Research Matters. February 25, The Nonlinear Eigenvalue. Director of Research School of Mathematics

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1 Research Matters February 25, 2009 The Nonlinear Eigenvalue Nick Problem: HighamPart II Director of Research School of Mathematics Françoise Tisseur School of Mathematics The University of Manchester Woudschoten Conference / 6

2 Outline Two lectures: Part I: Mathematical properties of nonlinear eigenproblems (NEPs) Definition and historical aspects Examples and applications Solution structure Part II: 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, University of Manchester Françoise Tisseur The nonlinear eigenvalue problem 2 / 36

3 Solvers Based on Newton s Method Newton s method is a natural approach to compute e vals/e vecs of NEPs provided good initial guesses are available. Local quadratic convergence. Initial guess is the only crucial parameter great advantage over other NEP eigensolvers. Two broad ways NEP F(λ)v = 0 can be tackled by a Newton-type method: Apply Newton s method to a scalar equation f (z) = 0 whose roots are the wanted e vals of F. Apply Newton s method directly to the vector problem F(λ)v = 0 together with some normalization condition on v. University of Manchester Françoise Tisseur The nonlinear eigenvalue problem 3 / 36

4 Newton s Method for Scalar Function Most obvious approach: Find roots of f (z) = detf(z). Combining Newton s method with Jacobi s formula λ (k+1) = λ (k) f (λ(k) ) f (λ (k) ) f (z) = detf(z)trace ( F(z) 1 F (z) ), we obtain the Newton-trace iteration [Lancaster 1966] λ (k+1) = λ (k) 1 trace ( F(λ (k) ) 1 F (λ (k) ) ). University of Manchester Françoise Tisseur The nonlinear eigenvalue problem 4 / 36

5 Newton s Method for Scalar Function Most obvious approach: Find roots of f (z) = detf(z). Combining Newton s method with Jacobi s formula λ (k+1) = λ (k) f (λ(k) ) f (λ (k) ) f (z) = detf(z)trace ( F(z) 1 F (z) ), we obtain the Newton-trace iteration [Lancaster 1966] λ (k+1) = λ (k) 1 trace ( F(λ (k) ) 1 F (λ (k) ) ). Potential problems: Inverse of nearly singular F(λ (k) ) as λ (k) λ. Requires F (z) explicitly. Computationally expensive. Initialization? University of Manchester Françoise Tisseur The nonlinear eigenvalue problem 4 / 36

6 Newton s Method for Scalar Function (Cont.) Kublanovskaya f (z) = r nn (z), where r nn (z) is (n, n) entry of R in rank-revealing QR decomposition of F(z), F(z)Π(z) = Q(z)R(z). This yields the Newton-QR iteration for a root of r nn (z), λ (k+1) = λ (k) 1/(e T n Q kf ( λ (k)) Π k R 1 k e n ). At convergence, we can take x = Π k R 1 k e n, y = Q k e n as approx for the right and left e vecs of the converged e val. University of Manchester Françoise Tisseur The nonlinear eigenvalue problem 5 / 36

7 Newton s Method for Scalar Function (Cont.) Kublanovskaya f (z) = r nn (z), where r nn (z) is (n, n) entry of R in rank-revealing QR decomposition of F(z), F(z)Π(z) = Q(z)R(z). This yields the Newton-QR iteration for a root of r nn (z), λ (k+1) = λ (k) 1/(e T n Q kf ( λ (k)) Π k R 1 k e n ). At convergence, we can take x = Π k R 1 k e n, y = Q k e n as approx for the right and left e vecs of the converged e val. Garret, Bai and Li (2016) propose an efficient implementation for large banded NEPs. MATLAB and C++ implementations including deflation are publicly available. University of Manchester Françoise Tisseur The nonlinear eigenvalue problem 5 / 36

8 Convergence basins [ ] e iz 2 1 F(z) = has e vals 0, ± 2π, ±i 2π in 1 1 Ω = {z C : 3 Re(z) 3, 3 Im(z) 3}. Newton-trace method Newton-QR method University of Manchester Françoise Tisseur The nonlinear eigenvalue problem 6 / 36

