Convergence Rates and Explicit Error Bounds of Hill's Method for

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May 15, 2012 - Section 5 summarizes the proofs of theorems in Section 3. Section 6 ...... p ) and λN,1,...,λN,k σ(Sµ p,N ) satisfy Assump- tion 3b. Then for an ...

Convergence Rates and Explicit Error Bounds of Hill’s Method for Spectra of Self-Adjoint Differential Operators

arXiv:1205.3308v1 [math.NA] 15 May 2012



Ken’ichiro Tanaka∗ † , Sunao Murashige∗ ‡ School of Systems Information Science, Future University Hakodate, 116-2 Kamedanakano-cho, Hakodate, Hokkaido, 041-8655, Japan † [email protected], ‡ [email protected] May 15, 2012 Abstract

We present the convergence rates and the explicit error bounds of Hill’s method, which is a numerical method for computing the spectra of ordinary differential operators with periodic coefficients. This method approximates the operator by a finite dimensional matrix. On the assumption that the operator is selfadjoint, it is shown that, under some conditions, we can obtain the convergence rates of eigenvalues with respect to the dimension and the explicit error bounds. Numerical examples demonstrate that we can verify these conditions using Gershgorin’s theorem for some real problems. Main theorems are proved using the Dunford integrals which project an eigenvector to the corresponding eigenspace. Keywords Hill’s method, Convergence rate, Error bound, Differential Operator, Eigenvalue Problem Mathematics Subject Classification (2000) 65L15, 65L20, 65L70

1

Introduction

This paper considers Hill’s method [8], which is a numerical method for computing spectra of ordinary differential operators Sp with periodic coefficients: p−1

Sp =

X dp dj f˜j (x) j , + p dx dx j=0

(1.1)

where x ∈ R, and the functions f˜j (j = 0, 1, . . . , p − 1) are C ∞ and satisfy f˜j (x + L) = f˜j (x) for some real positive constant L. The eigenvalue problem for Sp is described as Sp φ = λφ, where λ is an eigenvalue and φ is an eigenvector. This eigenvalue problem often appears in physical problems such as linear stability analysis of periodic solutions of nonlinear wave equations [7]. The set of eigenvalues λ is included in the spectrum σ(Sp ). Accordingly, computation of σ(Sp ) is important in both theoretical and practical points of view. It was reported in [7] that Hill’s method can produce very good computed results of σ(Sp ) for some problems. Note that implementation of Hill’s method is straightforward. The ideas of Hill’s method are to employ the Floquet-Bloch decomposition of the spectrum σ(Sp ) and to approximate the eigenvector φ by a finite Fourier series. This Fourier series approximation generates an eigenvalue problem of finite dimension D corresponding to the operator Sp . The aims of this paper are to show the convergence rates of the approximate eigenvalues with respect to the dimension D, and to explicitly obtain the error bounds of them. On the convergence property of Hill’s method, Curtis and Deconinck [4] proved that the exact eigenvalues exist near the computed ones for general cases of Sp , and showed that all exact eigenvalues can be approximated by the computed ones for the case of self-adjoint Sp . Also, they obtained the convergence rate for the case of self-adjoint Sp and the constant coefficients f˜1 , . . . , f˜p−1 . Johnson and Zumbrun [10] investigated Hill’s method using the Evans function, of which the roots correspond to eigenvalues, for general cases of Sp . They showed that the approximate eigenvalues converge to the exact ones, but did not get the convergence rate of them. Vainikko 1

[11][14] examined an approximation method for the eigenvalue problem Sp φ = λφ in an abstract framework, and obtained the convergence rate, which is based on the resolvent norm convergence of approximate operators, for general cases of Sp . We can apply Vainikko’s results to Hill’s method. But it is difficult to directly evaluate the value of the convergence rate. It should be noted that, although all of these convergence rates yield some error bounds of approximate eigenvalues with unknown coefficients, any explicit error bounds have not been shown. In this paper, we give a priori estimates of the convergence rate and a posteriori explicit error bounds of Hill’s method for self-adjoint Sp with the two cases of coefficient functions f˜j in (1.1), namely f˜j ∈ C ∞ and f˜j being analytic on some strip region containing the real line. These classes of f˜j are more general than [4]. The key ideas of these estimations are to project eigenvectors using the Dunford integrals (5.1) and (5.20), and to specify the disks around the exact eigenvalues using Gershgorin’s theorem (Theorem 3.3). This paper is organized as follows. Section 2 describes Hill’s method. In Section 3, we give the convergence rates and the explicit error bounds of Hill’s method for self-adjoint differential operators. Section 4 presents numerical examples which supports our results. Section 5 summarizes the proofs of theorems in Section 3. Section 6 concludes this paper.

2

Hill’s method

This section describes Hill’s method [8] which is a numerical method for computing the spectum σ(Sp ) of the ordinary differential operator Sp with periodic coefficients f˜j defined by (1.1). This operator Sp can be regarded as an operator Sp : H p (R) → L2 (R), where L2 (R) is the Lebesgue space of square integrable functions on R and H p (R) ⊂ L2 (R) is the Sobolev space of functions whose derivatives up to p-th order are square integrable. Hill’s method approximates the elements of σ(Sp ) by the following two steps. Step 1: Floquet-Bloch decomposition. In order to apply the Floquet theory, we introduce a new operator Spµ : H p ([0, L])per → L2 ([0, L])per defined by Spµ = e−iµx Sp eiµx ,

(2.1)

where µ ∈ [0, 2π/L), and L2 ([0, L])per and H p ([0, L])per are the Lebesgue space and the Sobolev space of periodic functions on [0, L], respectively. Note that Sp is defined on R, whereas Spµ is on [0, L]. If Sp is self-adjoint, then Spµ is also self-adjoint. Moreover, Spµ is explicitly written as p−1

Spµ =

X dj dp f (x) + j dxp j=0 dxj

(2.2)

for some periodic functions f0 , . . . , fp−1 with period L. It is known that the spectrum σ(Spµ ) consists of only the eigenvalues of Spµ , and the Floquet theory yields σ(Sp ) =

[

σ(Spµ ).

