Research Matters. February 25, The Nonlinear Eigenvalue Problem. Nick Higham. Part III. Director of Research School of Mathematics

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1 Research Matters February 25, 2009 The Nonlinear Eigenvalue Problem Nick Higham Part III Director of Research School of Mathematics Françoise Tisseur School of Mathematics The University of Manchester ESSAM, Kácov, / 6

2 Outline Two lectures: Part I: Mathematical properties of nonlinear eigenproblems (NEPs) Definition and historical aspects Examples and applications Solution structure Part II and Part III: Numerical methods for NEPs Solvers based on Newton s method Solvers using contour integrals Linear interpolation methods S. GÜTTEL AND F. TISSEUR, The nonlinear eigenvalue problem. Acta Numerica 26:1 94, Françoise Tisseur The nonlinear eigenvalue problem 2 / 26

3 Methods based on linear interpolation Instead of solving F(λ)v = 0 directly, we may approximate F by simpler function R m on Σ C and solve R m (λ)v = 0. Françoise Tisseur The nonlinear eigenvalue problem 3 / 26

4 Methods based on linear interpolation Instead of solving F(λ)v = 0 directly, we may approximate F by simpler function R m on Σ C and solve R m (λ)v = 0. Practical approach: Polynomial or rational form R m (z) = b 0 (z)d 0 + b 1 (z)d b m (z)d m where D j C n n are constant coefficient matrices and b j are polynomials or rational functions of type (m, m). Françoise Tisseur The nonlinear eigenvalue problem 3 / 26

5 Methods based on linear interpolation Instead of solving F(λ)v = 0 directly, we may approximate F by simpler function R m on Σ C and solve R m (λ)v = 0. Practical approach: Polynomial or rational form where R m (z) = b 0 (z)d 0 + b 1 (z)d b m (z)d m D j C n n are constant coefficient matrices and b j are polynomials or rational functions of type (m, m). It is crucial that R m F is some sense, for otherwise eigenpairs of F and R m are not related. Françoise Tisseur The nonlinear eigenvalue problem 3 / 26

6 Basic Properties of Approximant R m We impose that the approximation R m on Σ Ω to F H(Ω, C n n ) satisfies for some small ε. F R m Σ := max z Σ F(z) R m(z) 2 ε, Françoise Tisseur The nonlinear eigenvalue problem 4 / 26

7 Basic Properties of Approximant R m We impose that the approximation R m on Σ Ω to F H(Ω, C n n ) satisfies for some small ε. F R m Σ := max z Σ F(z) R m(z) 2 ε, Let λ Σ and v C n s.t. v 2 = 1 and R m (λ)v = 0. Then F(λ)v 2 = ( F(λ) R m (λ) ) v 2 F(λ) R m (λ) 2 ε, i.e., bounded residual. Ideally, R m should not any ei vals in Σ that are in the resolvent set of F (i.e., R m should be free of spurious ei vals). Françoise Tisseur The nonlinear eigenvalue problem 4 / 26

8 Rational Newton Basis Functions Approximate F H(Ω, C n n ) by R m (z) = b 0 (z)d 0 + b 1 (z)d b m (z)d m H(Σ, C n n ), where Σ Ω, D j C n n and b j are rational functions. Particularly useful: (scaled) rational Newton basis b 0 (z) 1 β 0, b j+1 (z) = z σ j β j+1 (1 z/ξ j+1 ) b j(z) with interpolation points σ j Σ, poles ξ j C \ Σ, and scaling factors β j 0. Françoise Tisseur The nonlinear eigenvalue problem 5 / 26

9 Rational Newton Basis Functions Approximate F H(Ω, C n n ) by R m (z) = b 0 (z)d 0 + b 1 (z)d b m (z)d m H(Σ, C n n ), where Σ Ω, D j C n n and b j are rational functions. Particularly useful: (scaled) rational Newton basis b 0 (z) 1 β 0, b j+1 (z) = z σ j β j+1 (1 z/ξ j+1 ) b j(z) with interpolation points σ j Σ, poles ξ j C \ Σ, and scaling factors β j 0. Shifted and scaled monomials: σ j = σ, ξ j =. Scaled Newton polynomials: ξ j =. Françoise Tisseur The nonlinear eigenvalue problem 5 / 26