9 Convergence basins [ ] e iz 2 1 F(z) = has e vals 0, ± 2π, ±i 2π in 1 1 Ω = {z C : 3 Re(z) 3, 3 Im(z) 3}. Newton-trace method Newton-trace (secant) method University of Manchester Françoise Tisseur The nonlinear eigenvalue problem 6 / 36

10 Newton-QR for Banded NEP Consider the loaded_string problem defined by ( F(λ)v = C 1 λc 2 + λ ) λ σ C 3 v = 0 with C 1, C 2 tridiagonal and C 3 = e n e T n. n = 100. Use NQR4UB [Garret and Li (2013)] to compute the 5 e vals in [4, 296] with lambda0 = 4. Residuals r nn (λ) / F(λ) F at each iter η F (λ i, v i ) = F (λ i )v i 2 F (λ i ) F v i 2 i λ i η F (λ i, v i ) η F (λ i, wi ) e e e e e e e e e e University of Manchester Françoise Tisseur The nonlinear eigenvalue problem 7 / 36

11 Other Variants Many variations based on scalar root finding exist: Newton-LU [Yang 1983, Wobst 1987] BDS (bordered, deletion, substitution) method [Andrew/Chu/Lancaster 1995] implicit determinant method [Spence/Poulton 2005] University of Manchester Françoise Tisseur The nonlinear eigenvalue problem 8 / 36

12 Newton s Method for Vector Equation Applying Newton s method to N [ v λ] = 0 where [ ] [ ] v F(λ)v N = λ u v 1 leads to the Newton s iteration [ ] [ ] [ ] v (k+1) v (k) F(λ λ (k+1) = λ (k) (k) ) F (λ (k) )v (k) 1 [ F(λ (k) )v (k) u 0 u v (k) 1 Other variants include nonlinear inverse iteration [Unger 1950, Ruhe 1973] two-sided Rayleigh functional iteration [Schreiber 2008] residual inverse iteration [Neumaier 1985] ]. University of Manchester Françoise Tisseur The nonlinear eigenvalue problem 9 / 36

13 Iterative Projection Methods for NEPs Let U C n k with k n and U U = I k (search space) and Q C n k, Q Q = I k (test space). Instead of solving F(λ)v = 0, solve k k projected NEP Q F(ϑ)Ux = 0 ( ) University of Manchester Françoise Tisseur The nonlinear eigenvalue problem 10 / 36

14 Iterative Projection Methods for NEPs Let U C n k with k n and U U = I k (search space) and Q C n k, Q Q = I k (test space). Instead of solving F(λ)v = 0, solve k k projected NEP Q F(ϑ)Ux = 0 ( ) Let (ϑ, x) be an e pair of ( ). If F(ϑ)Ux is small enough, accept (ϑ, Ux) as an approximate e pair for F. If not, extend search space into span{u, v} by one step of Newton iteration with initial guess (ϑ, Ux). v solves the Jacobi Davidson correction eqn: (I n F (ϑ)vq )F(ϑ)(I n vv ) v = F(ϑ)v. (Does not need to be solved accurately.) University of Manchester Françoise Tisseur The nonlinear eigenvalue problem 10 / 36

15 Iterative Projection Methods for NEPs Let U C n k with k n and U U = I k (search space) and Q C n k, Q Q = I k (test space). Instead of solving F(λ)v = 0, solve k k projected NEP Q F(ϑ)Ux = 0 ( ) Let (ϑ, x) be an e pair of ( ). If F(ϑ)Ux is small enough, accept (ϑ, Ux) as an approximate e pair for F. If not, extend search space into span{u, v} by one step of Newton iteration with initial guess (ϑ, Ux). v solves the Jacobi Davidson correction eqn: (I n F (ϑ)vq )F(ϑ)(I n vv ) v = F(ϑ)v. (Does not need to be solved accurately.) Variations of Jacobi Davidson for NEPs are proposed by [Hochstenbach and Sleijpen (2003)], [Betcke and Voss (2004)], [Voss (2007)] and [Effenberger (2013)]. University of Manchester Françoise Tisseur The nonlinear eigenvalue problem 10 / 36

