The Rayleigh-Ritz method, refinement and Arnoldi process for periodic matrix pairs

(1)

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Journal of Computational and Applied

Mathematics

journal homepage:www.elsevier.com/locate/cam

The Rayleigh–Ritz method, refinement and Arnoldi process for periodic

matrix pairs

Eric King-Wah Chu

a,∗

, Hung-Yuan Fan

b

, Zhongxiao Jia

c

, Tiexiang Li

d

, Wen-Wei Lin

e

a_{School of Mathematical Sciences, Building 28, Monash University, VIC 3800, Australia} b_{Department of Mathematics, National Taiwan Normal University, Taipei 116, Taiwan} c_{Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China} d_{Department of Mathematics, Southeast University, Nanjing 211189, China}

e_{CMMSC and NCTS, Department of Applied Mathematics, National Chiao Tung University, Hsinchu 300, Taiwan}

a r t i c l e i n f o Article history: Received 2 December 2009 Keywords: Arnoldi process Periodic eigenvalues Periodic matrix pairs Rayleigh–Ritz method Refinement Ritz values

a b s t r a c t

We extend the Rayleigh–Ritz method to the eigen-problem of periodic matrix pairs. Assuming that the deviations of the desired periodic eigenvectors from the corresponding periodic subspaces tend to zero, we show that there exist periodic Ritz values that converge to the desired periodic eigenvalues unconditionally, yet the periodic Ritz vectors may fail to converge. To overcome this potential problem, we minimize residuals formed with periodic Ritz values to produce the refined periodic Ritz vectors, which converge under the same assumption. These results generalize the corresponding well-known ones for Rayleigh–Ritz approximations and their refinement for non-periodic eigen-problems. In addition, we consider a periodic Arnoldi process which is particularly efficient when coupled with the Rayleigh–Ritz method with refinement. The numerical results illustrate that the refinement procedure produces excellent approximations to the original periodic eigenvectors.

Let Ej

,

Aj

∈

Cn×n

(

j

=

1

, . . . ,

p

)

, where Ej+p

=

Ejand Aj+p

=

Ajfor all j. We denote the periodic matrix pairs of

periodicity p by

{

(

Aj

,

Ej

)}

p

j=1. In this paper, the indices j for all periodic coefficient matrices are chosen in

{

1

, . . . ,

p

}

modulo

p. The equations

β

jAjxj−1

=

α

jEjxj

(

j

=

1

,

2

, . . . ,

p

)

(1)

with x0

=

xpdefine the nonzero periodic right eigenvectors

{

xj

}

pj=1for complex ordered pairs

{

(α

j

, β

j

)}

pj=1. Similarly, the

equations

β

j−1yHjAj

=

α

jyHj−1Ej−1

(

j

=

1

,

2

, . . . ,

p

)

(2)

with y0

=

ypdefine the nonzero periodic left eigenvectors

{

yj

}

pj=1. The ordered pairs

(π

α

, π

β

) ≡ ∏

p

j=1

α

j

, ∏

pj=1

β

j



then constitute the spectrum, with the traditional eigenvalues being the quotients

π

_α

/π

_β. Because of the possibility of infinite eigenvalues, we shall deal with spectra in their ordered pair representation, with equality interpreted in the sense of the corresponding equivalent relationship for quotients. Using the notation col

[

xj

]

pj=1

≡ [

x

⊤

1

, . . . ,

x

⊤

p

]

⊤_and

∗_{Corresponding author. Tel.: +61 412 596430; fax: +61 3 99054403.}

E-mail addresses:eric.chu@sci.monash.edu.au,eric.chu@monash.edu(E.K.-W. Chu),hyfan@ntnu.edu.tw(H.-Y. Fan),jiazx@tsinghua.edu.cn(Z. Jia), txli@seu.edu.cn(T. Li),wwlin@am.nctu.edu.tw(W.-W. Lin).

(2)

C



α

1

, . . . , α

p

β

1

, . . . , β

p



≡







α

1E1

−

β

1A1

−

β

₂A2

α

2E2

...

−

β

_pAp

α

pEp







,

(3)

the eigen-equations(1)and(2)can also be written as the multivariate eigen-problems, respectively,

C



α

1

, . . . , α

p

β

1

, . . . , β

p



col

[

xj

]

pj=1

=

0 (4) and



col

[

yj

]

pj=1



H C



α

2

, . . . , α

p

;

α

1

β

p

;

β

1

. . . , β

p−1



=

0⊤

.

(5)

In this paper, we consider only regular periodic matrix pairs for which

det C



α

1

, . . . , α

p

β

1

, . . . , β

p



=

n

−

k=0 ck

π

_αk

π

_βn−k

̸≡

0

,

(6)

and consequently all eigenvalues

(π

_α

, π

_β

) ̸≡ (

0

,

0

)

. For regular periodic matrix pairs, at least one of the coefficients ck

̸=

0

and there are exactly n eigenvalues for

{

(

Aj

,

Ej

)}

pj=1, counting multiplicities. The spectrum, or the set of all eigenvalue pairs,

of

{

(

Aj

,

Ej

)}

pj=1is denoted by

λ({(

Aj

,

Ej

)}

pj=1

)

.