10 Rational Newton Basis Functions Approximate F H(Ω, C n n ) by R m (z) = b 0 (z)d 0 + b 1 (z)d b m (z)d m H(Σ, C n n ), where Σ Ω, D j C n n and b j are rational functions. Particularly useful: (scaled) rational Newton basis b 0 (z) 1 β 0, b j+1 (z) = z σ j β j+1 (1 z/ξ j+1 ) b j(z) with interpolation points σ j Σ, poles ξ j C \ Σ, and scaling factors β j 0. Shifted and scaled monomials: σ j = σ, ξ j =. Scaled Newton polynomials: ξ j =. Choice of σ j, ξ j, β j by NLEIGS sampling [Güttel et al 2014]. Françoise Tisseur The nonlinear eigenvalue problem 5 / 26

11 NLEIGS sampling Assume F is holomorphic on Ω = C \ Ξ and we target the eigenvalues in Σ Ω. Assume we have chosen nodes σ 0, σ 1,..., σ m Σ and poles ξ 1,..., ξ m Ξ. Define s m (z) := (z σ m )b m (z). By the Hermite Walsh formula we have F(z) R m (z) = 1 s m (z) F(ζ) 2πi s m (ζ) ζ z dζ, and so the uniform approximation error on Σ satisfies F R m Σ := max z Σ F(z) R m(z) 2 C s m Σ s 1 m Γ Γ Aim: Make s m small on Σ and large on Γ. Françoise Tisseur The nonlinear eigenvalue problem 6 / 26

12 NLEIGS sampling of F on Σ Ξ Σ Françoise Tisseur The nonlinear eigenvalue problem 7 / 26

13 NLEIGS sampling of F on Σ 1. Discretize boundaries of Σ and Ξ sufficiently fine. Ξ Σ Françoise Tisseur The nonlinear eigenvalue problem 7 / 26

14 NLEIGS sampling of F on Σ 1. Discretize boundaries of Σ and Ξ sufficiently fine. 2. Choose interpolation node σ 0 Σ and set R 0 (z) F(σ 0 ). Ξ Σ Françoise Tisseur The nonlinear eigenvalue problem 7 / 26

15 NLEIGS sampling of F on Σ 1. Discretize boundaries of Σ and Ξ sufficiently fine. 2. Choose interpolation node σ 0 Σ and set R 0 (z) F(σ 0 ) error F R j Σ Σ Ξ Françoise Tisseur The nonlinear eigenvalue problem 7 / 26

16 NLEIGS sampling of F on Σ 1. Discretize boundaries of Σ and Ξ sufficiently fine. 2. Choose interpolation node σ 0 Σ and set R 0 (z) F(σ 0 ). 3. For j = 1, 2,..., m choose σ j and ξ j such that max s j 1(z) = s j 1 (σ j ) and min s j 1(z) = s j 1 (ξ j ). z Σ z Ξ 10 0 error F R j Σ Σ Ξ Françoise Tisseur The nonlinear eigenvalue problem 7 / 26

17 NLEIGS sampling of F on Σ 1. Discretize boundaries of Σ and Ξ sufficiently fine. 2. Choose interpolation node σ 0 Σ and set R 0 (z) F(σ 0 ). 3. For j = 1, 2,..., m choose σ j and ξ j such that max s j 1(z) = s j 1 (σ j ) and min s j 1(z) = s j 1 (ξ j ). z Σ z Ξ Choose β j such that s j Σ = 1 and set D j = F (σ j ) R j 1 (σ j ) b j (σ j ) error F R j Σ Σ Ξ Françoise Tisseur The nonlinear eigenvalue problem 7 / 26

18 NLEIGS sampling of F on Σ 1. Discretize boundaries of Σ and Ξ sufficiently fine. 2. Choose interpolation node σ 0 Σ and set R 0 (z) F(σ 0 ). 3. For j = 1, 2,..., m choose σ j and ξ j such that max s j 1(z) = s j 1 (σ j ) and min s j 1(z) = s j 1 (ξ j ). z Σ z Ξ Choose β j such that s j Σ = 1 and set D j = F (σ j ) R j 1 (σ j ) b j (σ j ) error F R j Σ Σ Ξ Françoise Tisseur The nonlinear eigenvalue problem 7 / 26