(2.3)

µ∈[0, 2π/L)

This decomposition (2.3) is called the Floquet-Bloch decomposition. Accordingly, it suffices to consider the eigenvalue problem Spµ φ = λφ for µ ∈ [0, 2π/L), where λ ∈ C is an eigenvalue and φ ∈ L2 ([0, L])per is an eigenvector. In addition, it should be noted that σ(Spµ ) is a discrete set of the eigenvalues of Spµ without accumulation points. This discreteness follows from the compactness of the resolvent of Spµ provided ρ(Spµ ) 6= ∅, where ρ(Spµ ) = C \ σ(Spµ ). See e.g. Lemma 2 in [6, Chapter XIX Section 2], Lemma 3 in [6, Chapter XIX Section 3] and Lemma 16 in [5, Chapter XIII Section 2]. Step 2: Fourier series approximation. Since φ ∈ L2 ([0, L])per has the Fourier series expansion:   ∞ 1 X ˆ 2πnx φ(x) = √ , φn exp −i L L n=−∞ 2

(2.4)

φ can be approximated by the truncation of this series:   N 1 X ˆ 2πnx (PˆN φ)(x) = √ . φn exp −i L L n=−N

(2.5)

This truncation reduces the eigenvalue problem Spµ φ = λφ to a finite dimensional problem. More precisely, the problem µ Sp,N φN = λN φN

µ with Sp,N = PˆN Spµ PˆN

(2.6)

gives approximate eigenvalues λN ’s for the original problem Spµ φ = λφ. Since the problem (2.6) is equivalent to a matrix eigenvalue problem, we can obtain the approximate values µ of the eigenvalues of Spµ using some standard numerical method. In the following sections, let σ(Sp,N ) denote µ the set of eigenvalues of Sp,N .

3

Convergence Rates and Error Bounds of Hill’s Method

In this section, we present theorems about a priori estimates of the convergence rate and a posteriori explicit error bounds of Hill’s method for the eigenvalue problem Spµ φ = λφ with self-adjoint Spµ on some assumptions. Their proofs are given in Section 5. In what follows, we use the notations defined in Section 2.

3.1

Assumptions

First, we assume the self-adjointness of the operator Spµ . Assumption 1. The operator Spµ is self-adjoint, i.e. (Spµ )∗ = Spµ , where



represents the adjoint of operators.

µ in (2.6) is also self-adjoint. On Assumption 1, Sp,N Next, we specify the smoothness of the coefficient functions fj ’s of Spµ in (2.2).

Assumption 2a. For the operator Spµ , the coefficients f0 , f1 , . . . , fp−1 are C ∞ . In addition, as a special case of Assumption 2a, the following assumption is prepared. Assumption 2b. For the operator Spµ , the coefficients f0 , f1 , . . . , fp−1 are analytic on a complex domain Dd = {z ∈ C | |Im z| < d}

(3.1)

for some d > 0. Assumption 2b is often satisfied in real problems. µ approximates Spµ very well, we may expect that approximate eigenvalues λN ’s for large enough N If Sp,N should be included in the neighborhood Bλ (rλ ) of the corresponding exact eigenvalue λ, where Bζ (r) = {z ∈ C | |z − ζ| ≤ r}.

(3.2)

Then we prepare the following two assumptions. Assumption 3a. For λ ∈ σ(Spµ ), there exists a positive real number rλ such that the following holds true: for µ some sequence {λN | λN ∈ σ(Sp,N )} and some positive integer N0 , if N > N0 then |λ − λN | ≤

rλ 2

(3.3)

and n o µ 2rλ ≤ min dist(λ, σ(Spµ ) \ {λ}), dist(λ, σ(Sp,N ) \ {λN }) . 3

(3.4)

Γλ (Spµ ) rλ /2

2rλ



λ

λN

|

{z } Elements of σ(Spµ ) \ {λ} or µ σ(Sp,N ) \ {λN }

Figure 1: Illustration of the conditions (3.3) and (3.4) in Assumption 3a. This figure depicts locations of an µ exact eigenvalue λ ∈ σ(Spµ ), the corresponding approximate eigenvalue λN ∈ σ(Sp,N ) and some other eigenvalues on the complex plane C. The eigenvalue λN is located inside the circle with center λ and radius rλ /2 (shown by the smaller dotted circle), and the other eigenvalues than λ and λN are outside the circle with center λ and radius 2rλ (shown by the larger dotted circle). The directed contour Γλ (Spµ ) is used to prove the theorems in Sections 3.2 and 3.3. Figure 1 illustrates these conditions (3.3) and (3.4). When a single sequence {λN } for some λ ∈ σ(Spµ ) is detected, Assumption 3a is effective for estimation of the convergence rate and the error bound of {λN }. In general, there exist more than one sequence of approximate eigenvalues for some λ ∈ σ(Spµ ). Then we set another assumption as follows: Assumption 3b. Let k be an integer with k ≥ 2. For λ ∈ σ(Spµ ), there exists a positive real number rλ such µ that the following holds true: for some sequences {λN,i | λN,i ∈ σ(Sp,N )} (i = 1, 2, . . . k) and some positive integer N0 , if N > N0 then |λ − λN,i | ≤

rλ 2

(i = 1, 2, . . . , k)

(3.5)

and 2rλ ≤ min

(

dist(λ, σ(Spµ )

\ {λ}), dist λ,

µ σ(Sp,N )

\

k [

i=1

{λN,i }

!!)