For the periodic matrix pairs

{

(

Aj

,

Ej

)}

pj=1, we have the periodic Schur decomposition of

{

(

Aj

,

Ej

)}

pj=1[1–3].

Theorem 1.1 (Periodic Schur Decomposition). Let

{

(

Aj

,

Ej

)}

jp=1be regular matrix pairs. There exist unitary matrices Qj

,

Zj

(

j

=

1

,

2

, . . . ,

p

)

such that

Q_jHAjZj−1

= ˆ

Aj

,

QjHEjZj

= ˆ

Ej

(

j

=

1

,

2

, . . . ,

p

)

are all upper triangular, with Z0

=

Zp. Moreover, the diagonal parts

{[

diag

(α

j1

, . . . , α

jn

),

diag

(β

j1

, . . . , β

jn

)]}

pj=1

of

{

(ˆ

Aj

, ˆ

Ej

)}

pj=1determine all the eigenvalues

∏

p

j=1

α

jk

, ∏

pj=1

β

jk



n

k=1of

{

(

Aj

,

Ej

)}

p

j=1, which can be arranged in any order.

We can also generalize the concept of deflating subspaces as follows [4,3].

Definition. LetXj

,

Yj

(

j

=

1

,

2

, . . . ,

p

)

be subspaces in Cnof equal dimension. The pairs

{

(

Xj

,

Yj

)}

p

j=1are called the periodic

deflating subspaces of

{

(

Aj

,

Ej

)}

pj=1if

AjXj−1

⊂

Yj

,

EjXj

⊂

Yj

(

j

=

1

,

2

, . . . ,

p

)

withX0

=

Xp. Furthermore, the subspaces

{

Xj

}

pj=1are called the periodic invariant subspaces of

{

(

Aj

,

Ej

)}

pj=1.

We list some further results and definitions from [3]. (i) Theorem 1.1implies that

λ({(

Aj

,

Ej

)}

pj=1

) = λ({(

A

⊤ j

,

E ⊤ j

)}

p j=1

)

.

(ii) An eigenvalue is said to be simple if it appears in a linear factor of the characteristic polynomial. (iii) Let Z₁(j)

,

Q₁(j)

∈

_Cn×k_satisfy

₍

_Z(j)

1

)

HZ (j) 1

=

(

Q (j) 1

)

HQ (j)

1

=

Ik, and letXj

=

span

(

Z1(j)

),

Yj

=

span

(

Q1(j)

)

for all j. It can be

verified [3] that

{

(

Xj

,

Yj

)}

p

j=1are periodic deflating subspaces of the regular matrix pairs

{

(

Aj

,

Ej

)}

p

j=1if and only if there

exist unitary matrices Zj

= [

Z1(j)

,

Z

(j) 2

]

,

Qj

= [

Q1(j)

,

Q (j) 2

] ∈

Cn ×n_{such that} Q_jHAjZj−1

=

[

A(₁₁j) A(₁₂j) 0 A(₂₂j)

]

,

Q_jHEjZj

=

[

E₁₁(j) E₁₂(j) 0 E₂₂(j)

]

,

(7)

where A₁₁(j)

,

E₁₁(j)

∈

_Ck×k, and both

{

(

A(₁₁j)

,

E₁₁(j)

)}

p_j₌₁and

{

(

A₂₂(j)

,

E₂₂(j)

)}

p_j₌₁are regular for all j. Furthermore, if the intersection of the spectra of the two sub-matrix pairs is empty, the periodic deflation subspaces

{

(

Xj

,

Yj

)}

pj=1are called simple

periodic deflating subspaces, and

{

Xj

}

pj=1simple periodic invariant subspaces.

From the periodic Schur decomposition inTheorem 1.1, we also obtain the periodic Kronecker canonical form [5–7] of

{

(

Aj

,

Ej

)}

p j=1.

Theorem 1.2 (Periodic Kronecker Canonical Form). Suppose that the periodic matrix pairs

{

(

Aj

,

Ej

)}

pj=1are regular. Then there

exist nonsingular matrices Xjand Yj

(

j

=

1

,

2

, . . . ,

p

)

such that

Y_jHEjXj

=

[

I 0 0 E0_j

]

,

Y_jHAjXj−1

=

[

Af_j 0 0 I

]

,

(8)

(3)

where Af_jand E0_j are all upper triangular,

J(j)

≡

Af_j₊_p₋₁Af_j₊_p₋₂

. . .

Af_j

(

j

=

1

,

2

, . . . ,

p

)

(9)

are Jordan canonical forms corresponding to the finite eigenvalues of

{

(

Aj

,

Ej

)}

p j=1, and

N(j)

≡

E0_jE_j0₊₁

. . .

E_j0₊_p₋₁

(

j

=

1

,

2

, . . . ,

p

)

(10)

are nilpotent Jordan canonical forms corresponding to the infinite eigenvalues.

Remarks.