19 NLEIGS sampling of F on Σ 1. Discretize boundaries of Σ and Ξ sufficiently fine. 2. Choose interpolation node σ 0 Σ and set R 0 (z) F(σ 0 ). 3. For j = 1, 2,..., m choose σ j and ξ j such that max s j 1(z) = s j 1 (σ j ) and min s j 1(z) = s j 1 (ξ j ). z Σ z Ξ Choose β j such that s j Σ = 1 and set D j = F (σ j ) R j 1 (σ j ) b j (σ j ) error F R j Σ Σ Ξ Françoise Tisseur The nonlinear eigenvalue problem 7 / 26

20 NLEIGS sampling of F on Σ 1. Discretize boundaries of Σ and Ξ sufficiently fine. 2. Choose interpolation node σ 0 Σ and set R 0 (z) F(σ 0 ). 3. For j = 1, 2,..., m choose σ j and ξ j such that max s j 1(z) = s j 1 (σ j ) and min s j 1(z) = s j 1 (ξ j ). z Σ z Ξ Choose β j such that s j Σ = 1 and set D j = F (σ j ) R j 1 (σ j ) b j (σ j ) error F R j Σ Σ Ξ Françoise Tisseur The nonlinear eigenvalue problem 7 / 26

21 NLEIGS sampling of F on Σ 1. Discretize boundaries of Σ and Ξ sufficiently fine. 2. Choose interpolation node σ 0 Σ and set R 0 (z) F(σ 0 ). 3. For j = 1, 2,..., m choose σ j and ξ j such that max s j 1(z) = s j 1 (σ j ) and min s j 1(z) = s j 1 (ξ j ). z Σ z Ξ Choose β j such that s j Σ = 1 and set D j = F (σ j ) R j 1 (σ j ) b j (σ j ) error F R j Σ Σ Ξ Françoise Tisseur The nonlinear eigenvalue problem 7 / 26

22 NLEIGS sampling of F on Σ 1. Discretize boundaries of Σ and Ξ sufficiently fine. 2. Choose interpolation node σ 0 Σ and set R 0 (z) F(σ 0 ). 3. For j = 1, 2,..., m choose σ j and ξ j such that max s j 1(z) = s j 1 (σ j ) and min s j 1(z) = s j 1 (ξ j ). z Σ z Ξ Choose β j such that s j Σ = 1 and set D j = F (σ j ) R j 1 (σ j ) b j (σ j ) error F R j Σ /cap(Σ,Ξ) Σ Ξ Françoise Tisseur The nonlinear eigenvalue problem 7 / 26

23 Other Interpolation Techniques These include Chebyshev interpolation: for NEPs with ei vals located on the real line or on pre-specified curves in C [Effenberger and Kressner 2012], Automatic rational approximation using the adaptive Antoulas-Anderson (AAA) algorithm [Nakatsukasa et al. (2017)], [Lietaert et al. (2018)]: F(λ) R m (λ) = P(λ) + m (A i λb i )r i (λ), i=1 where P(λ) is a matrix polynomial and r i (λ) are rational approximants in barycentric form. Françoise Tisseur The nonlinear eigenvalue problem 8 / 26

24 Eigenvalue Solution Interpolation techniques can be combined with a linearization of R m, i.e., R m (λ)x = 0 is rewritten as a structured generalized eigenproblem (GEP) of larger dimension. Solve the GEP: Av = λbv e.g., QZ algorithm for small dense problems, (rational) Krylov algorithms for large sparse problems. Françoise Tisseur The nonlinear eigenvalue problem 9 / 26

25 Polynomial Eigenvalue Problems (PEPs) Consider n n matrix polynomial P(λ) = λ d A d + + λa 1 + A 0 (monomial form) = φ d (λ)p d + + φ 1 (λ)p 1 + φ 0 (λ)p 0 (non-monomial form), where φ i (λ) is a polynomial of degree i. Françoise Tisseur The nonlinear eigenvalue problem 10 / 26