.

(3.6)

µ Remark 3.1. Assumptions 3a and 3b mean that, for sufficiently large N , the number of eigenvalues of Sp,N µ approximating λ ∈ σ(Spµ ) is at most finite, and the other elements of σ(Spµ ) and σ(Sp,N ) are relatively far from λ. The finiteness of the approximate eigenvalues is based on the fact that the eigenspace of Spµ for any λ is finite dimensional. S µ ). Thus When Assumption 3a or 3b are satisfied, there exist some convergent subsequences in N σ(Sp,N µ these assumptions are closely related to the convergence property of the eigenvalues of Sp,N . On this convergence property, the following two theorems are known [4][12]. Theorem 3.1 gives so-called “no-spurious-mode condition”. µ Theorem 3.1 ([4, Theorem 9]). Let D ⊂ C be a compact set and {λN } be a sequence in D with λN ∈ σ(Sp,N ). Then for any ε > 0 there exists a positive integer M such that the following holds true: for any N with N ≥ M , λN is contained in the ε-neighborhood of some λ ∈ D ∩ σ(Spµ ).

Theorem 3.2 ([4, Corollary of Lemma 10 and Theorem 16]). Assume that Spµ is self-adjoint. Then for any µ ) such that λNj → λ. λ ∈ σ(Spµ ) there exists some subsequence {λNj } of {λN } with λN ∈ σ(Sp,N In Section 3.3, a sufficient condition for (3.3) and (3.4) in Assumption 3a is given by Theorem 3.6. We can verify this sufficient condition using Theorems 3.1, 3.2 and the following theorem for a concrete problem as shown in Section 4.

4

Theorem 3.3 (Gershgorin’s Theorem [15]). For any n × n matrix V = (vij ) with vij ∈ C, all eigenvalues of Sn V are contained in i=1 Ci , where C1 , . . . , Cn are disks defined as     X Ci = z ∈ C |z − vii | ≤ |vij | . (3.7)   j6=i S Moreover, each connected component of ni=1 Ci contains as many eigenvalues of V as the disks composing it.

This theorem is used to estimate the radius rλ in the conditions (3.3) and (3.4), whereas Theorems 3.1 and 3.2 guarantee that the sequence of the approximate eigenvalues converges to an exact eigenvalue.

3.2

A Priori Estimates of Convergence Rates

When Assumptions 1, 2 and 3 are satisfied, we can obtain a priori estimates of the convergence rates as follows. Theorem 3.4. Let Assumptions 1 and 3a be satisfied. Then, when Assumption 2a is satisfied, namely fj being C ∞ , for a sufficiently large N and any positive integer q, there exists a positive constant C1 such that |λ − λN | ≤ C1 N −q ,

(3.8) Spµ

where C1 depends only on p, f0 , . . . , fp−1 , λ, rλ , q and an eigenvector φ of corresponding to λ. In addition, when Assumption 2b is satisfied, namely fj being analytic, for a sufficiently large N and any ε with 0 < ε < d, there exists a positive constant C2 such that   2π(d − ε) N , (3.9) |λ − λN | ≤ C2 N p+1/2 exp − L where C2 depends only on p, f0 , . . . , fp−1 , λ, rλ , d, ε and φ. Theorem 3.5. Let Assumptions 1 and 3b be satisfied. Then, when Assumption 2a is satisfied, namely fj being C ∞ , for a sufficiently large N , some choice of an index iN ∈ {1, . . . , k} and any positive integer q, there exists a positive constant C1 such that |λ − λN,iN | ≤ C1 N −q ,

(3.10)

where C1 depends only on p, f0 , . . . , fp−1 , λ, rλ , q and an eigenvector φ of Spµ corresponding to λ. In addition, when Assumption 2b is satisfied, namely fj being analytic, for a sufficiently large N , some choice of an index iN ∈ {1, . . . , k} and any ε with 0 < ε < d, there exists a positive constant C2 such that   2π(d − ε) N , (3.11) |λ − λN,iN | ≤ C2 N 2p+1 exp − kL where C2 depends only on p, f0 , . . . , fp−1 , λ, rλ , d, ε and φ.

3.3

A Posteriori Error Bounds

When Assumptions 1 and 2a are satisfied, we can explicitly obtain a posteriori estimates of the error bounds as follows. Here, for simplicity, only the case of Assumption 3a is considered. Theorem 3.6. Let Assumptions 1 and 2a be satisfied. Assume that there exists ζ ∈ C and r > 0 such that, for any N ,   µ λN ∈ Bζ (r) and Bζ (9r) ∩ σ(Sp,N ) \ {λN } = ∅, (3.12)

µ where λN ∈ σ(Sp,N ) and Bζ (r) is defined by (3.2). Then there uniquely exists λ ∈ σ(Spµ ) ∩ Bζ (r) satisfying   3|ζ| (2πN )p |λ − λN | ≤ 5 + r Lp+1/2   p l+N X X X X  · (3.13) (fˆj )m  , (fˆj )m (φˆN )l−m + (2N + 1) j=0

N N0

2 µ k(Spµ − Sp,N )φk2 < 1. rλ

(5.12)

Then for N > N0 we have |λ − λN | ≤

  8|λ| µ k(Spµ − Sp,N )φk2 . 3+ rλ

(5.13)

Proof. By Lemma 5.1, for N > N0 we can define φN =

1 Pλ (S µ )φ. µ )φk2 N p,N kPλN (Sp,N

(5.14)

Then φN is an eigenvector corresponding to λN with kφN k2 = 1. Then noting λ = (Spµ φ, φ)2 and λN = µ (Sp,N φN , φN )2 we have µ λ − λN = (Spµ φ, φ)2 − (Sp,N φN , φN )2

µ µ = (Spµ φ, φ − φN )2 + ((Spµ − Sp,N )φ, φN )2 + (φ − φN , Sp,N φN )2 ,

(5.15)

µ where the last equality follows from the self-adjointness of Sp,N . Therefore we have µ |λ − λN | ≤ (|λ| + |λN |) kφ − φN k2 + k(Spµ − Sp,N )φk2 .

