size predicted by previous step) Result: E n+1,v n+1,q n+1,x n+1,y n+1,λ n+1 and ρ n+1 begin 1 Set h = ρ n, Y n = (Jx n,y n ) and T n = V ny n PY nv n C Compute û n+1 = u n 1 2 hq nt n u n, u n+1 = ûn+1 û n+1 V n+1 = V n +h(i n V n V n)y n PY nv n Q n, V n+1 = Orth ( Vn+1 ) Q n+1 = I 2 2u n+1 u n+1. 3 Set E n+1 = J T V n+1 Q n+1 V n+1 4 Compute the rightmost eigentriple λ, x and ŷ of A+εE n+1 5 if Re( λ) Re(λ n ) then reject the step, reduce the stepsize to h := h/γ and repeat from 2 else accept the step: set h n+1 = h, λ n+1 = λ, x n+1 = x and y n+1 = ŷ 6 if h n+1 = ρ n then increase the stepsize to ρ n+1 := γρ n else set ρ n+1 = ρ n 7 Proceed to next step. For our purposes it is not important that E n+1 gives an accurate approximation of E(t n+1 ), the solution - at time t n+1 - of the ODEs (2.8) or (3.2)

17 Low rank ODEs for Hamiltonian matrix nearness problems 17 with initial datum E n as we need not follow the exact solution. We are only interested in increasing Re(λ n+1 ) with respect to Re(λ n ), as is ensured by Theorem 2.3 for sufficiently small step size h n > 0. We remark that the most expensive step of the algorithm is the computation of a leading eigentriple, which for a large sparse matrix can be done efficiently by using the implicitly restarted Arnoldi method, as implemented in ARPACK [LeSY98]. Initial conditions. We often make use of the following initial condition: Ẽ 0 = J T (Jx 0 y 0) herm, V 0 S 0 V 0 = JẼ0, where x 0 and y 0 are the left and right eigenvectors associated to the rightmost eigenvalue of A and S 0 is diagonal. Then we choose, according to the sign of the (diagonal) entries of S 0, and E 0 = J T V 0 Q 0 V 0. Q 0 = ( ) ±1 0 0 ±1 Terminating at a globally rightmost point. When it is crucial to find a globally (and not just locally) rightmost point in the Hamiltonian ε-pseudospectrum, we run the algorithm with a set of different initial values to reduce the possibility of getting trapped in a point that is only locally rightmost, which cannot be excluded from following a single trajectory and which we have indeed occasionally observed in our numerical experiments. According to Theorems a trajectory could even run into a stationary point for which the corresponding rightmost eigenvalue is a boundary point of the pseudospectrum that has a vertical tangent but is not locally rightmost, such as a locally leftmost boundary point on a non-convex right boundary. In view of the monotonicity of the real part of the eigenvalue along a trajectory (Theorem 2.3), it appears that only exceptional trajectories would run into a stationary point that does not correspond to a locally rightmost point of the pseudospectrum, and in fact we never observed such a situation in our extensive numerical experiments. For conciseness we do not formulate the algorithm for the real Hamiltonian case, which has a structure very similar to Algorithm 1. 5 Algorithms for Problem (A) In this section, we present two methods, making use of Algorithm 1, for addressing a generalization of Problem (A) from the introduction: Given a Hamiltonian matrix A with no eigenvalues on the imaginary axis and a given constant δ > 0, find a nearest (in the 2-norm) Hamiltonian matrix

