SOLVING RATIONAL EIGENVALUE PROBLEMS VIA LINEARIZATION

Transcription

1 SOLVNG RATONAL EGENVALUE PROBLEMS VA LNEARZATON YANGFENG SU AND ZHAOJUN BA Abstract Rational eigenvalue problem is an emerging class of nonlinear eigenvalue problems arising from a variety of physical applications n this paper, we propose a linearization-based method to solve the rational eigenvalue problem The proposed method converts the rational eigenvalue problem into a well-studied linear eigenvalue problem, and meanwhile, exploits and preserves the structure and properties of the original rational eigenvalue problem For example, the low-rank property leads to a trimmed linearization We show that solving a class of rational eigenvalue problems is just as convenient and efficient as solving linear eigenvalue problems of slightly larger sizes Key words Rational eigenvalue problem, linearization AMS subject classifications 65F5, 65F5, 5A8 ntroduction n recent years, there are a great deal of interest to study the rational eigenvalue problem REP) ) Rλ)x =, where Rλ) is an n n matrix rational function of the form 2) Rλ) = P λ) s i λ) q i λ) E i, P λ) is an n n matrix polynomial in λ of degree d, s i λ) and q i λ) are scalar polynomials of degrees n i and d i, respectively, and E i are n n constant matrices The REP 2) arises from optimization of acoustic emissions of high speed trains [3, free vibration of plates with elastically attached masses [2, vibration of fluid-solid structure [2, free vibrations of a structure with a viscoelastic constitutive relation describing the behavior of a material [6 and electronic structure calculations of quantum dots [23, A brute-force approach to solve the REP ) is to multiply equation ) by the scalar polynomial k q iλ) to turn it into a polynomial eigenvalue problem PEP) of the degree d = d + d + + d k Subsequently, the PEP is converted into a linear eigenvalue problem LEP) by the process known as linearization This approach is noted in [6 and employed in [, 9 for electronic structure calculations of quantum dots The recent study of the linearization techniques of the PEP can be found in [3, 4, 7, 8 and references therein This is a practical approach only if the number k of the rational terms and the degree d i of the polynomials q i λ) are small, say k = d = However, when k and/or d i are large, it leads to a PEP of much higher degree and becomes impractical for large-scale problems Furthermore, the possible low-rank property of the matrices E i is lost in the linearized eigenvalue problem An alternative approach is to treat the REP ) as a general nonlinear eigenvalue problem NEP), and solve it by a nonlinear eigensolver, such as Picard iteration self-consistent iteration), Newton s method, nonlinear Rayleigh quotient method, nonlinear Jacobi-Davidson method, and nonlinear Arnoldi method [6, 8, 2, 22 This approach limits the exploitation of the underlying rich structure and property of the REP ), and is challenging in convergence analysis and validation of computed eigenvalues School of Mathematical Sciences, Fudan University, Shanghai 2433, China Affiliated with Scientific Computing Key Laboratory of Shanghai Normal University, Division of Computational Science, E-nstitute of Shanghai Universities, Peoples Republic of China yfsu@fudaneducn Department of Computer Science and Mathematics, University of California, Davis, CA 9566 USA bai@csucdavisedu