(i) From [4], the matrices Af_j and E_j0in(8)can be further reduced to block-upper triangular. Each individual block in Af_j or

E0

j relates to the corresponding Jordan block of a multiple eigenvalue of

{

(

Aj

,

Ej

)}

pj=1.

(ii) For different values of j, the Jordan canonical forms J(j)_{and N}(j)_in₍₉₎_and₍₁₀₎_{may have different structures. Thus, an}

eigenvalue with a certain algebraic multiplicity may have different geometric multiplicities dependent on j.

The eigen-problem of the periodic matrix pairs

{

(

Aj

,

Ej

)}

pj=1reflects the behavior of the linear discrete-time periodic

systems

Ejxj+1

=

Ajxj

(

j

=

1

,

2

, . . . ,

p

)

(11)

with respect to solvability and stability [8–12]. There has been much recent interest in periodic systems. It arises in a large variety of applications, including queueing network [13,14], analysis of bifurcations and computation of multipliers [15,16], multirate sampled-data systems, chemical processes, periodic time-varying filters and networks and seasonal phenomena; see [8,9] and the references therein for further information. Note that the periodic matrix eigen-problem is mathematically equivalent to the product matrix eigen-problem and the cyclic matrix eigen-problem [17,18]. Recently, some reliable numerical algorithms have been designed for the computation of the periodic stable invariant subspaces [1,19]. Perturbation analysis of eigenvalues and periodic deflating subspaces of periodic matrix pairs have been extensively studied in [20,5,4,21]. For the large product matrix eigen-problems and the periodic matrix eigen-problems with Ej

=

I

(

j

=

1

,

2

, . . . ,

p

)

,

Kressner [17] presents a periodic Arnoldi process that generates orthonormal bases of certain periodic Krylov subspaces. Based on it, he proposes a periodic Arnoldi method for the product matrix eigen-problem and develops a periodic Arnoldi algorithm and a periodic Krylov–Schur algorithm.

The Rayleigh–Ritz method is widely used for the computation of approximations to an eigen-spaceXof an ordinary large matrix eigen-problem Ax

=

λ

x, from an approximating subspace_X

˜

_{. The harmonic Rayleigh–Ritz method is an alternative}

for solving the interior eigen-problem (see, e.g., [22, Chapter 4]). Furthermore, when one is concerned with eigenvalues and eigenvectors, one can compute certain refined (harmonic) Ritz vectors whose convergence is guaranteed [23–27]; see also [22].

The purpose of this paper is to generalize the concept of the Rayleigh–Ritz approximation for the periodic matrix pairs, leading to the periodic Rayleigh–Ritz approximation. We study the convergence of the periodic Ritz values and the corresponding periodic Ritz vectors and extend some of the results in [26,27,22] to the periodic Rayleigh–Ritz approximation. Similar to the ordinary eigen-problem case (when p

=

1) in [26,27,22], periodic Ritz vectors may fail to converge even if the corresponding periodic projection subspaces contain sufficiently accurate approximations to the desired periodic eigenvectors. It is thus necessary to refine the periodic Ritz vectors, as described in Section5. We shall prove the convergence of the refined periodic Ritz vectors and propose an algorithm for their computation. All the convergence results are nontrivial generalizations of some of the known ones for Rayleigh–Ritz approximations and their refinement for the ordinary eigenvalue problem in [26,27]; see also [22]. As an important special case when the periodic Arnoldi process [17] is employed to generate the periodic orthonormal bases of the periodic Krylov subspaces, the refinement can be realized much more efficiently.

In the rest of the paper,

‖ · ‖

denotes both the Euclidean vector norm and the subordinate spectral matrix norm, unless otherwise stated. The conjugate of a complex number

α

is denoted by

α

¯

and the unit imaginary number is denoted by

ι =

√

−

1.

The paper is organized as follows. We first consider the Rayleigh–Ritz procedure for the periodic eigen-problem(1)in Section2. The convergence of the Ritz value pairs and their corresponding periodic Ritz vectors will be treated in Sections3

and4, respectively. In Section5, we shall establish the convergence of the refined periodic Ritz vectors and propose a numerical method to compute them. In Section6, we consider the special case when the periodic Krylov subspaces are generated by the periodic Arnoldi process. In Section7, some numerical examples are given to illustrate the accuracy of the refined periodic Ritz vectors and the sharpness of their convergence bounds. The paper concludes with a brief summary in Section8.

2. The periodic Rayleigh–Ritz approximation

As is known [17,18], the eigen-problem of the periodic matrices

{

Aj

}

pj=1 is very closely related to the product matrix

(4)

algorithm and its Krylov–Schur version have been developed for solving eigenvalue problems associated with products of large and sparse matrices [17]. One of the central problems in this method is how to extract approximations to the desired eigenvalues and periodic eigenvectors from the given periodic subspaces

{ ˜

Xj

}

pj=1. The algorithm is based on a variant of the

Rayleigh–Ritz procedure applied to the eigen-problems(1)for the periodic matrix pairs

{

(

Aj

,

Ej

)}

pj=1. It performs restarts

and deflations via reordered periodic Schur decompositions and generates an approximate sequence of periodic subspaces

{ ˜

Xj

}

p

j=1containing increasingly accurate approximations to the desired periodic eigenvectors.