16 Deflation of Computed Eigenpairs Prevent the iteration from converging to already computed eigenpairs. Maps the already computed eigenvalues to infinity. Suppose we have computed l simple e vals of F, λ 1,..., λ l and let x i, y i C n be s.t. yi x i = 1, i = 1,..., l. Let l ( ) F(z) = F(z) i=1 I z λ i 1 z λ i Then Λ( F) = Λ(F) { } \ {λ 1,..., λ l }. i=1 y i x i If ṽ is an e vec of F with e val λ then l ( v = I z λ i 1 ) y z λ i xi ṽ i is an e vec of F associated with the e val λ. [Ferng et al. (2001)], [Huang et al. (2016)]. University of Manchester Françoise Tisseur The nonlinear eigenvalue problem 11 / 36.

17 Robust Succesive Computation of Eigenpairs Suppose we have already computed a minimal invariant pair (V, M) C n m C m m for F. Extend (V, M) into one size larger minimal invariant pair ( [ ]) ( V, M) M b = [ V x ], C n (m+1) C (m+1) (m+1). 0 λ [Effenberger (2013)] shows that ( λ, [ x b]) is an eigenpair of an (n + m) (n + m) NEP F(λ) [ x b] = 0. Solve F(λ) [ x b] = 0 by any of the previous methods. University of Manchester Françoise Tisseur The nonlinear eigenvalue problem 12 / 36

18 Block-Newton Method (V, M) C n m C m m is an invariant pair for F H(Ω, C n n ) if F(V, M) = C 1 Vf 1 (Λ) + C 2 Vf 2 (Λ) + + C l Vf l (Λ) = 0 n m. Add normalization condition: N (V, M) = 0 m m. [Kressner (2009)] showed that (V, M) is complete iff Jacobian M : C n m C m m C n m C m m ( V, M) ( L F ( V, M), L N ( V, M) ) is invertible. Newton correction ( V, M) satisfies M( V, M) = ( F ( V, M ), O m m ). University of Manchester Françoise Tisseur The nonlinear eigenvalue problem 13 / 36

19 Safeguarded Iteration for Hermitian NEPs Assume F( z) = F(z) for all z C and that if λ k is a kth eigenvalue of F(z) (i.e., µ = 0 is the kth largest eigenvalue of the Hermitian matrix F(λ k )), then (see Part I) λ k = min V S k V K(p) max p(x) I. x V K(p) x 0 This suggests the safeguarded iteration [Werner 1970]: 1 Choose an initial approx λ (0) to jth e val of F. 2 For k = 0, 1,... until convergence 3 Compute e vec x (k) of jth largest e val of F ( λ (k)). 4 Compute real root ρ of x (k) F(ρ)x (k) = 0 closest to λ (k) 5 set λ (k+1) = ρ. 6 end University of Manchester Françoise Tisseur The nonlinear eigenvalue problem 14 / 36

20 Safeguarded Iteration for Hermitian NEPs Assume F( z) = F(z) for all z C and that if λ k is a kth eigenvalue of F(z) (i.e., µ = 0 is the kth largest eigenvalue of the Hermitian matrix F(λ k )), then (see Part I) λ k = min V S k V K(p) max p(x) I. x V K(p) x 0 This suggests the safeguarded iteration [Werner 1970]: 1 Choose an initial approx λ (0) to jth e val of F. 2 For k = 0, 1,... until convergence 3 Compute e vec x (k) of jth largest e val of F ( λ (k)). 4 Compute real root ρ of x (k) F(ρ)x (k) = 0 closest to λ (k) 5 set λ (k+1) = ρ. 6 end Local quadratic convergence [Niendorf and Voss (2010)]. University of Manchester Françoise Tisseur The nonlinear eigenvalue problem 14 / 36

21 Methods Based on Contour Integration Given F H(Ω, C n n ). By Keldysh s theorem we have F(z) 1 = V (zi J) 1 W + R(z) on some closed set Σ Ω, where J is an m m Jordan block matrix containing all the eigenvalues Λ(F) Σ. University of Manchester Françoise Tisseur The nonlinear eigenvalue problem 15 / 36