2 2 Y SU AND Z BA n this paper, we propose a linearization-based approach to solve the REP ) Similarly to the linearization of the PEP, the new approach converts the REP ) into a linear eigenvalue problem LEP), exploits and preserves the structure and property of the REP ) as much as possible t has a number of advantages For example, the low-rank property of matrices E i as frequently encountered in applications leads to a trimmed linearization, namely, only small increase of the size comparing to the size of the original REP ) The symmetry of the REP can also be preserved in the LEP We show that under mild assumptions, the problem of solving a class of the REPs is just as convenient and efficient as the problem of solving an LEP of slightly larger size The rest of this paper is organized as follows n section 2, we formalize the definition of the REP ) and the assumptions we will use throughout n section 3, we present a linearization scheme n section 4, we show how to use the proposed linearization scheme to a number of REPs from different applications Numerical examples are given in section 5 2 Settings We assume throughout that the matrix rational function Rλ) is regular, that is, detrλ)) The roots of q i λ) are the poles of Rλ) Rλ) is not defined on these poles A scalar λ such that detrλ)) = is referred to as an eigenvalue, and the corresponding nonzero vector x satisfying the equation ) is called an eigenvector The pair λ, x) is referred to as an eigenpair Let us denote the matrix polynomial P λ) of degree d in λ as 2) P λ) = λ d A d + λ d A d + + λa + A, where A i are n n constant matrices We assume throughout that the leading coefficient matrix A d is nonsingular, which is equivalent to the assumption of a monic matrix polynomial A d = ) in the study of matrix polynomial [5 The treatment in the presence of singular A d is beyond the scope of this paper We assume that s i λ) and q i λ) are coprime, that is, having no common factors Furthermore, the rational functions siλ) q are proper, that is, s iλ) iλ) having smaller degree than q i λ) Otherwise by the polynomial long division, an improper rational function can be written as the sum of a polynomial and a proper rational function: s i λ) q i λ) = p iλ) + ŝiλ) q i λ) with ŝ i λ) having smaller degree than q i λ) Subsequently, the term p i λ)e i can be absorbed into the matrix polynomial term P λ): P λ) := P λ) + p i λ)e i f it is necessary, we assume that the leading coefficient matrix of the updated matrix polynomial is still nonsingular Since s i λ) and q i λ) are coprime, the proper rational function siλ) q iλ) can be represented as 22) s i λ) q i λ) = at i C i λd i ) b i, for some matrices C i, D i R di di, and vectors a i, b i R di Moreover, D i is nonsingular The process of constructing the quadruple C i, D i, a i, b i ) satisfying 22) is called a minimal realization in the theory of control system, see for example [, pp9 98 and [9 Finally, we assume that coefficient matrices E i have the rank-revealing decompositions 23) E i = L i Ui T, where L i, U i R n ri are of full column rank r i n section 4, we will see that the decompositions 23) are often immediately available in the practical REPs The rank of E i is typically much smaller than the size n, that is, r i n The algorithms for computing such sparse rank-revealing decompositions can be found in [7

3 RATONAL EGENVALUE PROBLEMS 3 3 Linearization Under the assumptions discussed in the previous section, let us consider a linearization method for solving the REP ) By the realizations 22) of the rational functions s i λ)/q i λ) and the factorizations 23) of the coefficient matrices E i, the rational terms of the matrix rational function Rλ) can be rewritten as the following s i λ) q i λ) E i = = = a T i C i λd i ) b i L i Ui T [ L i a T i C i λd i ) b i ri U T i L i ri a i ) T ri C i λ ri D i ) ri b i )Ui T, where is the Kronecker product Define C = diag r C, r2 C 2,, rk C k ), D = diag r D, r2 D 2,, rk D k ), L = [ L r a ) T L 2 r2 a 2 ) T L k rk a k ) T, U = [ U r b ) T U 2 r2 b 2 ) T U k rk b k ) T, where the size of C and D is m m, the size of L and U is n m, and m = r d + r 2 d r k d k Then the rational terms of Rλ) can be compactly represented in a realization form 3) s i λ) q i λ) E i = LC λd) U T We note that the matrix D is nonsingular since the matrices D i in 22) are nonsingular The eigenvalues of the matrix pencil C λd are the poles of Rλ) Using the representation 3), the REP ) can be equivalently written in the following compact form [ 32) P λ) LC λd) U T x =, and the matrix rational function Rλ) is written as 33) Rλ) = P λ) LC λd) U T f P λ) is linear and is denoted as P λ) = A λb, then the REP 32) is of the form [ A λb LC λd) U T x = 34) By introducing the auxiliary vector y = C λd) U T x, the equation 32) can be written as the following LEP: 35) where [ A L A = U T C A λb)z =, [ B, B = D [ x, z = y

4 4 Y SU AND Z BA n general, if the matrix polynomial P λ) is of the form 2), we can first write the REP 32) as a semi-pep of the form ) 36) λ d A d + λ d A d + + λa + Ãλ) x =, where Ãλ) A LC λd) U T Then by symbolically applying the well-known first) companion form linearization to the semi-pep 36), we have 37) A d A d 2 Ã λ) λ A d λ d x λ d 2 x x =, which can be equivalently written as A d A d 2 A λ A d L C λd) U T λ d x λ d 2 x x = The above equation is of the same form as 34) Therefore, by introducing the variable λ d x y = C λd) [ U T λ d 2 x = C λd) U T x, x we derive the following linearization of the REP ): 38) A λb)z =, where A = A d A d 2 A L U T C, B = A d D, z = λ d x λ d 2 x x y The size of matrices A and B is nd + m, where m = r d + r 2 d r k d k n the case that all the coefficient matrices E i are of full rank, ie, r i = n, the LEP 38) is of the size nd, where d = d+d + +d k This is the same size as the one derived by the brute-force approach However, it is typical that r i n in practice, then nd + m nd The LEP 38) is a trimmed linearization of the REP ) This will be illustrated by four REPs from applications in section 4 Note that under the assumption of nonsingularity of the matrix A d, the matrix B is nonsingular Therefore all eigenvalues of the LEP 38) are finite There is no infinite eigenvalue