For the periodic subspaces

{ ˜

Xj

}

p

j=1, suppose that they are spanned by the periodic orthonormal bases

{

Uj

}

p j=1 with

dim

( ˜

Xj

) =

k

(

j

=

1

,

2

, . . . ,

p

)

. Compute the (thin or compact) QR-decompositions

EjUj

=

VjNj

(

j

=

1

,

2

, . . . ,

p

)

(12)

where VH

j Vj

=

Ikand Njis upper triangular. Let, for all j,

V_jHAjUj−1

=

Mj

.

(13)

Then(12)and(13)define the periodic Rayleigh–Ritz pairs

{

(

Mj

,

Nj

)}

p

j=1with respect to

{

Uj

}

p

j=1. The following theorem shows

that for any periodic orthonormal bases

{

Uj

}

pj=1, the periodic Rayleigh–Ritz pairs yield minimal residuals.

Theorem 2.1. Let

{

(

Mj

,

Nj

)}

pj=1be the periodic Rayleigh–Ritz pairs with respect to the periodic bases

{

Uj

}

pj=1. Suppose that Njis

nonsingular for all j. Then the residuals Rj

≡

AjUj−1

−

EjUj

(

N

−1

j Mj

) (

j

=

1

,

2

, . . . ,

p

)

(14)

are minimal in the matrix 2-norm:

min

Cj∈Ck×k

‖

AjUj−1

−

EjUjCj

‖ = ‖

Rj

‖

.

(15)

Proof. Let Pj

≡

Nj−1Mj

(

j

=

1

,

2

, . . . ,

p

)

. For any Cj

∈

Ck×k, denote∆j

≡

Pj

−

Cj. From

(

I

−

VjVjH

)

EjUj

=

0 and

C_jHU_jHE_jHEjUjPj

=

CjHU H j E H jAjUj−1, we have

‖

AjUj−1

−

EjUjCj

‖

2

=

ρ(

UjH−1A H jAjUj−1

−

CjHU H j E H jAjUj−1

−

UjH−1A H jEjUjCj

+

CjHU H j E H j EjUjCj

)

=

ρ(

U_jH−1AHjAjUj−1

+

∆HjUjHEjHEjUj∆j

−

PjHUjHEjHEjUjPj

)

=

ρ[(

U_jH₋₁A_jH

−

P_jHU_jHE_jH

)(

AjUj−1

−

EjUjPj

) +

∆HjU H jE H j EjUj∆j

]

≥

ρ(

RH_jRj

) = ‖

Rj

‖

2

,

where

ρ(·)

denotes the spectral radius.

Remark. The residuals Rj’s in(14)possess the following geometric meaning

Rj

=

AjUj−1

−

EjUj

(

VjHEjUj

)

−1

(

VjHAjUj−1

)

= [

I

−

EjUj

(

VjHEjUj

)

−1VjH

]

AjUj−1

=

(

I

−

PEjUj

)

AjUj−1

,

where PEjUjis the orthogonal projector onto the subspace span

(

EjUj

)

. Furthermore, if Njis nonsingular, it is easily verified

that PEjUj

=

VjV

H

j . So

‖

Rj

‖

is the distance of AjUj−1from span

(

Vj

)

and should be minimal over all projections of AjUj−1onto

span

(

Vj

) = (

EjUj

)

.

We now describe the periodic Rayleigh–Ritz (pRR) procedure with respect to

{

Uj

}

pj=1 to approximate an eigen-pair

((π

α

, π

β

); {

xj

}

pj=1

)

of the periodic matrix pairs

{

(

Aj

,

Ej

)}

pj=1.

(i) Construct the periodic orthonormal bases

{

Uj

}

p

j=1, where Uj

∈

Cn×k.

(ii) Compute the QR-decompositions EjUj

=

VjNjwith VjHVj

=

Ik

(

j

=

1

,

2

, . . . ,

p

)

.

(iii) Compute Mj

=

VjHAjUj−1

(

j

=

1

,

2

, . . . ,

p

)

.

(iv) Compute a desired eigenvalue pair

(π

_µ

, π

_ν

) ≡ ∏

p_j₌₁

µ

j

, ∏

pj=1

ν

j



of the periodic Rayleigh–Ritz matrix pairs

{

(

Mj

,

Nj

)}

pj=1and the corresponding periodic right eigenvectors

{

zj

}

pj=1with

‖

zj

‖ =

1, by using the periodic QZ algorithm

with eigenvalue reordering techniques [1,2], such that

ν

jMjzj−1

=

µ

jNjzj

(

j

=

1

,

2

, . . . ,

p

).

(v) Withx

˜

j

≡

Ujzj, take the Ritz value pair and periodic Ritz vectors

((π

µ

, π

ν

); {˜

xj

}

p

j=1

)

as an approximate eigenvalue pair

(5)

3. Convergence of Ritz value pairs

Let

(π

_α

, π

_β

) ≡ ∏

p_j=1

α

j

, ∏

p j=1

β

j



be a simple eigenvalue pair of

{

(

Aj

,

Ej

)}

p

j=1and

{

xj

}

p

j=1be the corresponding periodic

right eigenvectors with

‖

xj

‖ =

1

(

j

=

1

,

2

, . . . ,

p

)

. That is, we have



β

jAjxj−1

=

α

jEjxj

,

‖

xj

‖ =

1

(

j

=

1

,

2

, . . . ,

p

).