(5.16)

Here kφ − φN k2 is bounded by µ µ )φ − φN k2 . )φk2 + kPλN (Sp,N kφ − φN k2 ≤ kφ − PλN (Sp,N

(5.17)

µ )φ − φN k2 in (5.17) is bounded as Since kPλN (Sp,N

!

1

µ µ PλN (Sp,N )φ kPλN (Sp,N )φ − φN k2 = 1 − µ

kPλN (Sp,N )φk2 2 µ = kPλN (Sp,N )φk2 − 1 µ )φ − φk2 , ≤ kPλN (Sp,N

we have µ )φk2 . kφ − φN k2 ≤ 2kφ − PλN (Sp,N

(5.18)

Applying the inequality (5.5) of Lemma 5.1 to (5.18), we have kφ − φN k2 ≤

4 µ k(Spµ − Sp,N )φk2 . rλ

(5.19)

Then combining (5.16) and (5.19), we have   4 µ (|λ| + |λN |) + 1 k(Spµ − Sp,N |λ − λN | ≤ )φk2 . rλ Finally, noting |λN | ≤ |λ − λN | + |λ| ≤ rλ /2 + |λ|, we have the conclusion. Remark 5.1. Assumption 1, i.e. the self-adjointness of Spµ is used (i) to estimate the norm of the resolvent µ µ of the approximate operator as k(νI − Sp,N )−1 k2 = dist(ν, σ(Sp,N ))−1 in (5.8), (ii) to make an approximate eigenvector as (5.14) and (iii) to deduce the equality (5.15). 11

µ What remains is to estimate k(Spµ − Sp,N )φk2 in the RHS of (5.13). Since µ k(Spµ − Sp,N )φk2 = k(Spµ − PˆN Spµ )φ + (PˆN Spµ − PˆN Spµ PˆN )φk2

≤ k(I − PˆN )Spµ φk2 + kPˆN Spµ (I − PˆN )φk2 ≤ (|λ| + kPˆN Spµ k2 )k(I − PˆN )φk2 ,

we consider estimation of kPˆN Spµ k2 and k(I − PˆN )φk2 . Their estimates are given by the following lemmas, whose proofs are shown in Appendix A. It suffices to consider the case of Assumption 2a for kPˆN Spµ k2 , whereas the both cases of Assumptions 2a and 2b need to be considered for k(I − PˆN )φk2 which depends on the smoothness of φ. Lemma 5.3. Let Spµ satisfy Assumption 2a. Then there exits a positive real constant C depending only on p and f0 , f1 , . . . , fp−1 such that kPˆN Spµ k2 ≤ C N p+1/2 . Lemma 5.4. Let Spµ satisfy Assumption 2a and φ ∈ H p ([0, L])per be an eigenvector of Spµ corresponding to λ ∈ σ(Spµ ) with kφk2 = 1. Then for any positive integer q there exists a positive constant C depending only on φ and q such that k(I − PˆN )φk2 ≤ C (1 + N )−q . Lemma 5.5. Let Spµ satisfy Assumption 2b for d > 0 and φ ∈ H p ([0, L])per be an eigenvector of Spµ corresponding to λ ∈ σ(Spµ ) with kφk2 = 1. Then for any ε with 0 < ε < d there exists a positive constant C depending only on φ, d and ε such that   2π (d − ε) ˆ k(I − PN )φk2 ≤ C exp − N , L where L is the period of the coefficient functions of Spµ . Here we prove Theorem 3.4. Proof of Theorem 3.4. Since it follows from Lemmas 5.3 and 5.4 that µ k(Spµ − Sp,N )φk2 → 0 as N → ∞, the condition (5.12) in Lemma 5.2 is satisfied. Therefore we can deduce the conclusion (3.8) from Lemmas 5.2, 5.3 and 5.4, and (3.9) from Lemmas 5.2, 5.3 and 5.5.

5.2

Proof of Theorem 3.5

µ )φ We use almost the same methods as the ones of Lemmas 5.1 and 5.2. On Assumption 3b, we consider PΛN (Sp,N µ for an eigenvector φ corresponding to λ ∈ σ(Sp,N ), where ΛN = {λN,1 , . . . , λN,k }, µ )= PΛN (Sp,N

1 2πi

I

ΓΛ (Spµ )

µ Rµ (Sp,N ) dµ,

(5.20)

and ΓΛ (Spµ ) is a directed circle with center λ and radius rλ , which contains ΛN . Here it should be noted that (5.2), (5.3) and (5.4) can be used for Assumption 3b when λN is replaced by λN,i (i = 1, 2, . . . , k). In this µ µ )φ is not an eigenvector corresponding to a single eigenvalue of Sp,N but a linear combination of case PΛN (Sp,N eigenvectors corresponding to λN,1 , . . . , λN,k . Then we can estimate the convergence rate of one of them after appropriate rearrangement of them explained later. µ Lemma 5.6. Let Spµ satisfy Assumptions 1 and 2a, λ ∈ σ(Spµ ) and λN,1 , . . . , λN,k ∈ σ(Sp,N ) satisfy Assumption 3b. Then for an eigenvector φ corresponding to λ we have µ )φk2 ≤ kφ − PΛN (Sp,N