5 RATONAL EGENVALUE PROBLEMS 5 The following theorem shows the connection between eigenvalues of the REP ) and the LEP 38) Theorem 3 LEP 38) a) f λ is an eigenvalue of the REP ), then it is an eigenvalue of the b) f λ is an eigenvalue of the LEP 38) and is not a pole of Rλ), then it is an eigenvalue of the REP ) Proof Define dn-by-dn matrices V L λ) = V R λ) = λa d A d A d 2 A λ λ d λ λ, We have detv L λ)) = detv R λ)) = and λa d + A d A d 2 A λ λ V Rλ) = V L λ) P λ) Therefore, 39) [ VR λ) det A λb) = det A λb) m λa d + A d A d 2 A L λ [ = det VR λ) λ U T C λd P λ) L [ VL λ) = det m U T C λd P λ) L = det, U T C λd ) m

6 6 Y SU AND Z BA where for the third equality, we use the identities [ U T V R λ) = [ U T and L = V Lλ) L By exploiting the block structure of the matrix in the determinant of the right-hand-side of the equation 39), we derive that [ ) P λ) L 3) det A λb) = det U T C λd f λ is an eigenvalue of the REP ), then it is not a pole of the REP t implies that λ is not an eigenvalue of C λd and C λd is nonsingular Therefore, we have the block factorization [ [ [ P λ) L LC λd) Rλ) U T = C λd U T, C λd where Rλ) is defined by 33) The proof of part a) immediately follows the identity 3) det A λb) = det Rλ)) detc λd) For the part b), if λ is an eigenvalue of the pencil A λb and is not the pole of Rλ), then the matrix C λd is nonsingular By the identity 3), λ is an eigenvalue of the REP ) We note that the condition that λ is not a pole of the Rλ) in Theorem 3b) is necessary Consider the following example: λ 2 ) 32) λ e 2e T 2 x =, where 2 is a 2 by 2 identity matrix, and e 2 is the second column of 2 Since detrλ)) = λλ λ ), the REP 32) has two eigenvalues and λ = is a pole Let y = λ e T 2 x, then the corresponding LEP is given by A λb)z = [ e2 e T 2 [ 2 λ ) [ x y = t has three eigenvalues,, and But λ = is not an eigenvalue of the REP 32) Two additional remarks are in order First, the realization of a rational function can be represented in different forms For example, the realization of σ λ) could be given by 2 σ λ) 2 = [ [ σ σ ) [ λ or in a symmetric form σ λ) 2 = [ [ σ σ [ λ ) [ The study of realization can be found in [, pp9 98 and references therein Second, there are many different ways of linearization for the matrix polynomials, including recent work [7, 8, 3, 4 Many of these linearizations can be easily integrated into the proposed

7 RATONAL EGENVALUE PROBLEMS 7 linearization of the REP For example, if the original REP ) is symmetric, namely A i are symmetric, matrices C and D are symmetric in the realization and L = U, then we can use a symmetric linearization proposed in [7 Specifically, let us consider a symmetric matrix polynomial of degree d = 3, then a symmetric linearization is given by 33) A A U U T C where y = C λd) U T x + λ A 3 A 3 A 2 A 3 A 2 A D λ 2 x λx x y =, 4 Applications n this section, we apply the proposed linearization in section 3 to four REPs from applications 4 Loaded elastic string We consider the following rational eigenvalue problem arising from the finite element discretization of a boundary value problem describing the eigenvibration of a string with a load of mass attached by an elastic spring: 4) A λb + λ λ σ E ) x =, where A and B are tridiagonal and symmetric positive definite, and E = e n e T n, e n is the last column of the identity matrix σ is a parameter [2, 2 By the linearization proposed in section 3, the first step is to write the rational function λ/λ σ) in a proper form t results that the REP 4) becomes 42) A + e n e T n λb + σ λ σ e ne T n ) x = Then it can be easily written in the realization form 34): [A + e n e Tn λb e n λ ) 43) e T n x = σ By defining the auxiliary vector y = λ ) e T n x, σ we have the linear eigenvalue problem 44) A λb)z =, where A = [ A + en e T n e n e T n [ B, B = /σ [ x, z = y The n + ) n + ) matrices A and B have the same structure and property as the coefficient matrices A and B in the original REP 4) They are tridiagonal and symmetric positive definite An alternative way to solve the REP 4) is to first transform it to a quadratic eigenvalue problem QEP) by multiplying the linear factor λ σ on the equation 4): 45) Qλ)x = [ λ 2 B λa + σb + e n e T n ) + σa x =,