18 18 Nicola Guglielmi et al. B having some eigenvalue δ-close to the imaginary axis. The algorithm has the goal to compute ε = inf{ε 0 : Λ ε (A,M, 2 ) S δ }, (5.1) where M is the set of Hamiltonian matrices and S δ = {z C : Re(z) δ}. Algorithm 1 is used to determine the extremizer E such that the rightmost eigenvalue of B = A + ε E in the left half-plane has real part δ. Since Algorithm 1 finds locally rightmost points, the algorithms we present here are guaranteed only to find upper bounds on ε ; nevertheless they typically find good approximations for the cases we have tested. 5.1 The left pseudospectral abscissa In order to compute the value of ε defined in (5.1), we make use of the following definition. The left ε-pseudospectral abscissa of a Hamiltonian matrix A is the real part of the right-most point of the Hamiltonian ε-pseudospectrum in the left complex plane C = {z : Re(z) 0}, i.e., α l ε(a) = max{rez : z Λ ε (A,M, ) C }. (5.2) Starting from ε > 0 such that α l ε(a) < 0, we want to compute a root ε of the equation α l ε(a) = δ. (5.3) By the monotonicity of pseudospectral sets, α l ε(a) is monotonically increasing with ε. Under a generic assumption, it is also strictly increasing, and then ε of (5.1) is the unique solution of (5.3). Remark 5.1 Since the function ε α l ε(a) is monotonically increasing, we can easily apply a bisection method in order to compute a zero of Equation (5.3). Having said this, our aim is to obtain a fast method to compute such a zero, which turns out to be possible under some additional assumptions, as we are going to explain. When such assumptions are not satisfied, we can always fall back to the bisection method. In this section we use the following notation: all quantities written as g(ε), like λ(ε), E(ε) and so on, are intended to be associated to stationary points (i.e., local extremizers) of (2.8) (or (3.2)). We make the following generic assumption for all ε near ε, for ε < ε. Assumption 5.1 Let λ(ε) with Reλ(ε) 0 be a locally rightmost point in the Hamiltonian ε-pseudospectrum of A and E(ε) the corresponding extremizer (with E(ε) of unit norm and Hamiltonian), that is λ(ε) is a rightmost eigenvalue of A+εE(ε). Then we assume that λ(ε) is simple and the mapping ε E(ε) is smooth. This clearly implies that also the mapping ε λ(ε) is smooth and the same holds for suitably normalized eigenvectors x(ε) and y(ε) (see e.g. [Kat95, MS88]).

19 Low rank ODEs for Hamiltonian matrix nearness problems 19 Remark 5.2 Assumption 5.1 requires the matrix E(ε), which is obtained from solving the differential equation (2.8) for a given ε, to depend smoothly on ε. Although the Newton method proposed in this section relies on this requirement, we will combine it with a bisection method to guarantee convergence also in cases where the solutions are not smooth. On the other hand, we note that in all numerical experiments we have performed, we have always observed the fast convergence resulting from such a smoothness assumption. This leads us to conjecture that Assumption 5.1 holds true generically. In order to derive an equation for ε to approximate ε, we compute the derivative of the left ε-pseudospectral abscissa, with respect to ε. α l ε(a) = Reλ(ε), Theorem 5.1 Let λ(ε) be a locally rightmost point of the complex or real Hamiltonian ε-pseudospectrum of A, and let E(ε) the corresponding extremizer (with E(ε) of unit norm and Hamiltonian) for ε [ε,ε). Suppose that Reλ(ε) 0 for all ε [ε,ε), and that Assumption 5.1 holds. Let x(ε) and y(ε) be left and right eigenvectors of A + εe(ε) belonging to the eigenvalue λ(ε). We then have dre λ(ε) dε = Re x(ε) E(ε)y(ε) x(ε) y(ε) 0. (5.4) Proof By Lemma 1.1, λ (ε) = x(ε) ( E(ε)+εE (ε) ) y(ε) x(ε), y(ε) where indicatesdifferentiationwithrespecttoε.letλ ε (t),forsmall t,denote the path of eigenvalues of A+εE(ε+t) with λ ε (0) = λ(ε). The maximality property of the real part of the eigenvalue λ(ε) of A+εE(ε) yields and hence 0 = Re dλ ε dt (0) = Re x(ε) E (ε)y(ε) x(ε) y(ε) Re(x(ε) E (ε)y(ε)) = 0. (5.5) This yields the stated formula. The non-negativity of monotonicity of pseudospectral sets. d Reλ(ε) is due to the dε Intuitively, the real part of a locally right-most point λ(ε) C should provide an indication how much ε needs to be increased for λ(ε) to hit the imaginary axis. In order to investigate this, we make the following assumption.