(16)

We assume that the periodic subspaces

{ ˜

Xj

}

pj=1contain accurate approximations to the periodic eigenvectors

{

xj

}

pj=1. For

given periodic orthonormal bases

{

Uj

}

pj=1with

[

Uj

,

Uj⊥

]

being unitary, we define, for all j,

θ

j

=

̸

(

xj

, ˜

Xj

)

(17)

v

j

=

UjHxj

,

v

⊥ j

=

(

U ⊥ j

)

H xj

.

(18)

Then it holds for all j that

‖

v

_j⊥

‖ =

sin

θ

_j

,

‖

v

_j

‖ =



1

−

sin2

_θ

j

=

cos

θ

j

,

(19)

assuming without loss of generality that all

θ

jare in the first quadrant. We now show that the spectrum of the periodic

Rayleigh–Ritz matrix pairs

(

Mj

,

Nj

) = (

VjHAjUj−1

,

VjHEjUj

) (

j

=

1

,

2

, . . . ,

p

)

(20)

obtained by (iv) in the pRR approximation contains a Ritz value pair

(π

_µ

, π

_ν

) ≡ ∏

p_j₌₁

µ

j

, ∏

pj=1

ν

j



that converges to

(π

α

, π

β

)

when sin

θ

j

→

0 for all j.

{

(

Mj

,

Nj

)}

pj=1be the periodic Rayleigh–Ritz matrix pairs defined by(20). Then for all j, there exist matricesEMj

andENjwhich satisfy

‖

EMj

‖ ≤

|

β

_j

|

α

_j

|

2

_{+ |}

β

j

|

2 min

{

ϵ

_j(1)

, ϵ

_j(2)

}

(21) and

‖

ENj

‖ ≤

|

α

_j

|

α

_j

|

2

_{+ |}

β

j

|

2 min

{

ϵ

_j(1)

, ϵ

(_j2)

}

(22) with

ϵ

(1) j

= |

α

j

|‖

Ej

‖



1

−

cos

θ

j cos

θ

j−1



+ |

α

_j

|‖

Ej

‖

sin

θ

j cos

θ

j−1

+ |

β

_j

|‖

Aj

‖

tan

θ

j−1 (23) and

ϵ

(2) j

= |

β

j

|‖

Ej

‖



1

−

cos

θ

j−1 cos

θ

j



+ |

α

_j

|‖

Ej

‖

tan

θ

j

+ |

β

j

|‖

Aj

‖

sin

θ

j−1 cos

θ

j (24)

such that

(π

_α

, π

_β

)

is an eigenvalue pair of the periodic matrix pairs

{

(

Mj

+

EMj

,

Nj

+

ENj

)}

p j=1. Proof. As

[

Uj

,

Uj⊥

]

(

j

=

1

,

2

, . . . ,

p

)

are unitary, pre-multiplying the equations in(16)by V H j produces

β

jVjHAj

[

Uj−1

,

Uj⊥−1

]



U_jH₋₁

(

U_j⊥₋₁

)

H



xj−1

−

α

jVjHEj

[

Uj

,

Uj⊥

]



U_jH

(

U_j⊥

)

H



xj

=

0

.

From(18)and(20), it follows that

β

j

(

Mj

v

j−1

+

VjHAjUj⊥−1

v

⊥

j−1

) − α

j

(

Nj

v

j

+

VjHEjUj⊥

v

⊥

j

) =

0

.

(25)

Let

v

ˆ

_j

≡

v

_j

/‖v

_j

‖

(

j

=

1

,

2

, . . . ,

p

)

. Dividing(25)by

‖

v

_j−1

‖

, we obtain, for all j,

β

jMj

v

ˆ

j−1

−

α

jNj

v

ˆ

j

=

α

jNj

v

ˆ

j

‖

v

_j

‖

v

_j−1

‖

−

α

_jNj

v

ˆ

j

+

α

jVjHEjUj⊥

v

⊥ j

‖

v

_j−1

‖

−

β

_jV_jHAjUj⊥−1

v

⊥ j−1

‖

v

_j−1

‖

.

(26)

(6)

If we define the residuals

rj

≡

β

jMj

v

ˆ

j−1

−

α

jNj

v

ˆ

j

(

j

=

1

,

2

, . . . ,

p

),

(27)

then(26),(19)and(23)imply

‖

rj

‖ ≤

ϵ

(

1)

j . Similarly, dividing(25)by

‖

v

j

‖

yields

‖

rj

‖ ≤

ϵ

(

2)

j , and consequently

‖

rj

‖ ≤

min

{

ϵ

_j(1)

, ϵ

(_j2)

}

. Next we define, for all j,

EMj

≡

− ¯

β

_j

|

α

_j

|

2

_{+ |}

β

j

|

2 rj

v

ˆ

Hj−1

,

ENj

≡

¯

α

j

|

α

_j

|

2

_{+ |}

β

j

|

2 rj

v

ˆ

jH

.