8 8 Y SU AND Z BA and then convert the QEP 45) into an equivalent linear eigenvalue problem by the linearization process A symmetric linearization is [ [ ) [ A + σb + en e T n ) σa B λx 46) + λ = σa σa x This is a generalized symmetric indefinite eigenvalue problem f Qa) < for some scalar a, then by letting λ = µ + a, the QEP 45) becomes [ µ 2 B µa + σ 2a)B + e n e T n ) + Qa) x = Consequently, it can be linearized to a generalized symmetric definite eigenvalue problem: [ ) [ ) [ A + σ 2a)B + en e T n Qa) B µx 47) + µ = Qa) Qa) x A practical issue is on the existence of the shift a and how to find it numerically, see recent work [6 We note that the size of the LEP 46) and 47) is 2n On the other hand, the size of the LEP 44) by the new linearization process is only n + 48) 42 Quadratic eigenvalue problem with low-rank stiffness matrix Consider the QEP: λ 2 M + λd + K)x = f K is singular, then zero is an eigenvalue since Kx = for some nonzero vector x Let us consider how to compute nonzero eigenvalues of the QEP 48) by exploiting the fact that K is singular and is of low rank Let K = LU T be the full-rank decomposition, where L, U R n r, r is the rank of K By the linearization discussed in section 3, the QEP 48) can be written in the REP form ) λm + D + λ LU T ) x = n the compact form 32), it becomes [ λm + D L λ) U T x = Note that the polynomial term P λ) = λm + D is linear Hence, by introducing the auxiliary vector we have the LEP 49) y = λ) U T x = λ U T x, [ M λ [ ) [ D L x + U T = y f L = K, U =, then the LEP 49) is a linearization of the QEP in the first companion form f L =, U = K T, then the LEP 49) is a linearization in the second companion form [ f K has rank r < n, the linearization 49) is not a linearization of the QEP 48) under the standard definition of linearization of the PEPs [5 However, by Theorem 3, we can conclude that the nonzero eigenvalues of QEP 48) and LEP 49) are the same The LEP 49) is a trimmed linearization of the QEP The order of the LEP 49) n + r could be significantly smaller than the size 2n of the linearization in the companion forms f M is singular or more general, the leading coefficient matrix A d is singular in the matrix polynomial, the PEP P λ) = has infinite eigenvalues and/or corresponding singular structure t is discussed in [3 that how to exploit the structure of the zero blocks in the coefficient matrices to obtain a trimmed linearization by partially) deflating those infinite eigenvalues and/or singular structures n the linearization 49), by using a full-rank factorization of the stiffness matrix K, we derived a trimmed linearization such that those zero eigenvalues are explicitly deflated

