A Study on the Convergence Analysis of the Inexact Simplified Jacobi–Davidson Method

,e Jacobi–Davidson iteration method is very efficient in solving Hermitian eigenvalue problems. If the correction equation involved in the Jacobi–Davidson iteration is solved accurately, the simplified Jacobi–Davidson iteration is equivalent to the Rayleigh quotient iteration which achieves cubic convergence rate locally. When the involved linear system is solved by an iteration method, these two methods are also equivalent. In this paper, we present the convergence analysis of the simplified Jacobi–Davidson method and present the estimate of iteration numbers of the inner correction equation. Furthermore, based on the convergence factor, we can see how the accuracy of the inner iteration controls the outer iteration.


Introduction
Let A be a sparse and Hermitian matrix. en, we are supposed to compute the smallest eigenvalue λ of A and the associated eigenvector x of λ with large, i.e., Ax � λx, with ‖x‖ � 1. (1) Here and in the following, ‖ · ‖ indicates the induced Euclidean norm for either a vector or a matrix. In the literature, there have been many methods developed involving gradient-type methods and subspace methods. e Lanczos, Arnoldi, Davidson, and Jacobi-Davidson methods are classical and effective methods for solving eigenproblem (1), and there have been many convergence results developed for these methods. It should be mentioned that, for the Lanczos and Arnoldi methods, the involved projection subspace should be restricted to the Krylov subspace. For more details about these methods as well as their variants, refer to in [1][2][3][4][5][6][7]. e main framework for the subspace methods is generating a sequence of enlarging subspaces V k , which contain more and more information for the desired eigenvalue or eigenvector of matrix A. e central problem for this method, which can be accomplished by the Rayleigh-Ritz procedure, is to extract the approximation to the desired eigenvalue or eigenvector from the projection subspace. For more details of the Rayleigh-Ritz procedure, refer to in [4]. As we know, for Hermitian matrices, the exact Rayleigh quotient iteration (RQI) [4] converges cubically. However, an ill-conditioned linear system of equations should be solved exactly which is expensive in each step of the iteration as the approximation is close to the target eigenvalue. e idea replacing the exact solution by a cheaper approximate solution results in an inexact Rayleigh quotient iteration (IRQI) [8,9]; however, this replacement may destroy the local convergence property of the RQI.
e Jacobi-Davidson (JD) iteration method [6] proposed by Sleijpen and van der Vorst can overcome these difficulties. A so-called correction equation As we know, if (2) is solved exactly, the JD method converges as fast as the RQI method. However, what we are interested in is the iterative solver with an ideal preconditioner for it. Although the shifted matrix A − θI becomes ill-conditioned as the approximation θ is near to the desired eigenvalue, correction equation (2) remains well conditioned, thanks to the projection onto the orthogonal complement of u. As Notay in [10] pointed out, with proper implementations, the potential indefiniteness of the coefficient matrix in system (2) cannot spoil the method.
At present, we have not seen the analysis of the connection between the solution of the correction equation and the convergence of the approximate vector u towards the wanted eigenvector. In this paper, we study the convergence of the simplified JD iteration.
rough the convergence factor, we try to analyze how the inner linear system controls the outer iteration. Moreover, we can also see that, under some assumptions, the JD method attains quadratic convergence rate locally. en, we can gain higher convergence rate through increasing the accuracy precision of the inner linear system. In the last part of this paper, we give the convergence analysis in terms of the residual norms, from which we can see that these results are asymptotically identical to those derived in the former section; we also gave the analysis on the iteration number of the inner linear system.
We should mention that some analyses for the JD method or, more generally, the Newton method have been established so far; see, e.g., [7,[11][12][13] and the references therein. In particular, Bai and Miao in [11] presented the convergence of the JD iteration method, and they proved that the JD iteration method attains quadratic convergence locally when the involved correction equation is solved by a Krylov subspace method and attains cubic convergence rate when the correction equation is solved to a prescribed precision proportional to the norm of the current residual vector. In this paper, we will use a completely different technique to demonstrate the convergence of the JD method from another point of view. In addition, we have further studied the connection between the iteration steps of the inner correction equation with the convergence of the outer iteration.
is paper is organized as follows. In Section 2, we give some preliminaries of the JD method. In Section 3, we give some known results and then built several new results concerning the convergence property of the simplified JD method whose involved correction equation is inexactly solved by Krylov solvers. In Section 4, we give the estimate on the iteration number of the inner linear system. Finally, we give some concluding remarks.