(28)

It then follows from(27)and(28)that

α

j

(

Nj

+

ENj

)ˆv

j

=

β

j

(

Mj

+

EMj

)ˆv

j−1

(

j

=

1

,

2

, . . . ,

p

)

(29)

withEMjandENjsatisfying(21)and(22)by construction.

Remark.

(i) Though

ϵ

_j(1)

, ϵ

_j(2)

→

0 as

θ

j

→

0 for j

=

1

,

2

, . . . ,

p, they are somehow complex and less clear. We shall simplify them,

first by defining

ϵ =

maxj=1,2,...,psin

θ

j. Applying Taylor expansions and observing that



1

−

cos

θ

j cos

θ

j−1



,



1

−

cos

θ

j−1 cos

θ

j



=

O

(ϵ

2

),

we obtain, by ignoring higher order small terms,

ϵ

(1)

j

, ϵ

(2)

j

≤

(|α

j

| ‖

Ej

‖ + |

β

j

| ‖

Aj

‖

)ϵ.

(30)

FromTheorem 3.1and the continuity of the eigenvalues of

{

(

Mj

,

Nj

)}

p

j=1, we immediately have the following corollary. Corollary 3.2. There exists a Ritz value pair

(π

_µ

, π

_ν

)

that converges to the simple eigenvalue pair

(π

_α

, π

_β

)

when sin

θ

j

→

0 for

all j.

4. Convergence of periodic Ritz vectors

FromTheorem 1.1, there are unitary matrices

[

xj

,

Xj

]

and

[

yj

,

Yj

]

, with Xj

,

Yj

∈

Cn×(n−1), such that

[

yH_j Y_jH

]

Aj

[

xj−1

,

Xj−1

] =

[

α

j lHj 0 Lj

]

,

[

yH_j Y_jH

]

Ej

[

xj

,

Xj

] =

[

β

j kHj 0 Kj

]

,

(31)

where the matrices Ljand Kjare

(

n

−

1

) × (

n

−

1

)

for j

=

1

,

2

, . . . ,

p. The periodic eigenvalue pairs of the periodic

matrix pairs

{

(

Lj

,

Kj

)}

pj=1are the periodic eigenvalue pairs of

{

(

Aj

,

Ej

)}

pj=1other than

(π

α

, π

β

)

. Also,(31)implies the spectral

decompositions Aj

=

α

jyjxHj−1

+

yjljHXjH−1

+

YjLjXjH−1 (32) and Ej

=

β

jyjxHj

+

yjkHjX H j

+

YjKjXjH

.

(33)

For any approximate eigen-pair, we have the following residual bound for the approximate eigenvectors.

{

(

Aj

,

Ej

)}

pj=1 have the spectral representations(32)and(33)with

[

xj

,

Xj

]

and

[

yj

,

Yj

]

being unitary for all

j

, ((π

_α˜

, π

_β˜

); {˜

xj

}

pj=1

)

be an approximation to the simple eigen-pair

((π

α

, π

β

); {

xj

}

pj=1

)

,

τ

j

≡ ˜

α

jEj

˜

xj

− ˜

β

jAj

˜

xj−1

(

j

=

1

,

2

, . . . ,

p

),

(34) and sep

((π

α˜

, π

_β˜

), {(

Lj

,

Kj

)}

pj=1

) ≡ ‖

C −1

_‖

−1 ₍₃₅₎ with C

≡







˜

α

1K1

− ˜

β

1L1

− ˜

β

₂L2

α

˜

2K2

...

− ˜

β

_pLp

α

˜

pKp







.

(7)

If sep

((π

α˜

, π

_β˜

), {(

Lj

,

Kj

)}

p j=1

) >

0, then







p

−

j=1 sin2̸

(

_x j

, ˜

xj

) ≤



p

∑

j=1

‖

τ

_j

‖

2 sep

((π

_α˜

, π

_β˜

), {(

Lj

,

Kj

)}

pj=1

)

.

(36)

Proof. Pre-multiplying(34)by Y_jH, we get, with the help of(32)and(33),

Y_jH

τ

j

= ˜

α

jYjHEjx

˜

j

− ˜

β

jYjHAj

˜

xj−1

= ˜

α

jKjXjHx

˜

j

− ˜

β

jLjXjH−1x

˜

j−1

.

(37) This implies C







X₁Hx

˜

1

...

X_pHx

˜

p







=







Y₁H

τ

1

...

Y_pH

τ

p





 .

(38)

Note thatCis invertible in the neighborhood of

(π

_α

, π

_β

)

if and only if the eigenvalue pair

(π

_α

, π

_β

)

is simple. As

[

xj

,

Xj

]

is

unitary, we have sin̸

(

_x

j

, ˜

xj

) ≡ ‖

XjHx

˜

j

‖

for all j. Note that

‖

YjH

τ

j

‖ ≤ ‖

τ

j

‖

for all j. The theorem then follows from inverting

Cin(38)and taking norms.