22 Methods Based on Contour Integration Given F H(Ω, C n n ). By Keldysh s theorem we have F(z) 1 = V (zi J) 1 W + R(z) on some closed set Σ Ω, where J is an m m Jordan block matrix containing all the eigenvalues Λ(F) Σ. Let Γ Ω be a contour enclosing the eigenvalues of J, let X C n r be a probing matrix, and f H(Ω, C) (filter function). Then A f := 1 f (z)f(z) 1 Xδz 2πi Γ = 1 f (z)v (zi J) 1 W Xδz 2πi Γ = Vf (J)W X. University of Manchester Françoise Tisseur The nonlinear eigenvalue problem 15 / 36

23 Beyn s Integral Approach Construct 1 A 0 := F(z) 1 Xδz = V W X 2πi Γ A 1 := 1 z F(z) 1 Xδz = V J W X. 2πi Γ Assuming that V, W, and W X are of full rank m, can show that e vals of λa 0 A 1 are e vals of F inside Γ. [Asakura 2009] [Beyn 2012] [Yokota/Sakurai 2013] University of Manchester Françoise Tisseur The nonlinear eigenvalue problem 16 / 36

24 Beyn s Integral Approach Construct 1 A 0 := F(z) 1 Xδz = V W X 2πi Γ A 1 := 1 z F(z) 1 Xδz = V J W X. 2πi Γ Assuming that V, W, and W X are of full rank m, can show that e vals of λa 0 A 1 are e vals of F inside Γ. [Asakura 2009] [Beyn 2012] [Yokota/Sakurai 2013] Other filter functions f can be used via A f = 1 f 2πi Γ (z)f(z) 1 Xδz leading to higher-order moments. [Murakami 10] [Güttel et al 15] [Austin/Trefethen 15] [Van Barel 16] University of Manchester Françoise Tisseur The nonlinear eigenvalue problem 16 / 36

25 Quadrature The contour integrals involved in A 0 and A 1 are approximated by numerical quadrature. Let γ : [0, 1] Γ be a parameterization, then n c A j,nc = ω l z j F(γ(t l )) 1 X A j l=1 is a quadrature approximation with n c nodes (j = 1, 2). Note that the n c solves F(γ(t l )) 1 X are completely decoupled and can be assigned to different processors. = Great potential for parallelization! University of Manchester Françoise Tisseur The nonlinear eigenvalue problem 17 / 36

26 Care has to be taken with the quadrature rule The quality of the quadrature approximation determines the accuracy of the computed e vals [Beyn 2012] [Güttel/T. 2017]. Example: Quadrature errors A j A j,nc as n c increases (right) with different quadrature rules on (almost) square contour (left) eigenvalues domain + conformal square Bernstein ellipse 10-2 O(ρ nc ) conformal trapezoid Gauss-Legendre O(n 2 c ) University of Manchester Françoise Tisseur The nonlinear eigenvalue problem 18 / 36

27 Methods based on linear interpolation Instead of solving F(λ)v = 0 directly, we may approximate F R m on Σ C by simpler NEP and solve R m (λ)v = 0. University of Manchester Françoise Tisseur The nonlinear eigenvalue problem 19 / 36

28 Methods based on linear interpolation Instead of solving F(λ)v = 0 directly, we may approximate F R m on Σ C by simpler NEP and solve R m (λ)v = 0. Practical approach: 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 fixed and b j are rational functions. University of Manchester Françoise Tisseur The nonlinear eigenvalue problem 19 / 36

29 Methods based on linear interpolation Instead of solving F(λ)v = 0 directly, we may approximate F R m on Σ C by simpler NEP and solve R m (λ)v = 0. Practical approach: 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 fixed and b j are rational functions. Particularly useful: (scaled) rational Newton basis b 0 (z) 1, b j+1 (z) = z σ j β j+1 (1 z/ξ j+1 ) b j(z) with interpolation points σ j Σ and poles ξ j C \ Σ. Choice of σ j, ξ j, β j by NLEIGS sampling [Güttel et al 2014]. University of Manchester Françoise Tisseur The nonlinear eigenvalue problem 19 / 36

30 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. University of Manchester Françoise Tisseur The nonlinear eigenvalue problem 20 / 36