Preliminaries
Let x i n i�1 be the eigenvectors of the Hermitian matrix A associated with the eigenvalues λ i n i�1 with the ascending order λ 1 < λ 2 ≤ . . . ≤ λ n . In the following discussion, we want to compute the eigenvector x 1 associated with the simple smallest eigenvalue λ 1 . Denote by ϕ the angle between the current approximation u and the wanted eigenvector x 1 .
We first present the algorithmic description of the simplified JD method in Algorithm 1.
As we know, if the correction equation in Algorithm 1 is solved accurately, the simplified JD method is equivalent to the RQI method. In fact, according to (2) We can see that the new approximation u � u + t/‖u + t‖ is the Rayleigh quotient iteration vector. In fact, eorem 4.2 in [14] tells us that the inexact simplified JD method and the IRQI method can also be equivalent if the inner linear system is solved by Krylov subspace methods. Here, the 'inexact' method means that the inner linear system is solved by an iteration method.
Based on the property of the correction equation, the inner linear system in Algorithm 1 is solved approximately by Krylov subspace methods, and we will use the obtained vector to update the current eigenvector approximation.
Suppose that we have obtained an approximate eigenvector u which is close to the wanted eigenvector x in Algorithm 1; we decompose it in the following way: where w⊥x 1 with ‖w‖ � 1 and ϕ is the angle between vectors u and x 1 . Also, in this paper, the approximate vector u is close to the wanted eigenvector x 1 which means In the following, we give a lemma which reveals the relation between two Krylov subspaces. Lemma 2.1. Let u be a normalized vector, θ � u * Au, and r � Au − θu.
Denote by Π � I − uu * and A(θ, u) � Π(A − θI)Π. en, for k ≥ 2, the two Krylov subspaces V J k− 1 and V R k have the following relation: where and Proof. We prove this lemma by induction over k. Obvi- Next, we will prove that this relation satisfies for i � k. Denote v � A(θ, u) k− 3 r; based on the fact v⊥u and the induction hypothesis,

Convergence Analysis of the JD Iteration
In the following discussion, we solve the correction equation approximately by Krylov subspace methods, such as CG or MINRES with the initial vector being zero, to obtain an approximate solution t to update the new eigenvector approximation u. at is to say, solution t satisfies the following equation: where η m is the residual at step m.
In order to analyze conveniently, (7) can also be represented by other equivalent forms such as (30).
As we know, the JD method is one of the "inner-outer" type iterations. In the method of this type, it is essential to know how the inner iteration controls the outer iteration. In other words, we want to know how accuracy or how many steps should be solved for the inner iteration equation to ensure the convergence of the outer iteration. As Sleijpen and van der Vorst pointed out in [6], we cannot answer the question at present. All these problems may be based on the convergence analysis of the algorithm. In the following, we first give some convergence results. Lemma 3.1 (see [7]). If Algorithm 1 is applied to seek the smallest simple eigenvalue of the Hermitian matrix A and we assume that the approximate solution t satisfies the relation in (30), then for u � u + t/‖u + t‖, we asymptotically have Lemma 3.2 (see [13]). If Algorithm 1 is applied to seek the smallest simple eigenvalue of the Hermitian matrix A and we assume that the approximate solution t satisfies the relation in (30), then for u � u + t/‖u + t‖, we asymptotically have e two convergence results above are analysed based on the angle between vectors u and x 1 . Proposition 1 in [15] gives us another result for the convergence analysis based on the metric of ‖u − x 1 ‖. However, all these results cannot answer the question asked above, and we could not answer how the convergence order varies as the inner correction equation is being solved.
u be the associated residual. en, the following estimate holds: where ϕ is the angle between the new approximation u and x 1 and τ � u * (A − θI)t m .
Proof. Suppose that u is the current approximation to the wanted eigenvector x 1 , and we decompose it in the way of (3). If we solve the correction equation where q m (λ) � q m (λ) + 1. e residual of the correction equation at step m is with τ � u * (A − θI)t m and q m+1 (λ) � λq m (λ) − τ.
According to the decomposition in (3), we have us, we obtain Note that q m+1 (λ) � λq m (λ) − τ; that is, q m (λ) � q m+1 (λ) + τ/λ; then, Journal of Mathematics 3 Based on (3) and (12), we get us, we have In addition, it obviously holds that ereby, based on (15) □ eorem 3.1 gives us a preliminary convergence analysis of the simplified JD method. Since the simplified JD method is a JD method without subspace acceleration, the convergence factor of the former is an upper bound of the latter.
at is to say, in order to gain the convergence property of the JD method, we can analyze its simplified form.
Next, we analyze the convergence factor of eorem 3.1. e current approximation u is very near to the wanted vector x 1 ; that is to say, |cos ϕ| ≥ δ with δ being a constant smaller than one. Combining with the fact |λ 1 − w * Aw| ≤ |λ n − λ 1 |, it is clear that the second term of the convergence factor can be bounded; then, it is the first term which plays a vital role on the analysis of the convergence.
For the purpose of analysis, we present the following lemma.
en, the following estimate holds: where ϕ is the angle between vectors u and x 1 and σ min ⊥ (A) represents the smallest singular value of A restricted to the subspace span u It follows from x * u � 0 that n i�1 α i β i � 0; equivalently, we have α 1 β 1 � − n i�2 α i β i . Moreover, we have e first assumption in (21) indicates that λ 2 + λ 1 − 2θ < λ 2 − λ 1 and λ 2 − θ > λ 2 − λ 1 /2 hold; then, we have the estimate Here, we use the fact α 1 � cos ϕ. Combining the above estimate with the second assumption in (21), we frequently obtain the estimate in (22).