Theorem 4.1leads easily to the following corollary.

Corollary 4.2. For j

=

1

,

2

, . . . ,

p, we have

sin̸

(

_x j

, ˜

xj

) ≤

√

p max j=1,2,...,p

‖

τ

_j

‖

sep

((π

α˜

, π

_β˜

), {(

Lj

,

Kj

)}

pj=1

)

.

(39)

InCorollary 3.2, we see that there is a Ritz value pair

(π

_µ

, π

_ν

)

approaching the simple eigenvalue pair

(π

_α

, π

_β

)

when sin

θ

j

→

0 for all j. If, in addition, the p residual norms

‖

τ

j

‖

(

j

=

1

,

2

, . . . ,

p

)

defined in(34)approach zero, the periodic Ritz

vectors

{˜

xj

}

p

j=1converge to the periodic right eigenvectors

{

xj

}

p

j=1. Thus,Theorem 4.1andCorollary 4.2show that a converging

Ritz value pair and vanishing residuals imply the convergence of the periodic Ritz vectors since

‖

C−1

_‖

_{is uniformly bounded}

when

(π

_µ

, π

_ν

)

converges to the simple eigenvalue

(π

_α

, π

_β

)

.

When p

=

1, it has been proved that the Ritz vector may fail to converge for a (nearly) multiple Ritz value (see, e.g. [26,

27]). We now perform a convergence analysis of the periodic Ritz vectors and establish some a priori error bounds, showing why the periodic Ritz vectors can fail to converge. Let the periodic Ritz pair

((π

_µ

, π

_ν

); {˜

xj

}

pj=1

)

be used to approximate the

simple periodic eigen-pair

((π

_α

, π

_β

); {

xj

}

pj=1

)

.

Again, fromTheorem 1.1, there are unitary matrices

[

zj

,

Zj

]

and

[

w

j

,

Wj

]

, with Zj

,

Wj

∈

Cr×(r−1), such that

[

w

H j W_jH

]

Mj

[

zj−1

,

Zj−1

] =

[

µ

j dHj 0 Dj

]

,

[

w

H j W_jH

]

Nj

[

zj

,

Zj

] =

[

ν

j fjH 0 Fj

]

,

(40)

where the matrices Djand Fjare

(

k

−

1

) × (

k

−

1

)

for j

=

1

,

2

, . . . ,

p.

Since the only assumption on

{ ˜

Xj

}

p

j=1is that they contain accurate approximations to the periodic eigenvectors

{˜

xj

}

p j=1,

the eigenvalue pairs of

{

(

Dj

,

Fj

)}

pj=1 are not necessarily near the eigenvalue pairs of

{

(

Aj

,

Ej

)}

pj=1 rather than

(π

µ

, π

ν

)

.

Particularly, this means that an eigenvalue pair of

{

(

Dj

,

Fj

)}

pj=1could be arbitrarily near and even equal to the Ritz value

pair

(π

_µ

, π

_ν

)

. For a multiple

(π

_µ

, π

_ν

)

, there are more than one

{˜

xj

}

p

j=1 to approximate the unique periodic eigenvectors

{

xj

}

pj=1. It will be impossible for the periodic Rayleigh–Ritz method to tell which particular approximation is better. If

(π

µ

, π

ν

)

is near an eigenvalue of

{

(

Dj

,

Fj

)}

pj=1, we will get a unique periodic

{˜

x

}

p

j=1, but there is no guarantee that it converges

to

{

xj

}

pj=1.

The above analysis leads us to postulate that the periodic Ritz vectors

{˜

x

}

p_j₌₁ will converge provided that

(π

_µ

, π

_ν

)

is uniformly away from those eigenvalues (other Ritz values) of

{

(

Dj

,

Fj

)}

pj=1, independent of

θ

j

,

j

=

1

,

2

, . . . ,

p. We next

prove that this is indeed the case quantitatively.

Theorem 4.3. Assume that the periodic Rayleigh–Ritz pairs

{

(

Mj

,

Nj

)}

pj=1have the spectral decompositions(40)and

sep

((π

_α

, π

_β

), {(

Dj

,

Fj

)}

pj=1

) ≡ ‖ ˆ

C

(8)

with

ˆ

C

≡







α

1F1

−

β

1D1

−

β

₂D2

α

2F2

...

−

β

_pDp

α

pKp







.

Let

ϵ =

maxj=1,2,...,psin

θ

j. Then for j

=

1

,

2

, . . . ,

p, we have

sin̸

(

_x_j

, ˜

_x_j

) ≤

_sin

θ

_j

+

√

p max

{

min

{

ϵ

_j(1)

, ϵ

_j(2)

}}

sep

((π

_α

, π

_β

), {(

Dj

,

Fj

)}

pj=1

)

(42)

≤



1

+

√

p

(|α

j

| ‖

Ej

‖ + |

β

j

| ‖

Aj

‖

)

sep

((π

_α

, π

_β

), {(

Dj

,

Fj

)}

p j=1

)



ϵ

(43)

with

ϵ

_j(1)

, ϵ

_j(2)defined as in(23)and(24).