20 20 Nicola Guglielmi et al. Assumption 5.2 Let ε be such that for ε < ε, λ(ε) is a locally rightmost point λ(ε) with Reλ(ε) < 0 and limreλ(ε) = 0. We assume that the eigenvalue εրε λ(ε) of A+εE(ε) has algebraic multiplicity two. Moreover, we assume that λ(ε) is defective or, equivalently, that the zero singular value of A+εE(ε) λ(ε)i is simple. A few comments about Assumption 5.2 are needed. Here and in the following λ(ε), the left and right eigenvectors x(ε) and y(ε), and the extremizer E(ε) are considered as the limits of the corresponding quantities for ε ր ε. Assumption 5.2 ensures that these limits exist, see [ConH80] for the eigenvectors and part (c) of the proof of Theorem 5.2 for E(ε). Assumption 5.2 is generically valid for the set of unstructured matrices that have a multiple eigenvalue, see Remark 2.16 in [DP09]. We conjecture that Assumption 5.2 is also generically valid for the situation under consideration. We emphasize that Assumption 5.2 is useful to justify a fast algorithm for the computation of the distances we are interested in, through the next Theorem 5.2, but it is not essential for the two-level approach taken here. In fact, we have to compute the zero of a monotonically increasing function (see Theorem 5.1), which can be done - although less efficiently - by a bisection-like method, see also Remark 5.1. Theorem 5.2 Consider the setting of Theorem 5.1 with E(ε) being an extremizer having the form stated in Theorem 2.2. Under Assumption 5.2 we have 0 < lim εրε Reλ(ε) Reλ (ε) <. Moreover we have for some positive constant c. Reλ(ε) = c ε ε(1+o(1)), ε ր ε, (5.6) Proof (a) Let us consider the nonnegative function δ(ε) := x(ε) y(ε) > 0 ε [ε,ε), δ(ε) = 0, (recall that by the normalization considered in this paper x(ε) y(ε) > 0 for a simple eigenvalue). In order to prove the result we first study δ(ε) for ε close to ε and show that δ(ε) = O( ε ε). Then we show that Re(λ(ε)) is infinitesimal to the same order of δ(ε) in a neighbourhood of ε. In order to compute the derivative of δ(ε) we make use of the formulas in [MS88] for the derivative of the eigenvectors (x(ε) and y(ε) of unit 2-norm and such that x(ε) y(ε) > 0) associated to a simple eigenvalue of a smooth matrix valued function M(ε), x (ε) = x(ε) M (ε)g(ε)x(ε)x(ε) x(ε) M (ε)g(ε) y (ε) = y(ε) G(ε)M (ε)y(ε)y(ε) G(ε)M (ε)y(ε), (5.7)

21 Low rank ODEs for Hamiltonian matrix nearness problems 21 where G(ε) is the group inverse of M(ε) λ(ε)i. Exploiting the well-known property G(ε)y(ε) = 0, x(ε) G(ε) = 0, these formulas imply δ (ε) = x(ε) M (ε)g(ε)x(ε)δ(ε)+y(ε) G(ε)M (ε)y(ε)δ(ε). (5.8) Using a result in[go11] relating the group inverse and the pseudoinverse of amatrixwithasimplezeroeigenvalue,wehave(b indicatesthepseudoinverse of B) G(ε) = 1 δ(ε) 2Ĝ(ε) Ĝ(ε) = ( δ(ε)i y(ε)x(ε) )( M(ε) λ(ε)i ) ( δ(ε)i y(ε)x(ε) ). (5.9) Let M(ε) = A+εE(ε) so that M (ε) = E(ε)+εE (ε). By Assumption 5.2 we get that (M(ε) λ(ε)i) remains bounded as ε ր ε, and thus Ĝ(ε) = y(ε)x(ε) ( M(ε) λ(ε)i) y(ε)x(ε). Next define N(ε) = M(ε) λ(ε)i and Using (5.9) and (5.5), we obtain κ(ε) = x(ε) N(ε) y(ε). (5.10) x(ε) ( E(ε)+εE (ε))ĝ(ε)x(ε) = κ(ε)x(ε) E(ε)y(ε)+O(δ(ε)) (5.11) ( ) y(ε) Ĝ(ε) E(ε)+εE (ε) y(ε) = κ(ε)x(ε) E(ε)y(ε)+O(δ(ε)). (5.12) We show in parts (b) and (c) of the proof that the right-hand side in these equations is different from zero for ε close to ε. (b) We prove that κ(ε) 0. Since δ(ε) = 0, we have x(ε) y(ε), and since x(ε) spans the nullspace of N(ε) (by the defectivity condition in Assumption 5.2), we obtain y(ε) Ker(N(ε) ) = Range(N(ε)) = z 1 (ε) s.t. N(ε)z 1 (ε) = y(ε). (5.13) Assume, in a proof by contradiction, N(ε) y(ε) x(ε), which means N(ε) y(ε) Range(N(ε)) = z 2 (ε) s.t. N(ε)z 2 (ε) = N(ε) y(ε). (5.14) Premultiplying (5.14) by N(ε) 2 we get (omitting the argument ε in the following) N 3 z 2 = N 2 N y = N NN Nz 1 = NNz 1 = Ny = 0. The null-space of N 3 is two-dimensional by Assumption 5.2 and contains the two nonzero vectors y and N y, since Ny = 0 and N 2 N y = 0. Note that N y 0 because otherwise y would be in the nullspace of N, which is the