□
In the following, we give an estimate of |τ| with τ � u * (A − θI)t m .
us, combining (26) and the correct equation in (7), we have It further indicates that and, at last, we obtain To see the behavior of the convergence for the JD method clearly, the stopping criterion we adopt is that the norm of the current residual is reduced by a factor ξ from that of the initial residual. at is, t satisfies the following equation: where d is the residual direction and ξ is the stopping factor.
Theorem 3.2. Given ξ 1 , ξ 2 < 1, if ‖η m ‖ ≤ ξ 1 ‖r‖, then the JD method converges linearly as follows: under the assumption If ‖η m ‖ ≤ ξ 2 ‖r‖ 2 , then the JD method converges quadratically as follows: under the assumption and Proof. According to the properties of the Krylov subspace method, e.g., the conjugate gradient method, we have Specially, we have η m ⊥ r; combining with the factorization in (3), we have Journal of Mathematics en, we further have By straightforward computations, we have which implies that If ‖η m ‖ ≤ ξ 1 ‖r‖, according to the estimate of |τ|, we get ereby, we obtain On the contrary, under the assumption in (3), we have 6 Journal of Mathematics us, combining with the estimate in (10), we have the following estimate: Similar to the above proof, if ‖η m ‖ ≤ ξ 2 ‖r‖ 2 , according to the estimate of |τ|, we get Note that and combining with the assumption in (33), we obtain Utilizing the estimate in (10) again, we have the following estimate: □ Note that the assumptions in (32) and (34) will easily be satisfied if min 2≤j≤n | (λ j − θ)q m+1 (λ j − θ)| is not very small because the right terms of the two assumptions have the factor |λ 1 − θ| � O(sin 2 ϕ), which would be small if the current approximate eigenvector is near to the desired one.
From eorem 3.1, we cannot fully understand the convergence of the JD method because it just gives us a preliminary convergence analysis and includes some unknown factors to be explored. us, we further explored these unknown factors in eorem 3.2 and established the Journal of Mathematics convergence of the JD method, from which we can see clearly how the inner correction equation controls the convergence of the outer iteration.
In addition, from eorem 3.2, we can see that the JD method converges linearly if the accuracy of the correction equation is roughly O(‖r‖) and converges quadratically with the accuracy of the correction equation being O(‖r‖ 2 ). Moreover, observing the convergence factor, we can see that the method can gain cubic convergence rate ideally.

Estimate for the Iteration Number
In this section, we first give the bounds of the residual norms of the JD iteration method. rough this bound, we may analyze the relation between the outer iteration and the inner iteration; more clearly speaking, we can see how the inner iteration controls the convergence property of the outer iteration.  Let (θ, u) be the approximate eigenpair obtained by Algorithm (1) with ‖u‖ � 1, θ � u * Au, and the residual r � Au − θu. If we solve the JD correction equation by the Krylov subspace method with the zero initial vector, then we get the following estimate: where σ min ⊥ � σ min ⊥ (A − θI) and u � q m (A − θI)u is the new approximation defined as (11).
Proof. Using the minimal residual property of Ritz values (Fact 1.9 in [4]), we have Based on the relation in (12) of the residual equation, we have (52) By making use of the estimate of |τ|, we get (53) □ From the above theorem, we can choose a considerate stopping factor ξ to obtain a higher convergence rate as follows.
Given a normalized vector u arbitrary and a stopping factor ε For m � 1, 2, . . ., do (64) us, if 1/μ 1 ((1 + ξ m )‖r‖/σ min ⊥ + ξ m ) < 1, that is, the JD method converges. It indicates that the residual norm of the correction equation ‖res m ‖ satisfies Based on Lemma 4.2, we know that if 2k 3/2 ⊥ ( ����� � k ⊥ − 1 / ����� � k ⊥ + 1 ) m ‖r‖ ≤ μ 1 σ min ⊥ − ‖r‖/‖r‖ + σ min ⊥ ‖r‖, the JD method converges. By straightforward computations, we frequently get □ We remark that the convergence theories in eorem 4.1 and Corollary 4.1 are identical to those of eorem 3.2 to interpret the convergence of the JD method from the point of view of the residual norm. It indicates that, at the start of the iteration process, the inner correction equation only needs to be solved with a small number of iterations; however, once we obtain a good approximate eigenvector, the inner correction equation is recommended to be solved with a high accuracy.
From eorem 4.2, we can see that the iteration number of the correction equation solved by the CG method, which ensures the decrease of the outer iteration, depends on ‖r‖. If ‖r‖ does not shake vigorously, the inner iteration number will be roughly a constant as the outer iteration proceeds.

Concluding Remarks
We have proved that the inexact simplified Jacobi-Davidson iteration method for Hermitian eigenvalue problems can attain cubic convergence rate locally, and it is asymptotically convergent as fast as the Rayleigh quotient iteration.
us, both exact and inexact simplified Jacobi-Davidson methods are competitive with the exact and inexact Rayleigh quotient iterations. Moreover, we give an estimate of iteration numbers of the inner correction equation. Based on these theoretical results, we can see clearly how the accuracy of the inner correction equation controls the outer iteration.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare no conflicts of interest.