12

2 µ k(Spµ − Sp,N )φk2 , rλ

(5.21)

where ΛN = {λN,1 , . . . , λN,k }. Furthermore, if kφk2 = 1 and there exists a positive integer N0 such that for any N > N0 2 µ k(Spµ − Sp,N )φk2 < 1, rλ

(5.22)

µ )φ 6= 0 for N > N0 . we have PΛN (Sp,N

Proof. Noting the contour Γλ (Spµ ) = ∂Bλ (rλ ) surrounds λ and λN,1 , . . . , λN,k in the case of Assumption 3b, we µ ). Then we can obtain the conclusions in a similar manner to can use Γλ (Spµ ) to define Pλ (Spµ ) and PΛN (Sp,N Lemma 5.1. µ Lemma 5.7. Let Spµ satisfy Assumptions 1 and 2a, λ ∈ σ(Spµ ) and λN,1 , . . . , λN,k ∈ σ(Sp,N ) satisfy Assumption 3b. Furthermore, for an eigenvector φ corresponding to λ with kφk2 = 1, assume that there exists a positive integer N0 such that for any N > N0

2 µ k(Spµ − Sp,N )φk2 < 1. rλ

(5.23)

′ ′ Then for N > N0 there exist a choice of an index iN ∈ {1, . . . , k} and an integer kN with 2 ≤ kN ≤ k such that

where

′ o1/kN n µ , )φk2 |λ − λN,iN | ≤ Kλ,N k(Spµ − Sp,N

(5.24)

    o n rλ 2k µ max 1, kSp,N kk2 Kλ,N = (4|λ| + rλ ) k 2 max 1, |λ| + 2        rλ k rλ k 2 µ + max 1, . k2 + |λ| + + max 1, kSp,N rλ 2 2

µ ′ In particular, if there exists N1 ≥ N0 such that Kλ,N k(Spµ − Sp,N )φk2 < 1 for N > N1 , the integer kN in (5.24) can be replaced by k.

Proof. If there exist i, j ∈ {1, . . . , k} with i 6= j such that λN,i = λN,j , we reserve only one of them, and consequently we obtain the set of indices ΩN such that {λN,i | i ∈ ΩN } is identical to {λN,1 , . . . , λN,k } as a set and the elements of it are mutually distinct. Then we renumber the elements of {λN,i | i ∈ ΩN } and let it be ′ ′ ′ }, where k denoted by {λN,1 , . . . , λN,kN N = #ΩN . Without loss of generality we can assume that 2 ≤ kN . µ ′ It follows from Lemma 5.6 that PΛN (Sp,N )φ is a nonzero linear combination of eigenvectors φN,1 , . . . , φN,kN ′ ′ ∈ C corresponding to λN,1 , . . . , λN,kN for N > N0 . Then there exists a sequence of coefficients aN,1 , . . . , aN,kN ′ ) 6= (0, . . . , 0) such that with (aN,1 , . . . , aN,kN µ ′ φN,k′ )φ = aN,1 φN,1 + · · · + aN,kN PΛN (Sp,N N

(5.25)

for N > N0 . Here we define iN (1) as a minimizer of |λ − λN,i | for i with aN,i 6= 0 in (5.25), i.e. |λ − λN,iN (1) | = min{|λ − λN,i | | i : aN,i 6= 0 in (5.25)}.

(5.26)

′ ′ Moreover, we define {iN (2), . . . , iN (kN )} as a permutation of {1, . . . , kN } \ {iN (1)}. Consequently, we obtain ′ ) } for N > N0 . For simplicity, let λN,j , φN,j and aN,j denote λN,i (j) , the ordered set {λN,iN (1) , . . . , λN,iN (kN N ′ φN,iN (j) and aN,iN (j) for j = 1, . . . , kN , respectively. In the following we consider (5.25) under this renumbering. µ µ Here we begin the estimate of the convergence rate of λN,1 . Applying the operator (Sp,N − λN,2 I) · · · (Sp,N − ′ I) to the both side of (5.25), we have λN,kN µ µ ) φ = aN,1 gN (λN,1 ) φN,1 , ) PΛN (Sp,N gN (Sp,N

where gN is the polynomial defined as ′ ). gN (x) = (x − λN,2 ) · · · (x − λN,kN

13

(5.27)

′ Because aN,1 6= 0 and λN,1 6= λN,i (i = 2, . . . , kN ), we can define a normalized eigenvector φN corresponding to λN,1 as µ µ −1 ) φ, φN = CN gN (Sp,N ) PΛN (Sp,N

(5.28)

µ µ ) φk2 , CN = sign(gN (λ)) k gN (Sp,N ) PΛN (Sp,N

(5.29)

where

Using φN in (5.28), in the same manner as (5.16), we can derive an estimate: µ |λ − λN,1 | ≤ (|λ| + |λN,1 |)kφ − φN k2 + k(Spµ − Sp,N )φk2 .

(5.30)

Multiplying the both side of (5.30) by |gN (λ)|, we have ′

kN Y

i=1

µ |λ − λN,i | ≤ (|λ| + |λN,1 |) |gN (λ)| kφ − φN k2 + |gN (λ)| k(Spµ − Sp,N )φk2 .