22 22 Nicola Guglielmi et al. nullspace of N, which contradicts the above observation that y 0 is in the orthogonal complement of the nullspace of N. Moreover, y and N y are linearly independent, since otherwise the relation y = cn y would yield, on multiplication with N, that 0 = Ny = cnn y = cnn Nz 1 = cnz 1 = cy, which contradicts y 0. Therefore, the null-space of N 3 is spanned by y and N y, and since we have shown that it contains z 2, we obtain z 2 = c 1 y +c 2 N y. Premultiplying this equation with N then gives N y = Nz 2 = c 2 NN Nz 1 = c 2 Nz 1 = c 2 y, which contradicts the linear independence of y and N y. We have thus led the assumption N(ε) y(ε) x(ε) to a contradiction. Therefore, κ(ε) 0. (c) We now show that x(ε) E(ε)y(ε) 1 for ε ε. First we note that, for the eigenvalue λ(ε) ir, the left and right eigenvalues are related by y(ε) = Jx(ε). The characterization 2. of Theorem 2.2 yields that for ε < ε, where V(ε) diagonalizes JE(ε), Y(ε) = (Jx(ε),y(ε)) = V(ε)R(ε), JE(ε) = V(ε) ( ) 1 0 V(ε), 0 1 and R C 2 2. For ε ε we thus obtain ( ) ρ ρ R(ε) = with ρ 2 + σ 2 = 1. σ σ Since again by 2. of Theorem 2.2 we have for ε < ε ( ) ( ) 0 1 R(ε) R(ε) τ1 (ε) 0 = with τ τ 2 (ε) 1 (ε),τ 2 (ε) > 0, for ε ε we must have ( ) ρ 2 2 ρσ ρσ σ 2 = ( ) τ1 (ε) 0 0 τ 2 (ε) with τ 1 (ε),τ 2 (ε) 0, which implies σ = 0 and hence ( ) ρ ρ R(ε) = 0 0 with ρ 2 = 1.

23 Low rank ODEs for Hamiltonian matrix nearness problems 23 This yields which implies V(ε) y(ε) = ( ) ρ, 0 x(ε) E(ε)y(ε) = y(ε) JE(ε)y(ε) = y(ε) V(ε) (d) Inserting (5.11) and (5.12) into (5.8) we get δ (ε) = 2κ(ε) x(ε) E(ε)y(ε) δ(ε) ( ) 1 0 V(ε) y(ε) = (1+O(δ(ε))). (5.15) Equation (5.15) implies that lim εրε δ(ε)δ (ε) is finite and is different from zero. The solution of the initial value problem δ (ε) = d (1+o(1)), δ(ε) = 0 (5.16) δ(ε) where d is a positive constant (this is due to the fact that δ decreases in a left neighbourhood of ε), is Observing that (see Theorem 5.1) δ(ε) = 2d ε ε ( 1+o(1) ). (5.17) Reλ (ε) = 1 2Reκ(ε) δ (ε) ( 1+O(δ(ε)) ) with Reλ(ε) = 0 and Reκ(ε) bounded away from zero in a neighbourhood of ε, we get (5.6) and conclude the proof. 5.2 A Newton bisection method Recalling that we have introduced the notation α l ε(a) = Reλ(ε) (see (5.2)), according to Theorem 5.2 we have that the function ε α l ε(a) has a graph like that shown in Figure 5.2. For conciseness we omit the dependence on A in the coded algorithms and denote it simply by α l ε. We consider a given δ > 0. In view of Theorem 5.1, it seems very natural to make use of a Newton iteration, ε n+1 = ε n ( α l ε n +δ ) x(ε n ) y(ε n ) Re(x(ε n ) E(ε n )y(ε n )), n = 0,1,... possibly coupled with bisection (see Algorithm 2 where we denote by dα l ε the derivative of α l ε with respect to ε stated in Theorem 5.1). The method yields quadratic convergence to ε when this is a simple root.