26 Polynomial Eigenvalue Problems (PEPs) Consider n n matrix polynomial P(λ) = λ d A d + + λa 1 + A 0 (monomial form) = φ d (λ)p d + + φ 1 (λ)p 1 + φ 0 (λ)p 0 (non-monomial form), where φ i (λ) is a polynomial of degree i. Algorithms for dense and large sparse PEPs (monomial form). Algorithms for PEPs in non-monomial form. Françoise Tisseur The nonlinear eigenvalue problem 10 / 26

27 Condensed Forms P(λ) = d i=0 λi A i, A i C n n, A d 0. Generalized Schur decomposition: there exist U, V unitary s.t. U(λA 1 + A 0 )V = λt + S is upper triangular. λ j = s jj /t jj, j = 1: n. Can be computed by the QZ algorithm. Useful for purging and locking of e vals, implicit restart,... No analog of generalized Schur decomposition when d > 1. Françoise Tisseur The nonlinear eigenvalue problem 11 / 26

28 Condensed Forms P(λ) = d i=0 λi A i, A i C n n, A d 0. Generalized Schur decomposition: there exist U, V unitary s.t. U(λA 1 + A 0 )V = λt + S is upper triangular. λ j = s jj /t jj, j = 1: n. Can be computed by the QZ algorithm. Useful for purging and locking of e vals, implicit restart,... No analog of generalized Schur decomposition when d > 1. Rewrite P(λ)x = 0 as a linear eigenproblem (A λb)ξ = 0 of larger dimension (usually dn dn). Françoise Tisseur The nonlinear eigenvalue problem 11 / 26

29 Standard Solution Process (d = 2) Find all λ and x satisfying Q(λ)x = (λ 2 M + λd + K )x = 0. Commonly solved by linearization: Convert Q(λ)x = 0 into (A λb)ξ = 0, e.g., [ ] [ ] K 0 D M A λb = λ, ξ = 0 I I 0 [ x λx Solve (A λb)ξ = 0 with an eigensolver for generalized eigenproblem (e.g., QZ algorithm). Recover eigenvectors of Q(λ) from those of A λb. Solution process extend to degree d > 2. Numerical issues with this process. ]. Françoise Tisseur The nonlinear eigenvalue problem 12 / 26

30 Example 2: Beam Problem /////// L //////////// Transverse displacement u(x, t) /////// ρa 2 u t 2 + c(x) u t + EI 4 u x 4 = 0. u(0, t) = u (0, t) = u(l, t) = u (L, t) = 0. Finite element method leads to Q(λ)v = (λ 2 M + λd + K )v = 0 with symmetric M, D, K R n n. M > 0, K > 0, D 0 all e vals have Re(λ) 0. D is rank 1. Can show n pure imaginary e vals. [Higham et al, 2018]. Françoise Tisseur The nonlinear eigenvalue problem 13 / 26

31 Computed Spectra of C 1, L 1 and L 2 C 1 (λ) = λ [ ] [ ] M 0 D K +, 0 I I 0 [ ] [ ] [ ] [ ] M 0 D K 0 M M 0 L 1 (λ) = λ +, L 0 K K 0 2 (λ) = λ + M D 0 K Françoise Tisseur The nonlinear eigenvalue problem 14 / 26

32 Computed Spectra of C 1, L 1 and L 2 C 1 (λ) = λ [ ] [ ] M 0 D K +, 0 I I 0 [ ] [ ] [ ] [ ] M 0 D K 0 M M 0 L 1 (λ) = λ +, L 0 K K 0 2 (λ) = λ + M D 0 K C 1 (λ) L 1 (λ) L 2 (λ) 4 x x x Françoise Tisseur The nonlinear eigenvalue problem 14 / 26

33 Sensitivity and Stability of Linearizations Developed theory concerning the sensitivity and stability of linearizations [Higham, Mackey, T. 06, Higham, Li, T. 07, Grammont, Higham, T., 11]. Importance of scaling QEPs/PEPs before computing e vals via linearization. Eigenvalue parameter scaling: λ = γµ, Q(µ) := δq(γµ). Does not affect sparsity of matrix coeffs. γ, δ chosen to improve growth factors κ L (λ)/κ Q (λ) (conditioning), η Q (z i, λ)/η L (z, λ) (backward error). Françoise Tisseur The nonlinear eigenvalue problem 15 / 26