(5.31)

Noting Assumption 3b and (5.26), we can deduce from (5.31) that µ )φk2 . |λ − λN,1 |kN ≤ (|λ| + |λN,1 |) |gN (λ)| kφ − φN k2 + (rλ /2)kN −1 k(Spµ − Sp,N ′



(5.32)

Then what remains is to estimate |gN (λ)| kφ − φN k2 . By the triangle inequality, we have |gN (λ)| kφ − φN k2 ≤ kgN (λ)φ − CN φN k2 + k(CN − gN (λ)) φN k2 , and kgN (λ)φ − CN φN k2 µ µ ) φk2 = kgN (Spµ ) Pλ (Spµ ) φ − gN (Sp,N ) PΛN (Sp,N

µ µ µ )} φk2 . ){Pλ (Spµ ) − PΛN (Sp,N ≤ k{gN (Spµ ) − gN (Sp,N )} φk2 + kgN (Sp,N

(5.33)

Therefore setting E1,N = k (CN − gN (λ)) φN k2 ,

µ )} φk2 , E2,N = k{gN (Spµ ) − gN (Sp,N

µ µ )} φk2 ){Pλ (Spµ ) − PΛN (Sp,N E3,N = kgN (Sp,N

we have |gN (λ)| kφ − φN k2 ≤ E1,N + E2,N + E3,N .

(5.34)

We can estimate E1,N , E2,N and E3,N as follows. Using (5.28) and (5.29), we have E1,N = |CN − gN (λ)| = |CN − sign(gN (λ))kgN (λ)φk2 | = |kCN φN k2 − kgN (λ)φk2 | ≤ kCN φN − gN (λ)φk2 ≤ E2,N + E3,N ,

(5.35)

where the last inequality is due to (5.33). Noting that φ is an eigenvector of Spµ corresponding to λ, we have µ µ {gN (Spµ ) − gN (Sp,N )}φ = {gN (λ) − gN (Sp,N )}φ ′ kN −1

=

X

  ′ ′ µ α(i, ΛN ) λkN −1−i I − (Sp,N )kN −1−i φ

X

µ µ α(i, ΛN ) β(i, λ, Sp,N ) (λI − Sp,N )φ

X

µ µ α(i, ΛN ) β(i, λ, Sp,N ) (Spµ − Sp,N ) φ,

i=0

′ kN −2

=

i=0

′ kN −2

=

i=0

14

(5.36)

where α(i, ΛN ) = (−1)i fore noting

P

′ 2≤j1 N

p l+N j  X X X 2π(l − m) 1 µ −i (fˆj )m (φˆN )l−m √ k(Spµ − Sp,N )φN k2 ≤ L L |l|>N j=0 m=l−N p X  p X l+N X 1 2πN ≤√ (fˆj )m (φˆN )l−m . L L j=0

(5.46)

|l|>N m=l−N

Considering the case |l| ≥ 2N , we have X

l+N X

|l|≥2N m=l−N

X ˆ (fj )m (φˆN )l−m ≤

l+N X

|l|≥2N m=l−N

X ˆ (fˆj )m . (fj )m ≤ (2N + 1)

(5.47)

|m|≥N

Combining (5.45), (5.46) and (5.47), we obtain the error bound (3.13).

6

Concluding Remarks

We have considered the convergence rates and the explicit error bounds of Hill’s method, which is a numerical method for computing the spectra of self-adjoint differential operators Sp in (1.1) with periodic coefficient functions f˜j . On the assumption (Assumptions 3a and 3b) that the computed eigenvalue λN is close to the exact one λ, it is shown in Theorems 3.4 and 3.5 that the convergence rate of the computed eigenvalue is all order polynomial in the case of f˜j being C ∞ as shown in (3.8) and (3.10), and exponential in the case of f˜j being analytic as shown in (3.9) and (3.11). In addition, even if the exact eigenvalue λ is unknown, it is shown 17

in Theorems 3.4, 3.5 and 3.6 that, if the condition (3.12) is satisfied, we can obtain the convergence rate using (3.8)–(3.11) and the explicit error bound using (3.13). There are some cases in which the condition (3.12) can µ be checked using Gershgorin’s theorem. These theorems are proved using the Dunford integrals Pλ (Sp,N ) in µ (5.1) and Pλ (Sp ) in the proof of Theorem 3.6, which project an eigenvector to the corresponding eigenspace. This integral is suitable for proving these theorems, because the boundary of the neighborhood of an eigenvalue can be directly used as the contour of the integral. Numerical examples using Hill’s operator (4.1) support these theoretical results. As described in Remark 5.1, self-adjointness of the operator Spµ enables us to estimate the norm of the resolvent as shown in (5.8), and to project an eigenvector to the corresponding eigenspace using the Dunford integral. For the case of non-self-adjoint operators, we have developed some other approaches, which will be reported somewhere else soon.

Acknowledgments This work is partially supported by Grant-in-Aid for Scientific Research (S) 20224001.

References [1] Atkinson. K., The numerical solution of the eigenvalue problem for compact integral operators, Trans. Amer. Math. Soc. 129, 458–465 (1967) [2] Atkinson, K., Convergence rates for approximate eigenvalues of compact integral operators, SIAM J. Num. Anal. 12, 213–222 (1975) [3] Atkinson, K., Han, W., Theoretical numerical analysis: A functional analysis framework, second edition, Springer, New York (2005) [4] Curtis, C. W., Deconinck, B., On the convergence of Hill’s method, Math. Comp. 79, 169–187 (2010) [5] Dunford, N, Schwartz, J. T., Linear operators Part II Spectral theory, Self adjoint operators in Hilbert space, Wiley & Sons, New Jersey (1963) [6] Dunford, N, Schwartz, J. T., Linear operators Part III Spectral operators, Wiley & Sons, New Jersey (1971) [7] Deconinck, B., Kutz, J. N., Computing spectra of linear operators using Hill’s method, J. Computational Physics 219, 296–321 (2006) [8] Hill, G. W., On the part of the motion of the lunar perigee which is a function of the mean motions of the sun and moon, Acta Math. 8, 1–36 (1886) [9] Hille, E., Ordinary differential equations in the complex domain, Dover, New York (1997) [10] Johnson, M. A., Zumbrun, K., Convergence of Hill’s method for nonselfadjoint operators, SIAM J. Numer. Anal. 50, 64–78 (2012) [11] Krasnosel’skii, M. A., Vainikko, G. M., Zabreiko, P. P., Rutitskii, Ya. B., Stetsenko, V. Ya., Approximate solution of operator equations, Wolters-Noordhoff, Groningen (1972) [12] Reed, M., Simon, B., Methods of modern mathematical physics. I. Functional analysis, revised and enlarged edidition, Academic Press, New York (1980) [13] Reed, M., Simon, B., Methods of modern mathematical physics. IV. Analysis of operators, Academic Press, New York (1978) [14] Vainikko, G. M., A perturbed Galerkin method and the general theory of approximate methods for nonlinear equations, Zh. Vychisl. Mat. Mat. Fiz. 7, 723–751 (1967) (in Russian), U. S. S. R. Comput. Math. and Math. Phys. 7, 18–32 (1967) (in English). [15] Varga, R. S., Gershgorin and his circles, Springer, Berlin Heidelberg (2004) 18