9 RATONAL EGENVALUE PROBLEMS 9 43 Vibration of a fluid-solid structure Let us consider an REP arising from the simulation of mechanical vibrations of fluid-solid structures [5, 6, 2 t is of the form ) λ 4) A λb + E i x =, λ σ i where the poles σ i, i =, 2,, k are positive, matrices A and B are symmetric positive definite, and E i = C i C T i, C i R n ri has rank r i for i =, 2,, k By the linearization proposed in section 3, we first write the rational terms of 4) in the proper form 4) Let A + C i Ci T λb σ i σ i λ C ic T i ) x = C = [ C C 2 C k, Σ = diagσ r,, σ k rk ), where ri is the r i -by-r i identity Then the equation 4) can be written as [ A + CC T λb C λσ ) C T x = By introducing the variable y = λσ ) C T x, we have the following LEP: [ x 42) A λb) =, y where [ A + CC T C A = C T [ B, B = Σ are of the size n + k r i Note that the matrix A is symmetric, and B is symmetric positive definite The LEP 42) is a generalized symmetric definite eigenvalue problem, which can be essentially solved by a symmetric eigensolver, such as implicitly restarted Lanczos algorithm [2 or the thick-restart Lanczos method [24 Mazurenko and Voss [5 addressed the question to determine the number of eigenvalues of the REP 4) in a given interval α, β) by using the fact that the eigenvalues of 4) can be characterized as minimax values of a Rayleigh functional [5, p6 By using the linearization 42), the question can be answered through computing the inertias of the symmetric matrix pencil A λb First, we have the following proposition Proposition 4 f A and B are positive definite and the poles σ i >, then all eigenvalues of the REP 4) are real and positive Proof Since the matrix pencil A λb in 42) is a symmetric definite pencil, all eigenvalues are real By the fact that A is semi-positive definite, all eigenvalues are nonnegative Therefore, we just need to show that zero is not an eigenvalue of the pencil A λb By contradiction, let z = [ x T y T T be an eigenvector corresponding to the zero eigenvalue Then by A B)z = Az =, we have A + CC T )x + Cy =, C T x + y =

10 Y SU AND Z BA Note that A is positive definite, we have x = and y = This is a contradiction Therefore all eigenvalues of the LEP 42) are positive By Theorem 3, we conclude that all eigenvalues of the REP 4) are positive Based on Theorem 3 and Proposition 4, we conclude that the number of eigenvalues of the REP 4) in the interval α, β) is given by 43) κ = l l, where l is the number of eigenvalues of A λb in the interval α, β), l = σ i α,β) l i, and l i is the number of zero eigenvalues of A σ i B The quantities l and l can be computed using the Sylvester s law of inertia for the real symmetric matrices A τb for τ = α, β and poles σ i α, β) 44 Damped vibration of a structure This is an REP arising from the free vibrations of a structure if one uses a viscoelastic constitutive relation to describe the behavior of a material [6 The REP is of the form ) λ 2 44) M + K + b i λ G i x =, where the mass and stiffness matrices M and K are symmetric positive definite, b j are relaxation parameters over the k regions, G j is an assemblage of element stiffness matrices over the region with the distinct relaxation parameters We consider the case where G i = L i L T i and L i R n ri By defining L = [L, L 2,, L k, D = diagb r, b 2 r2,, b k rk ), the REP 44) can be written in the form 32): λ 2 M + K L + λd) L T )x = By linearizing the second-order matrix polynomial term λ 2 M + K in a symmetric form, we derive the following symmetric LEP: M M K L + λ M λx x =, L T D y where the auxiliary vector y = + λd) L T x The size of the LEP is 2n + r + r r k 5 Numerical examples n this section, we present two numerical examples to show computational efficiency of the proposed linearization process of the REP ) in sections 3 and 4 We do not compare the proposed approach with a general-purpose nonlinear eigensolver, such as Newton s method, nonlinear Arnoldi method[22 or preconditioned iterative methods [2 nstead, we compare the extra cost of solving the REP ) over the problem of solving the PEP P λ)x = without the rational terms in ) All numerical experiments were run in MATLAB 7 on a Pentium V PC with 26GHz CPU and GB of core memory Example We present numerical results for the REP 4) arising from vibration analysis of a loaded elastic string discussed in section 4 This REP is included in the collection of nonlinear eigenvalue problems NLEVP) [2 f the NLEVP is included in MATLAB, then the matrices A, B and E can be generated by calling coeffs = nlevp loaded_string,n); A = coeffs{}; B = coeffs{2}; E = coeffs{3}; where n is the size of the REP As in [2, the pole σ is set to be The interested eigenvalues are few smallest ones in the interval λ, + ) The following table records the computed smallest eigenvalues and the corresponding residual norms of the trimmed LEP 44) with the size n = by MATLAB function eig:

11 RATONAL EGENVALUE PROBLEMS i λi residual norm e e e e e e e e e e 3 The residual norm R λ) x 2 / x 2 is used to measure the precision of a computed eigenpair λ, x) of REP ), the same as in [6, Algorithm 5 We note that the first eigenvalue λ <, which is not of practical interest according to [2 Eigenvalues λ 2 to λ 6 match all significant digits of the computed eigenvalues by a preconditioned iterative method reported in [2 The following table reports the CPU elapsed time for solving the trimmed LEP 44) for different sizes n by using MATLAB dense eigensolver eig For comparison, we also report the CPU elapsed time of solving the QEP 45) by using MATLAB function polyeig, and the symmetric LEP 47) of size 2n by using eig n this particular case, it is known that Qa) < for a = 2 Therefore, the symmetric LEP 47) is a generalized symmetric definite eigenproblem solver n = 2 n = 4 n = 6 n = 8 Trimmed LEP 44) eig QEP 45) polyeig Full sym-lep 47) eig A λb eig t is clear that the trimmed LEP 44) is the most efficient linearization scheme to solve the REP 4) Although one can efficiently exploit the positive definiteness in the generalized symmetric definite eigenproblem 47) it is still slower than the trimmed LEP 44) due to the fact that the size of 47) is 2n By the table, we also see that the brute-force approach to convert the REP 4) into the QEP 45) and then solve it via a companion form linearization is most expensive Note that the matrix pair A, B) in the trimmed linearization 44) is symmetric tridiagonal and positive definite, the same properties as the matrices A and B Therefore, from the last row of the previous table we see that the CPU elapsed time of solving the REP via LEP 44) and the one of solving the eigenvalue problem of the pencil A λb are essentially the same Example 2 This is a numerical example for the REP 4) discussed in section 43 The size of matrices A and B is n = 36, 46 The number of nonzeros of A is nnz = 255, 88 B has the same sparsity as A There are nine rational terms, k = 9 The matrices C i in rational terms have two dense columns and are of rank r i = 2 The pole σ i = i for i =, 2,, k Our aim is to compute all eigenvalues in the interval α, β) =, 2), ie, between the first and second poles As we discussed in section 4, the linearization of the REP 4) leads to the LEP A λb)z =, where A and B are defined as in 42) with the size n + r d + + r k d k = n = n + 8 By the expression 43), we can conclude that there are 8 eigenvalues in the interval To apply the expression 43), we need to know the inertias of the matrices A τb for τ = α, β These can be computed using the LDL T decomposition of the matrix A τb Since C i are dense, the explicit computation of the LDL T decomposition of A τb is too expensive nstead, we can first perform the following congruence transformation [ A τb C A τb 5) = L τσ L T C T

12 2 Y SU AND Z BA = L L 2 A τb τσ F L T 2 L T, where F = C T A τb) C + Σ/τ, C L =, L 2 = C T A τb) τ Σ Under these congruence transformations, we see that the inertias of the matrices A τb can be computed from the the inertias of the matrices A τb, τ Σ and F Therefore, we only need to compute the LDL T decomposition of the sparse symmetric matrix A τb n MATLAB, the LDL T decomposition of a sparse symmetric matrix is computed by the function ldlsparse, which is based on [4 Let us turn to compute the 8 eigenvalues in the interval α, β) by applying MATLAB s sparse eigensolver eigs for the LEP 42) The function eigs is based on the implicitly restarted Arnoldi RA) algorithm [2 The following parameters are used for applying the function eigs with the shift-and-invert spectral transformation: tau = 5; % the shift num = 8; % number of wanted eigenvalues optsissym = true; optsisreal = true; optsdisp = ; optstol = e-3; % residual bound optsp = 4*num; % number of Lanczos basis Furthermore, we need to provide an external linear solver for the linear system [ [ ) [ [ A τb C τσ + [C T x b 52) = x 2 b 2 The following procedure is such a solver to compute the solution vectors x and x 2 based on the LDL T decomposition A τb = LDL T and the Sherman-Morrison-Woodbury formula : C := L C 2 b := L b 3 e = C T D b Σb 2 /τ 4 e := + C T D C Σ/τ) e 5 x = L T D b Ce), x 2 = Σb 2 e)/τ The linear system 52) needs to be solved repeatedly for different right-hand sides For computational efficiency, the matrix C := L C in step and the matrix + C T D C Σ/τ) in step 4 are computed only once and stored before calling eigs The CPU elapsed time is displayed in the following table, where ldlsparse is the time for computing the decomposition A τb = LDL T preprocessing is for computing the matrices C := L C and + C T D C Σ/τ), and the assemblage of the matrix B = diagb, Σ ) A λb A λb ldlsparse preprocessing 49 eigs Total The residuals for all 8 computed eigenpairs are less than 55 3 For comparison, in the third column of the previous table, we also record the CPU time to solve only the eigenvalue problem of A + BB T ) = A A B + B T A B) B T A