34 Spectrum of C 1, L 2 Before/After Scaling 4 x x x x Françoise Tisseur The nonlinear eigenvalue problem 16 / 26

35 quadeig for Q(λ) = λ 2 M + λd + K. Eigensolver for dense (small to medium size) quadratics quadeig. Incorporates: Appropriate [ choice ] [ of linearization: ] D I M 0 uses λ. K 0 0 I Deflation of 0 and eigenvalues. Eigenvalue parameter scaling (FLV/tropical). Careful recovery of the eigenvectors. Backward stable when D < ( M K ) 1/2. MATLAB and Fortran implementations (NAG, LAPACK) [Hammarling, Munro, T. 2013]. Françoise Tisseur The nonlinear eigenvalue problem 17 / 26

36 Eigensolvers for Large Sparse QEPs Second Order Arnoldi (SOAR) method [Bai & Su, 2005]. Projection applied directly to quadratic. Quadratic Arnoldi method [Meerbergen, 2008]. Two-level Orthogonal Arnoldi method [Lu, Su, Bai, 2016]. Françoise Tisseur The nonlinear eigenvalue problem 18 / 26

37 Eigensolvers for Large Sparse QEPs Second Order Arnoldi (SOAR) method [Bai & Su, 2005]. Projection applied directly to quadratic. Quadratic Arnoldi method [Meerbergen, 2008]. Two-level Orthogonal Arnoldi method [Lu, Su, Bai, 2016]. See Ivana Šain Glibic s talk for implicit restart. Extend to higher degree matrix polynomials. Françoise Tisseur The nonlinear eigenvalue problem 18 / 26

38 PEPs in Non-monomial Form Let P(λ) = P 0 φ 0 (λ) + P 1 φ 1 (λ) + + P d φ d (λ), where P k C n n and the φ k (λ) are rational functions/polynomials. shifted and scaled monomials, φ 0 (λ) = 1/β 0, φ j (λ) = s σφ j 1 (λ)/β j, j 1, orthogonal polynomials, φ 1 (λ) = 0, φ 0 (λ) = 1, sφ j (λ) = α j φ j+1 (λ) + β j φ j (λ) + γ j φ j 1 (λ), j 0, rational Newton basis functions,.... φ 0 (λ) = 1 β 0, φ j (λ) = s σ j β j (1 s/ζ j ) φ j 1(λ), j 1, Françoise Tisseur The nonlinear eigenvalue problem 19 / 26

39 Basis Functions Matrix Relation The linear recurrence relation between the basis functions can be rewritten in matrix form as M d Φ d (λ) = λn d Φ d (λ), where Φ d (λ) = [φ 0 (λ), φ 1 (λ),..., φ d (λ)] T and M d, N d C d (d+1) are well-defined matrices. E.g., for rational Newton basis, σ 1 β 1 1 β 1 /ζ 1 σ 2 β 2 1 β 2 /ζ 2 M d =......, N d = σ d β d 1 β d /ζ d Françoise Tisseur The nonlinear eigenvalue problem 20 / 26

40 CORK Pencil Rewrite P(λ) as d 1 g(λ)p(λ) = (A j λb j )φ j (λ) = (A λb)(φ d 1 (λ) I n ), j=0 A j, B j depend on coeffs P j of P(λ), g(λ) = 1 for poly. bases, g(λ) = (1 λ ζ d ) for rat. basis, A λb = [ A 0 λb 0 A d 1 λb d 1 ] C n nd, The CORK pencil [ ] A λb A λb = C nd nd (M d 1 λn d 1 ) I n satisfies (A λb)(φ d 1 (λ) I n ) = e 1 g(λ)p(λ). [Van Beeumen et al, 2015] Françoise Tisseur The nonlinear eigenvalue problem 21 / 26