24 24 Nicola Guglielmi et al. Algorithm 2: Newton-bisection method for Problem (A) Data: Matrix A Hamiltonian, k max (max number of iterations), tol (tolerance) ε lb and ε ub (starting values for the lower and upper bounds for ε ) Result: ε (upper bound for the measure) begin 1 Set λ(0) rightmost eigenvalue of A in the left complex plane, x(0) and y(0) the corresponding left and right eigenvectors Set ε 1 = 0 2 Initialize E(ε 1 ) (according to the setting) Compute α l ε 1, dα l ε 1 3 Set ε 0 = α l ε 1 /dα l ε 1 4 Set k = 0 5 Initialize lower and upper bounds: ε lb = 0, ε ub = + while α l ε k α l ε k 1 < tol do 6 Compute α l ε k, E(ε k ) by solving the ode (2.8) with initial datum E(ε k 1 ) 7 Update upper and lower bounds ε lb, ε ub 8 if α l ε k = 0 then Set ε ub = min(ε ub,ε k ) Compute ε k+1 = (ε lb +ε ub )/2 (bisection step) else Set ε lb = max(ε lb,ε k ) 9 Compute dα l ε k 10 Compute ε k+1 = ε k α l ε k /dα l ε k (Newton step) if k = k max then Halt else Set k = k Return ε = ε k This works very well when δ is not too small (see the following Example 5.1) since when δ is very small, that is when ε is very close to ε, the loss of differentiability of the function Re(λ(ε)) at ε leads to difficulties in the Newton iteration and results in a bisection-like behaviour of the method. A good choice for ε 0 is the unstructured distance from the imaginary axis, that is the norm of the smallest unstructured matrix F such that A+F has a purely imaginary eigenvalue. Such a distance can be effectively computed by combining the method proposed in [GO11, GL11] with a Newton iteration. Example 5.1 Consider the Hamiltonian matrix i 2 2+2i 1 i i 1 i 1+2i 2 2i 0 1+i A = 2+i 1 i 1+2i 1+i 1 i i 4 i 2+i i 1 1 i 1 i 1 i 1+2i 0 3 i 1+2i 1+2i

25 Low rank ODEs for Hamiltonian matrix nearness problems Fig. 5.1 Hamiltonian 2-norm pseudospectra of the matrix A in Example 5.1 for ε = (left picture) and ε = ε 10 in Table 5.2 (right picture). (i) We first consider the choice δ = 0.5 and obtain the results shown in Table 5.1. k ε k α l ε k Table 5.1 Computed values of ε and α l ε for Example 5.1 with δ = 0.5. The convergence appears to be quadratic, as expected, and ε 4 provides a very accurate estimate. (ii) Then we consider the case δ = In this case the method behaves like bisection as is reported by Table 5.2, where we indicate by 0 a situation when the considered ε is larger than ε, meaning that ε is in the part of the domain where α l ε(a) is identically zero. Table 5.2 shows that after 10 iterations the solution is not yet fully accurate (compare to the following Table 5.3). In order to obtain a faster convergence when δ is small, in the next section we propose a method which is based on an interpolation idea exploiting Theorem 5.2 (see Figure 5.2). 5.3 An algorithm for small δ Let the distance δ > 0 be given, with δ small, and let ε denote a value of ε such that the Hamiltonian pseudospectrum of A touches the imaginary axis. Note that ε is the limit of ε (see (5.1)) as δ 0 +.

26 26 Nicola Guglielmi et al. k ε k α l ε k Table 5.2 Computed values of ε and α l ε for Example 5.1 with δ = δ ε Fig. 5.2 The function α l ε(a) in a neighbourhood of ε. For ε ε it becomes identically zero. The computational problem consists in finding the value ε such that α l ε (A) = δ, as shown in the picture. Close to the imaginary axis we make use of Theorem 5.2 and use the expansion (which is meaningful only when ε ε), ε ε γα l ε(a) 2, for ε < ε, (5.18) with γ > 0. Given ε k, we use Theorem 5.1 and estimate γ and ε by γ k and ε k using the formulæ (the first of which is obtained by differentiating (5.18) with respect to ε), γ k = x (ε k )y(ε k ) 2 α l ε k (A) Re(x (ε k )(E(ε k ))y(ε k )) (5.19) ε k = ε k +γ k (α l ε k ) 2 (5.20)