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A Linear Eigenvalue Algorithm for the Nonlinear Eigenvalue Problem

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A Linear Eigenvalue Algorithm for the Nonlinear Eigenvalue Problem Noname manuscript No. (will be inserted by the editor) A linear eigenvalue algorithm for the nonlinear eigenvalue problem Elias Jarlebring Wim Michiels Karl Meerbergen · · Received: date / Accepted: date Abstract The Arnoldi method for standard eigenvalue problems possesses several at- tractive properties making it robust, reliable and efficient for many problems. The first result of this paper is a characterization of the solutions to an arbitrary (ana- lytic) nonlinear eigenvalue problem (N!") as the reciprocal eigenvalues of an infinite dimensional operator denoted . #e consider the Arn oldi method for the operator B B and show that with a particular choice of starting function and a particular choice of scalar product, the structure of the operator can be exploited in a very effective way. The structure of the operator is such that when the $rnoldi method is started with a constant function, the iterates will be polynomials. 'or large class of !"s, we show that we can carry out the infinite dimensional Arnoldi algorithm for the operator in arithmetic based on standard linear algebra operations on vectors and matrices B of finite size. This is achieved by representing the polynomials by vector coefficients. The resulting algorithm is by construction such that it is completely equivalent to the standard $rnoldi method and also inherits many of its attractive properties, which are illustrated with examples. Keywords onlinear eigenvalue problems The $rnoldi method Krylov subspaces · · · *pectral methods Chebyshev polynomia ls · 1 Introduction n n Consider a given functionT,Ω C × whereΩ C is an open disc centered at → ⊂ the origin andT is assumed to be anal ytic inΩ. #e will here consider the (analytic) n nonlinear eigenvalue problem given by findingλ Ω andx C - such that ∈ ∈ \{ } T(λ)x.-. (/) This general problem has been extensively studied. *ee, e.g., the standard references 01/,/23 and the problem collection 04]. 2 5n the literature there exists a number of nice methods specialized for different types of structures ofT , such as methods for t he (uadratic eigenvalue problem (Q!") 03,/83 and more generally the polynomial eigenvalue problem (P!") 011,/-3 and also other types of structures 017,/6,8,/7,//]. Apart from the specialized methods there also e%ist general methods which have a sense of local convergence for one or a few eigenvalues 01-,1:,/;,/1]. Our goal in this paper is to present a general algorithmic framework, applicable to a large class of !Ps allowing us, in a (uite automatized and reliable way, to find all eigenvalues of (1) close to a given target. #e will assume that the target is the origin. ote that there is no loss of generality to use the origin as a target in this sense, since a substitution allows us to shift the origin to an arbitrary complex point. 'undamental for the construction is an equivalence between (1) and the eigenvalues of an operator denoted . The operator is an integration-type operator and infinite B B dimensional in the sense that it is a map between infinite dimensional function spaces. 5n the equivalence (presented in *ection 1) we show that the reciprocal eigenvalues of are the solutions to (1). B #e will here make a construction with the $rnoldi method, which is a common method for standard and generalized eigenvalue problems. In *ection 7 we summarize the $rnoldi method, when applied to the operator . #e subsequently sho w that the B structure of is such that when the $rnoldi method for is started with a cons tant B B function, the iterates will be polynomials. =y representing the polynomials in a given basis (here, either with monomials or +hebyshev polynomials) we show how we can carry out the action of on a polynomial using only finite arithmetic, i.e., standard B linear algebra operations applied to matrices of finite size. With this construction, we show (in *ection 4) that with a particular choice of starting function and a particular scalar product, the $rnoldi method for the operator can be carried out in finite B arithmetic. 