31 NLEIGS sampling of F on Σ Ξ Σ University of Manchester Françoise Tisseur The nonlinear eigenvalue problem 21 / 36

32 NLEIGS sampling of F on Σ 1. Discretize boundaries of Σ and Ξ sufficiently fine. Ξ Σ University of Manchester Françoise Tisseur The nonlinear eigenvalue problem 21 / 36

33 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 University of Manchester Françoise Tisseur The nonlinear eigenvalue problem 21 / 36

34 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 University of Manchester Françoise Tisseur The nonlinear eigenvalue problem 21 /

35 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 University of Manchester Françoise Tisseur The nonlinear eigenvalue problem 21 /

36 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 University of Manchester Françoise Tisseur The nonlinear eigenvalue problem 21 /

37 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 University of Manchester Françoise Tisseur The nonlinear eigenvalue problem 21 /

38 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 University of Manchester Françoise Tisseur The nonlinear eigenvalue problem 21 /

39 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 University of Manchester Françoise Tisseur The nonlinear eigenvalue problem 21 /

40 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 University of Manchester Françoise Tisseur The nonlinear eigenvalue problem 21 /

41 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 University of Manchester Françoise Tisseur The nonlinear eigenvalue problem 21 / 36

42 Krylov solution Interpolation techniques can be combined with linearization of R m structured GEP A m x = λb m x rational Krylov algorithms for solving GEP 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]. University of Manchester Françoise Tisseur The nonlinear eigenvalue problem 22 / 36

43 Krylov solution Interpolation techniques can be combined with linearization of R m structured GEP A m x = λb m x rational Krylov algorithms for solving GEP 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]. University of Manchester Françoise Tisseur The nonlinear eigenvalue problem 22 / 36

44 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). University of Manchester Françoise Tisseur The nonlinear eigenvalue problem 23 / 36

45 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 University of Manchester Françoise Tisseur The nonlinear eigenvalue problem 23 / 36

46 References I A. L. Andrew, K. E. Chu, and P. Lancaster. On the numerical solution of nonlinear eigenvalue problems. Computing, 55:91 111, J. Asakura, T. Sakurai, H. Tadano, T. Ikegami, and K. Kimura. A numerical method for nonlinear eigenvalue problems using contour integral. JSIAM Letters, 1:52 55, University of Manchester Françoise Tisseur The nonlinear eigenvalue problem 24 / 36

47 References II M. Berljafa and S. Güttel. A Rational Krylov Toolbox for MATLAB. MIMS EPrint , Manchester Institute for Mathematical Sciences, The University of Manchester, UK, Available for download at T. Betcke and H. Voss. A Jacobi Davidson-type projection method for nonlinear eigenvalue problems. Future Generation Computer Systems, 20(3): , University of Manchester Françoise Tisseur The nonlinear eigenvalue problem 25 / 36

48 References III W.-J. Beyn. An integral method for solving nonlinear eigenvalue problems. Linear Algebra Appl., 436(10): , C. Campos and J. E. Roman. Parallel Krylov solvers for the polynomial eigenvalue problem in SLEPc. SIAM J. Sci. Comput., 38(5):S385 S411, C. Effenberger. Robust successive computation of eigenpairs for nonlinear eigenvalue problems. SIAM J. Matrix Anal. Appl., 34(3): , University of Manchester Françoise Tisseur The nonlinear eigenvalue problem 26 / 36

49 References IV W. R. Ferng, W.-W. Lin, D. J. Pierce, and C.-S. Wang. Nonequivalence transformation of λ-matrices eigenproblems and model embedding approach to model tuning. Numer. Linear Algebra Appl., 8(1):53 70, C. K. Garrett, Z. Bai, and R.-C. Li. A nonlinear QR algorithm for banded nonlinear eigenvalue problems. ACM Trans. Math. Software, 43(1):4:1 4:19, University of Manchester Françoise Tisseur The nonlinear eigenvalue problem 27 / 36

50 References V C. K. Garrett and R.-C. Li. Unstructurally banded nonlinear. eigenvalue solver software, http: // S. Güttel and F. Tisseur. The nonlinear eigenvalue problem. In Acta Numerica, volume 26, pages Cambridge University Press, University of Manchester Françoise Tisseur The nonlinear eigenvalue problem 28 / 36