41 Properties of CORK Pencil The CORK pencil [ ] A λb A λb = C nd nd (M d 1 λn d 1 ) I n satisfies right- and left-sided factorizations: (A λb)(φ d 1 (λ) I n ) = e 1 g(λ)p(λ), H(λ)(A λb) = e T 1 g(λ)p(λ). Françoise Tisseur The nonlinear eigenvalue problem 22 / 26

42 Properties of CORK Pencil The CORK pencil [ ] A λb A λb = C nd nd (M d 1 λn d 1 ) I n satisfies right- and left-sided factorizations: (A λb)(φ d 1 (λ) I n ) = e 1 g(λ)p(λ), H(λ)(A λb) = e T 1 g(λ)p(λ). If φ d 1 (λ) 0 at λ then Φ d 1 (λ) x is a right ei vec of A λb with ei val λ iff x is a right ei vec of P(λ) with ei val λ. [ ] y If H(λ) has full rank then is a left ei vec of R φ (λ)y A λb with ei val λ iff y is a left ei vec of P(λ) with ei val λ. Françoise Tisseur The nonlinear eigenvalue problem 22 / 26

43 Rational Arnoldi Algorithm Given A, B C dn dn, v C dn \ {0}, shifts (τ j ) k j=1 C. Set v 1 := v/ v 2. for j = 1, 2,..., k Compute w := (A τ j B) 1 Bv j. Orthogonalize ŵ := w j i=1 µ i,jv i, where µ i,j = v i w. Set µ j+1,j = ŵ 2 and normalize v j+1 := w/µ j+1,j. end Françoise Tisseur The nonlinear eigenvalue problem 23 / 26

44 Rational Arnoldi Algorithm Given A, B C dn dn, v C dn \ {0}, shifts (τ j ) k j=1 C. Set v 1 := v/ v 2. for j = 1, 2,..., k Compute w := (A τ j B) 1 Bv j. Orthogonalize ŵ := w j i=1 µ i,jv i, where µ i,j = v i w. Set µ j+1,j = ŵ 2 and normalize v j+1 := w/µ j+1,j. end If A λb is a CORK pencil then v j = (I d+1 Q j )u j, with u j of smaller length v j. Françoise Tisseur The nonlinear eigenvalue problem 23 / 26

45 Gun Problem F(λ)v = [ K λm + i(λ σ 2 1) 1 2 W1 + i(λ σ 2 2) 1 2 W2 ] v = 0. #104 gun: eigenvalues 9 target set ' 8 eigenvalues (a) eigenvalues (b) nodes < j poles 9 j #10 4 Françoise Tisseur The nonlinear eigenvalue problem 24 / 26

46 Krylov solution dynamic increase of degree m during Krylov iteration = infinite Arnoldi method [Jarlebring et al 2012] compact Krylov basis storage of [Lu, Su, Bai 2016] = CORK [Van Beeumen et al 2015]. Françoise Tisseur The nonlinear eigenvalue problem 25 / 26

47 Krylov solution dynamic increase of degree m during Krylov iteration = infinite Arnoldi method [Jarlebring et al 2012] compact Krylov basis storage of [Lu, Su, Bai 2016] = CORK [Van Beeumen et al 2015]. NLEIGS implementations are available in the SLEPc library version 3.7 [Campos & Roman 2016] Rational Krylov Toolbox [Berljafa & Güttel 2015]. S. GÜTTEL, R. VAN BEEUMEN, K. MEERBERGEN, W. MICHIELS, NLEIGS: A class of fully rational Krylov methods for nonlinear eigenvalue problems, SIAM J. Sci. Comput., V. HERNANDEZ, J. E. ROMAN, V. VIDAL, SLEPc: A scalable and flexible toolkit for the solution of eigenvalue problems, ACM TOMS, Françoise Tisseur The nonlinear eigenvalue problem 25 / 26

48 Concluding Remarks NEPs have interesting mathematical properties. They arise in many applications and their efficient solution requires ideas from numerical linear algebra, complex analysis, and approximation theory (among other fields). Françoise Tisseur The nonlinear eigenvalue problem 26 / 26