A

Proofs of Lemmas 5.3, 5.4, and 5.5

First, we prove Lemma 5.3. Proof of Lemma 5.3. We set fp ≡ 1 for simplicity. Let em (m = 0, ±1, ±2, . . .) be the Fourier bases defined as   1 2πm em (x) = √ exp −i x , L L and let (fˆj )m (j = 0, 1, . . . , p, m = 0, ±1, ±2, . . .) denote the Fourier coefficients of fj (j = 0, 1, . . . , p), i.e. (fˆj )m = (fj , em )2 . Then it is a standard fact that under Assumption 2a, for j = 0, 1, . . . , p, m = 0, ±1, ±2, . . . and any positive integer q, there exists a positive real constant Cj,q depending only on fj and q such that |(fˆj )m | ≤ Cj,q (1 + |m|)−q .

(A.1)

We estimate the norm of (PˆN Spµ ψ)(x) =

p N X X

n=−N j=0

(

) j  2π(n − m) ψˆn−m en (x) (fˆj )m −i L m=−∞ ∞ X

(A.2)

for N ≥ 1 and a nonzero vector ψ ∈ L2 ([0, L])per . Setting  j 2π(n − m) ˆ cj (n, m) = (fj )m −i L and q = p + 1 in (A.1), for j = 0, 1, . . . , p, m = 0, ±1, ±2, . . . and n with |n| ≤ N , we have  j j  2π |n| + |m| N −j |cj (n, m)| = |(fˆj )m | L N  j  j 2π |m| −p−1 ≤ 1+ Cj,p+1 (1 + |m|) L N  j 2π Cj,p+1 (1 + |m|)j−p−1 , ≤ L and therefore |cj (n, m)| ≤ Combining (A.2) and (A.3) we have

p X N X

µ ˆ kPN Sp ψk2 ≤

j=0 n=−N  p  X N X =  ≤



p X



j=0

j=0 p X j=0

2π L

j

Cj,p+1 j N . 1 + |m|



en

m=−∞ 2  2 1/2 ∞ X  cj (n, m)ψˆn−m  ∞ X

cj (n, m)ψˆn−m

(A.3)

!

n=−N m=−∞

j=0

p X



(

(

N X

n=−N

∞ X

m=−∞

|cj (n, m)|

2

 2j N X 2π 2 Cj,p+1 N 2j L

n=−N

C˜j,p+1 (2N + 1)1/2 N j kψk2 ,

19

!

∞ X

m=−∞ ∞ X

|ψˆn−m |

2

!)1/2

1 (1 + |m|)2 m=−∞

!

kψk22

)1/2 (A.4)

where C˜j,p+1 (j = 0, 1, . . . , p) are bounded constants defined as C˜j,p+1 =



2π L

j

∞ X

1 (1 + |m|)2 m=−∞

Cj,p+1

!1/2

.

Hence it follows from (A.4) that kPˆN Spµ k2 ≤

p X

C˜j,p+1 (2N + 1)1/2 N j .

(A.5)

j=0

Thus we have the conclusion. Next, we prove Lemma 5.4 and 5.5 with the aid of the following two lemmas. Since these lemmas are fundamental in the theory of differential equations, we only give the outlines of their proofs in Appendix B. Lemma A.1. Let Spµ satisfy Assumption 2a and φ ∈ H p ([0, L])per be an eigenvector of Spµ corresponding to λ ∈ σ(Spµ ). Then φ is C ∞ . Lemma A.2. Let Spµ satisfy Assumption 2b for d > 0 and φ ∈ H p ([0, L])per be an eigenvector of Spµ corresponding to λ ∈ σ(Spµ ). Then φ can be uniquely extended to an analytic function on Dd with φ(z + L) = φ(z) for any z ∈ C, where Dd is defined as (3.1) and L is the period of the coefficient functions of Spµ .