51 References VI 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, M. E. Hochstenbach and G. L. Sleijpen. Two-sided and alternating Jacobi Davidson. Linear Algebra Appl., 358(1): , T.-M. Huang, W.-W. Lin, and V. Mehrmann. A Newton-type method with nonequivalence deflation for nonlinear eigenvalue problems arising in photonic crystal modeling. SIAM J. Sci. Comput., 38(2):B191 B218, University of Manchester Françoise Tisseur The nonlinear eigenvalue problem 29 / 36

52 References VII E. Jarlebring, K. Meerbergen, and W. Michiels. The infinite Arnoldi method and an application to time-delay systems with distributed delay. In R. Sipahi, T. Vyhlídal, S.-I. Niculescu, and P. Pepe, editors, Time Delay Systems: Methods, Applications and New Trends, volume 423 of Lecture Notes in Control and Information Sciences, pages Springer-Verlag, Berlin, D. Kressner. A block Newton method for nonlinear eigenvalue problems. Numer. Math., 114(2): , University of Manchester Françoise Tisseur The nonlinear eigenvalue problem 30 / 36

53 References VIII V. N. Kublanovskaya. On an approach to the solution of the generalized latent value problem for λ-matrices. SIAM J. Numer. Anal., 7: , P. Lancaster. Lambda-Matrices and Vibrating Systems. Pergamon Press, Oxford, ISBN Reprinted by Dover, New York, University of Manchester Françoise Tisseur The nonlinear eigenvalue problem 31 / 36

54 References IX D. Lu, Y. Su, and Z. Bai. Stability analysis of the two-level orthogonal Arnoldi procedure. SIAM J. Matrix Anal. Appl., 37(1): , A. Neumaier. Residual inverse iteration for the nonlinear eigenvalue problem. SIAM J. Numer. Anal., 22(5): , V. Niendorf and H. Voss. Detecting hyperbolic and definite matrix polynomials. Linear Algebra Appl., 432: , University of Manchester Françoise Tisseur The nonlinear eigenvalue problem 32 / 36

55 References X A. Ruhe. Algorithms for the nonlinear eigenvalue problem. SIAM J. Numer. Anal., 10: , K. Schreiber. Nonlinear Eigenvalue Problems: Newton-type methods and Nonlinear Rayleigh Functionals. PhD thesis, Technischen Universitat Berlin, Germany, A. Spence and C. Poulton. Photonic band structure calculations using nonlinear eigenvalue techniques. J. Comput. Phys., 204(1):65 81, University of Manchester Françoise Tisseur The nonlinear eigenvalue problem 33 / 36

56 References XI H. Unger. Nichtlineare Behandlung von Eigenwertaufgaben. ZAMM Z. Angew. Math. Mech., 30(8-9): , M. Van Barel. Designing rational filter functions for solving eigenvalue problems by contour integration. Linear Algebra Appl., 502: , R. Van Beeumen, K. Meerbergen, and W. Michiels. Compact rational Krylov methods for nonlinear eigenvalue problems. SIAM J. Matrix Anal. Appl., 36(2): , University of Manchester Françoise Tisseur The nonlinear eigenvalue problem 34 / 36

57 References XII H. Voss. A Jacobi Davidson method for nonlinear and nonsymmetric eigenproblems. Computers and Structures, 85: , B. Werner. Das Spektrum von Operatorenscharen mit verallgemeinerten Rayleighquotienten. PhD thesis, Universität Hamburg, Hamburg, Germany, R. Wobst. The generalized eigenvalue problem and acoustic surface wave computations. Computing, 39:57 69, University of Manchester Françoise Tisseur The nonlinear eigenvalue problem 35 / 36

58 References XIII W. H. Yang. A method for eigenvalues of sparse λ-matrices. Internat. J. Numer. Methods Eng., 19: , S. Yokota and T. Sakurai. A projection method for nonlinear eigenvalue problems using contour integrals. JSIAM Letters, 5:41 44, University of Manchester Françoise Tisseur The nonlinear eigenvalue problem 36 / 36