49 Concluding Remarks NEPs have interesting mathematical properties. They arise in many applications and their efficient solution requires ideas from numerical linear algebra, complex analysis, and approximation theory (among other fields). There is more to be said, e.g., Structured NEPs? Higher-order integral moments Preconditioning/scaling of linearizations Implementation, software packages Françoise Tisseur The nonlinear eigenvalue problem 26 / 26

50 References I Z. Bai and Y. Su. SOAR: a second-order Arnoldi method for the solution of the quadratic eigenvalue problem. SIAM J. Matrix Anal. Appl., 26(3): , M. Berljafa and S. Güttel. A Rational Krylov Toolbox for MATLAB. MIMS EPrint , Manchester Institute for Mathematical Sciences, The University of Manchester, UK, pp. Available for download at Françoise Tisseur The nonlinear eigenvalue problem 20 / 26

51 References II C. Campos and J. E. Roman. Parallel iterative refinement in polynomial eigenvalue problems. Numer. Linear Algebra Appl., 23(4): , C. Effenberger and D. Kressner. Chebyshev interpolation for nonlinear eigenvalue problems. BIT, 52(4): , L. Grammont, N. J. Higham, and F. Tisseur. A framework for analyzing nonlinear eigenproblems and parametrized linear systems. Linear Algebra Appl., 435(3): , Françoise Tisseur The nonlinear eigenvalue problem 21 / 26

52 References III S. Güttel and F. Tisseur. The nonlinear eigenvalue problem. In Acta Numerica, volume 26, pages Cambridge University Press, S. Güttel, R. Van Beeumen, K. Meerbergen, and W. Michiels. NLEIGS: A class of fully rational Krylov methods for nonlinear eigenvalue problems. SIAM J. Sci. Comput., 36(6):A2842 A2864, S. Hammarling, C. J. Munro, and F. Tisseur. An algorithm for the complete solution of quadratic eigenvalue problems. ACM Trans. Math. Software, 39(3):18:1 18:19, Apr Françoise Tisseur The nonlinear eigenvalue problem 22 / 26

53 References IV V. Hernandez, J. E. Roman, and V. Vidal. SLEPc: A scalable and flexible toolkit for the solution of eigenvalue problems. ACM Transactions on Mathematical Software (TOMS), 31(3): , N. J. Higham, R.-C. Li, and F. Tisseur. Backward error of polynomial eigenproblems solved by linearization. SIAM J. Matrix Anal. Appl., 29(4): , N. J. Higham, D. S. Mackey, and F. Tisseur. The conditioning of linearizations of matrix polynomials. SIAM J. Matrix Anal. Appl., 28(4): , Françoise Tisseur The nonlinear eigenvalue problem 23 / 26

54 References V N. J. Higham, D. S. Mackey, F. Tisseur, and S. D. Garvey. Scaling, sensitivity and stability in the numerical solution of quadratic eigenvalue problems. Internat. J. Numer. Methods Eng., 73(3): , E. Jarlebring, K. Meerbergen, and W. Michiels. Computing a partial Schur factorization of nonlinear eigenvalue problems using the infinite Arnoldi method. SIAM J. Matrix Anal. Appl., 35(2): , Françoise Tisseur The nonlinear eigenvalue problem 24 / 26

55 References VI P. Lietaertand, J. Pérez, B. Vandereycken, and K. Meerbergen. Automatic rational approximation and linearization of nonlinear eigenvalue problems. Technical Report arxiv: v2, D. Lu, Y. Su, and Z. Bai. Stability analysis of the two-level orthogonal Arnoldi procedure. SIAM J. Matrix Anal. Appl., 37(1): , K. Meerbergen. Fast frequency response computation for Rayleigh damping. Internat. J. Numer. Methods Eng., 73(1):96 106, Françoise Tisseur The nonlinear eigenvalue problem 25 / 26

56 References VII Y. Nakatsukasa, O. SÃČÅate, and L. N. Trefethen. The aaa algorithm for rational approximation. SIAM J. Sci. Comput., 40(3):A1494 A1522, R. Van Beeumen, K. Meerbergen, and W. Michiels. Compact rational Krylov methods for nonlinear eigenvalue problems. SIAM J. Matrix Anal. Appl., 36(2): , Françoise Tisseur The nonlinear eigenvalue problem 26 / 26