13 RATONAL EGENVALUE PROBLEMS 3 the linear term A λb of REP 4) using the same parameters for calling eigs As we can see, it only takes about 33% extra time to solve the full REP than the simple eigenvalue problem of the pencil A λb The computed 8 eigenvalues λ, λ 2,, λ 8 of the REP 4) in the interval, 2) are plotted below, along with the computed 8 eigenvalues µ, µ 2,, µ 8 of the linear pencil A µb closest to the shift τ = 5 µ µ 2 µ 3 µ 4 µ 5 µ 6 µ 7 µ 8 2 λλ λ λ λ λ λ λ We observe that there are only 6 eigenvalues of the pencil A λb in the interval, 2) Among them, µ 6, µ 7, µ 8 are good approximations of λ 6, λ 7, λ 8, respectively f we treat the REP 4) as a general NEP and use an iterative method such as the nonlinear Arnoldi method [22) with the initial approximations µ 6, µ 7, µ 8, we would expect good convergence to the desired eigenvalues λ 6, λ 7, λ 8 t would be difficult to predict the convergence behavior of the iterative method with the initial approximations µ, µ 2,, µ 5 Acknowledgement We are grateful to Heinrich Voss for providing the data for Example 2 in section 5 We thank Françoise Tisseur for her numerous comments Support for this work has been provided in part by the National Science Foundation under the grant DMS-6548 and DOE under the grant DE-FC2-6ER25794 The work of Y Su was supported in part by China NSF Project 8749 and E-nstitutes of Shanghai Municipal Education Commission, N E34 REFERENCES [ A C Antoulas Approximation of Large-Scale Dynamical Systems SAM, 25 [2 T Betcke, N J Higham, V Mehrmann, C Schroder, and F Tisseur NLEVP: A collection of nonlinear eigenvalue problems Technical Report MMS EPrint 284, School of Mathematics, The University of Manchester, 28 [3 R Byers, V Mehrmann, and H Xu Trimmed linearizations for structured matrix polynomials Linear Alg Appl, 429: , 28 [4 T A Davis Algorithm 849: A concise sparse Cholesky factorization package ACM Trans Math Software, 34):587 59, Dec 25 [5 Gohberg, P Lancaster, and L Rodman Matrix Polynomials Academic Press, New York, 982 [6 C-H Guo, N J Higham, and F Tisseur Detecting and solving hyperbolic quadratic eigenvalue problems SAM J Matrix Anal Appl, 3:593 63, 29 [7 N J Higham, D S Mackey, N Mackey, and F Tisseur Symmetric linearizations for matrix polynomials SAM J Matrix Anal Appl, 29):43 59, 26 [8 N J Higham, D S Mackey, and F Tisseur The conditioning of linearizations of matrix polynomials SAM J Matrix Anal Appl, 284):5 28, 26 [9 T-M Hwang, W-W Lin, J-L Liu, and W Wang Jacobi-Davidson methods for cubic eigenvalue problems Numer Lin Alg Appl, 2:65 624, 25 [ T-M Hwang, W-W Lin, W-C Wang, and W Wang Numerical simulation of three dimensional quantum dot J Comp Physics, 96:28 232, 24 [ P Lancaster and M Tismenetsky The Theory of Matrices Academic Press, London, 2nd edition, 985 [2 R B Lehoucq, D C Sorensen, and C Yang ARPACK Users Guide: Solution of Large-Scale Eigenvalue Problems with mplicitly Restarted Arnoldi Methods SAM, Philadelphia, 998 [3 DS Mackey, N Mackey, C Mehl, and V Mehrmann Structured polynomial eigenvalue problems: Good vibrations from good linearization SAM J Matrix Anal Appl, 284):29 5, 26 [4 DS Mackey, N Mackey, C Mehl, and V Mehrmann Vector spaces of linearizations for matrix polynomials SAM J Matrix Anal Appl, 284):97 4, 26 [5 L Mazurenko and H Voss Low rank rational perturbations of linear symmetric eigenproblems Z Angew Math Mech, 868):66 66, 26 [6 V Mehrmann and H Voss Nonlinear eigenvalue problems: A challenge for modern eigenvalue methods GAMM- Reports, 27:2 52, 24 [7 M J O Sullivan and M A Saunders Sparse rank-revealing LU factorization via threshold complete pivoting

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