Proof of Lemma 5.4. Since φ is C ∞ by Lemma A.1, in a similar manner to (A.1), we can use the fundamental fact for the Fourier coefficients that there exists a positive constant Cφ,q depending only on φ and q such that |φˆn | ≤ Cφ,q (1 + |n|)−q−1

for any integer n. Then we have k(I − PˆN )φk22 = ≤

X

|n|>N

|φˆn |2 ≤

2 Cφ,q

(1 + N )2q

X

|n|>N

X

|n|>N

2 Cφ,q (1 + |n|)−2q−2

∞ 2 X Cφ,q 1 1 ≤ . (1 + |n|)2 (1 + N )2q n=−∞ (1 + |n|)2

Thus we obtain the conclusion. Proof of Lemma 5.5. Since φ can be regarded as an analytic function on Dd with φ(z+L) = φ(z) by Lemma A.2, there exists a positive constant Cφ,d,ε depending only on φ, d and ε such that   2πdε |n| (A.6) |φˆn | ≤ Cφ,d,ε exp − L for any integer n, where dε = d − ε. In fact, we can show this estimate by the Paley-Wiener type argument. Noting the periodicity φ(z + L) = φ(z), by Cauchy’s integral theorem we have   Z L 1 2πn φˆn = √ φ(x) exp −i x dx L L 0   Z L 1 2πn (x − sign(n) dε i) dx = √ φ (x − sign(n) dε i) exp −i L L 0     Z L 2πdε 1 2πn |n| √ x dx, = exp − φ (x − sign(n) dε i) exp −i L L L 0 where sign(n) is the sign of n. This expression implies (A.6). Therefore we have k(I − PˆN )φk22 =

X

|n|>N

|φˆn |2 ≤

X

|n|>N

2 Cφ,d,ε exp(−2 dε,L |n|) =

where dε,L = 2πdε /L. Thus we obtain the conclusion. 20

2 2 Cφ,d,ε exp(−2 dε,L (N + 1)) , 1 − exp(−2 dε,L )

B

Proofs of Lemmas A.1 and A.2

Proof of Lemma A.1. Since φ is an eigenvector of Spµ corresponding to λ, we have p−1 X

φ(p) (x) = −

fj (x)φ(j) (x) + λφ(x).

j=0

Then it follows from φ ∈ H p ([0, L])per that φ(p) ∈ H 1 ([0, L])per. Hence φ ∈ H p+1 ([0, L])per. Iterating a similar argument, we have φ ∈ H q ([0, L])per for any positive integer q. Then the conclusion follows from the fundamental embedding relation: H q ([0, L])per ֒→ C q−1 ([0, L])per for any positive integer q, where C q−1 ([0, L])per is the set of periodic functions on [0, L] which are q − 1 times continuously differentiable (see e.g. [3, Proposition 7.5.4]). Proof of Lemma A.2. We choose an arbitrary ε with 0 < ε < d and set dε = d − ε. Then f0 , f1 , . . . , fp−1 are analytic on Ddε and continuous on Ddε . Hence there exists a positive constant Mε such that |fj (z)| ≤ Mε for all j = 0, 1, . . . , p − 1 and z ∈ Ddε . By Lemma A.1, we can assume the existence of an eigenvector φ ∈ C ∞ ([0, L])per corresponding to λ ∈ σ(Spµ ). Then for the fixed λ and each x0 ∈ R, we consider the equation Spµ ψ = λψ as an initial value problem with the condition ψ (n) (x0 ) = φ(n) (x0 ) (n = 0, . . . , p − 1). By the matrix F (z) defined as   −fp−1 (z) −fp−2 (z) · · · −f1 (z) −(f0 (z) − λ)   1 0 ··· 0     ..   0 1 ··· . F (z) =     .. . . . . . .   . . . . 0 ··· ··· 1 0 and the vector w0 = (φ(p−1) (x0 ), φ(p−2) (x0 ), . . . , φ(x0 ))T , the initial value problem above is expressed as w′ (z) = F (z)w(z),

w(x0 ) = w0 ,

(B.1)

where w(z) corresponds to (ψ (p−1) (z), ψ (p−2) (z), . . . , ψ(z))T . The equation (B.1) is equivalently expressed as Z z F (s)w(s)ds. (B.2) w(z) = w0 + x0

By the fundamental method of successive approximations [9, Theorem 2.3.1]1 , it is shown that the equation (B.1) has a unique analytic solution w on Ddε for each x0 ∈ R. Then what remains is to prove the consistency of the solutions for different x0 ’s and the periodicity of the solution. First, note that for any x0 the solution w is identical to v = (φ(p−1) , φ(p−2) , . . . , φ)T on R. This follows from the condition

Z x

Z x



kw(x) − v(x)k = F (s)(w(s) − v(s))ds ≤ sup kF (z)k kw(s) − v(s)k |ds|, (B.3) z∈Ddε

x0

x0

(1)

(2)

where k · k is a norm of Rp . Then if w1 and w2 are the solutions for x0 and x0 , respectively, we have Z z Z z (1) (2) F (s)w2 (s)ds F (s)w1 (s)ds − w1 (z) − w2 (z) = w1 (x0 ) − w2 (x0 ) + (2)

(1)

x0

x0

(1)

(2)

= w1 (x0 ) − w2 (x0 ) − (1)

(2)

= v(x0 ) − v(x0 ) − =

Z

z (1)

x0

Z

Z

(1) x0 (2)

F (s)w2 (s)ds +

x0

(1)

x0

(2)

F (s)v(s)ds +

x0

Z

z (1)

x0

Z

F (s)(w1 (s) − w2 (s))ds

z (1)

x0

F (s)(w1 (s) − w2 (s))ds

F (s)(w1 (s) − w2 (s))ds

1

In [9, Theorem 2.3.1], the general form w ′ = F (z, w) is considered. Here we consider the special case that F is linear with respect to w.

21

for any z ∈ Ddε , which implies w1 ≡ w2 on Ddε by the same criterion as (B.3). Next, the expressions Z

z+L

w(z + L) = v(x0 + L) + F (s)w(s)ds = v(x0 ) + x0 +L Z z F (t)w(t + L)dt, = v(x0 ) + x Z z0 F (t)w(t)dt. w(z) = v(x0 ) +

Z

z

F (t + L)w(t + L)dt

x0

x0

guarantees the periodicity of w also by the same criterion as (B.3). Finally, since the all arguments above hold true for any ε with 0 < ε < d, we obtain the conclusion.

22

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