5n *ection : we characterize the algorithm in a different way. #e show an equiva- lence with the presented algorithm and the $rnoldi method on a large matrix, either consisting of a companion linearization of the truncated Taylor expansion or the pencil corresponding to a spectral discretization. Due to the fact that the algorithm only consists of standard linear algebra opera- tions, it is suitable for large-scale problems. This is illustrated in the examples section (*ection 9). #e finally wish to mention some other methods which have some components similar to our method. The method in 0193 is also motivated by the $rnoldi method and it is a generalization in the sense that it reduces to the standard $rnoldi method for the linear eigenvalue problem, i.e., ifT(λ) .A λI. The action of the op erator − is integration. There are other methods for !"s based on integration. ?nlike the B method presented in this paper, the methods in 02,:3 are based on a contour integral formulation. 2 Operator equivalence "ormulation 5n this paper it will be advantageous to slightly reformulate the nonlinear eigenvalue problem (/). #e will first of all assume thatλ . - is not an eigenval ue of the original problem and define, 1 T(-) T(λ) B(λ) ,.T(-) − − , λ.- (1) λ � 3 1 andB(-) ,. T(-) − T �(-). With this transformation, (/) yields that, − λB(λ)x.x. (7) ote thatB is analytic inΩ sinceT is analytic inΩ. 5n order to characterize the solutions to (7), we will in this paper take an approach where the iterates can be interpreted as functions. #e will commonly work with func- n tionsϕ, R C and denote the function variableθ. → 'or notational convenience we will use the concise functional analysis notation d B( dθ ), which can consistently defined (sinceB is analytic), e.g., with the Taylor ex- n pansion. 5n particular, if we applyB( d ) to a functionϕ, R C and evaluate at dθ → θ . -, we can express it as d ∞ / B( )ϕ (-) . B(i)(-)ϕ(i)(-). (4) dθ i@ i=0 � � � With the notation above we are now ready to present a characterization of (7) based n on an operator denoted , which maps functions in ( ) C (R, C ), i.e., infinitely B n D B ⊂ ∞n differentiable functionsϕ, R C , to functions inC (R, C ). 5n the following we → ∞ define the operator and characterize som e important properties. 5n particular, we B show that the non-zero eigenvalues of are the reciprocal solu tionsλ to (7). B #efinition 1 %&he operator ' Let denote the map defined by the domain ( ) ,. n (i) B (i) B D B ϕ C (R, C ), ∞ B (-)ϕ (-)/(i!) is finite and the action { ∈ ∞ i=0 } � θ ( ϕ)(θ). ϕ(θA)d θABC(ϕ), (:) B �0 where ∞ B(i)(-) d C(ϕ) ,. ϕ(i)(-) . B( )ϕ (-). (9) i@ dθ i=0 � � � Proposition 2 %)inearity' For any two functions ϕ, ψ ( ) and any two constant ∈D B scalars α,β C, the map satisfies the linearity property, ∈ B (αϕBβψ).α ϕBβ ψ. B B B B(i) (i) Proof Note thatC is linear in the sense that,C(αϕBβψ) ,. i∞=0 i! (αϕ (-) B βψ(i)(-)) .αC(ϕ)BβC(ψ). The proof is complet ed by noting that integration is linear and hence the sum in (:) is linear. � n Theorem * %Operator equi!alence' Letx C - ,λ Ω C and denote ∈ \{ } ∈ ⊂ ϕ(θ) ,.xe λθ. Then the following statements are equivalent. i) The pair(λ,x) is a solution to the no nlinear eigenvalue problem (3). ii) The pair(λ,ϕ) is a solution to the in finite dimensional eigenvalue problem λ ϕ. ϕ. (8) B Moreover, all eigenfunctions of depend exponentially onθ, i.e., ifλ ψ.ψ then B B ψ(θ).xe λθ. 4 Proof #e first show that an eigenfunction of always is exponential inθ. *uppose B ϕ ( ) satisfies (7) and cons ider the derivative ∈D B d d (λ ϕ).λ ( ϕ).ϕ �, (2) dθ B dθ B which exists since all functions of the domain of are differentiable. Du e to the fact B that the action of is integration, the left -hand side of (2) is λϕ. The result of (2) is B hence the differential equation λϕ.ϕ �, for which the solution always is of the form ϕ(θ). xeλθ. 5n order to show that i) implies ii), suppose (λ, x) is a solution to (7). Cetϕ(θ) ,. xeλθ and note thatϕ ( ) since ∈D B ∞ / ∞ λi B(i)(-)ϕ(i)(-) . B(i)(-)ϕ(0) .B(λ)x, i@ i@ i=0 i=0 � � is finite. #e here used thatλ Ω implies that the series is convergent. ow note that ∈ the derivative of the left-hand side of (7) isλ d ( ϕ). λϕ, since ϕ is a primitive dθ B B function ofϕ. 'rom the fact that an eigenfunction takes the formϕ(θ) . xe λθ it follows that the derivative of the right-hand side of (7) is λϕ. Dence, the derivative of the function equality (8) is satisfied. 5t remains to show that (7) holds in one point. Consider (7) evaluated atθ . -. The left-hand s ide isλ( ϕ)(-) . λC(ϕ) and the B right-hand sideϕ(0) .x. 5t follows that the diff erence is, d λC(ϕ) x.λ(B( )ϕ)(0) x.λB(λ)ϕ(0) x.-, − dθ − − where in the last step we used that (λ, x) is an eigenpair of (7). #e have shown i) implies ii). 5n order to show the converse, suppose (λ,ϕ) ( C, ( )) is a solution to (7). #e ∈ D B already know that a solution to (8) is an exponential times a vector (which we callx), i.e.,ϕ(θ).
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