Proof. Let the periodic Ritz pair

((π

_µ

, π

_ν

); {˜

xj

}

pj=1

)

be an approximation to the periodic eigen-pair

((π

α

, π

β

); {

xj

}

pj=1

)

. As in

the proof ofTheorem 3.1, let

v

ˆ

_j

=

UH

j xj

/‖

UjHxj

‖

. Then we get

rj

≡

β

jMj

v

ˆ

j−1

−

α

jNj

v

ˆ

j

(

j

=

1

,

2

, . . . ,

p

),

which is defined by(26). It is seen from the proof ofTheorem 3.1that

‖

rj

‖ ≤

min

{

ϵ

j(1)

, ϵ

(2)

j

}

with

ϵ

(_j1)

, ϵ

_j(2)in(23)and(24).

Note that we can regard

((π

_α

, π

_β

); {ˆv

j

}

jp=1

)

as an approximate periodic eigen-pair to the periodic eigen-pair

((π

µ

, π

ν

);

{

zj

}

pj=1

)

of

{

(

Mj

,

Nj

)}

pj=1. Then fromCorollary 4.2, it follows for j

=

1

,

2

, . . . ,

p that

sin̸

(

_z j

, ˆv

j

) ≤

√

p max j=1,2,...,p

‖

rj

‖

sep

((π

_α

, π

_β

), {(

Dj

,

Fj

)}

pj=1

)

≤

√

p max j=1,2,...,pmin

{

ϵ

_j(1)

, ϵ

(_j2)

}

sep

((π

_α

, π

_β

), {(

Dj

,

Fj

)}

pj=1

)

.

Since Ujis orthonormal for j

=

1

,

2

, . . . ,

p, we have from the definitions ofx

˜

j

, ˆv

jand

θ

jthat

sin̸

(

_z

j

, ˆv

j

) =

sin̸

(

Ujzj

,

Uj

v

ˆ

j

) =

sin̸

(˜

xj

,

UjUjHxj

).

Note the triangle inequality

̸

(

_x

j

, ˜

xj

) ≤

̸

(

xj

,

UjUjHxj

) +

̸

(

UjUjHxj

, ˜

xj

) =

̸

(

xj

, ˜

Xj

) +

̸

(˜

xj

,

UjUjHxj

)

with the equality holding when the vectors xj

, ˜

xjand UjUjHxjare linearly dependent. Therefore, we get

sin̸

(

_x j

, ˜

xj

) ≤

sin

θ

j

+

sin̸

(

zj

, ˆv

j

)

≤

sin

θ

j

+

√

p max j=1,2,...,p

{

min

{

ϵ

_j(1)

, ϵ

(_j2)

}}

sep

((π

_α

, π

_β

), {(

Dj

,

Fj

)}

pj=1

)

,

which proves(42). Furthermore, from(30)we get(43).

FromTheorem 3.1, since the Ritz value pair

(π

_µ

, π

_ν

)

approaches the eigenvalue pair

(π

_α

, π

_β

)

as

θ

j

→

0 for j

=

1

,

2

, . . . ,

p, by the continuity argument we have

sep

((π

_α

, π

_β

), {(

D_j

,

F_j

)}

p_j₌₁

) →

sep

((π

_µ

, π

_ν

), {(

D_j

,

F_j

)}

p_j₌₁

).

We see that a sufficient condition for the convergence of the periodic Ritz vectors

{˜

xj

}

pj=1is that sep

((π

µ

, π

ν

), {(

Dj

,

Fj

)}

pj=1

)

is uniformly bounded away from zero. This condition can be checked during the procedure. However, as we have argued above, sep

((π

_µ

, π

_ν

), {(

Dj

,

Fj

)}

p

j=1

)

can be arbitrarily small (and even be exactly zero) when

(π

µ

, π

ν

)

is arbitrarily near other

eigenvalue pairs (or is associated with a multiple eigenvalue pair) of

{

(

Mj

,

Nj

)}

pj=1. Consequently, while the periodic Ritz

value pair converges unconditionally once

θ

j

→

0 for j

=

1

,

2

, . . . ,

p, its corresponding periodic Ritz vectors may fail to

(9)

5. Refinement of periodic Ritz vectors

As we have seen, the periodic Ritz vectors may fail to converge. Since the Ritz value pair is known to converge to the simple eigenvalue pair

(π

_α

, π

_β

)

when sin

θ

j

→

0 for all j, this suggests that we can deal with non-converging Ritz vectors

by retaining the Ritz value pair while replacing the periodic Ritz vectors with a set of unit vectors

ˆ

xj

∈ ˜

Xj

=

span

(

Uj

) (

j

=

1

,

2

, . . . ,

p

)

with suitably small residuals. Thus, we constructx

ˆ

j

(

j

=

1

,

2

, . . . ,

p

)

from

min ˆ xj







p

−

j=1

‖

µ

_jEjx

ˆ

j

−

ν

jAj

ˆ

xj−1

‖

2 subject to

ˆ

xj

∈

span

(

Uj

), ‖ˆ

xj

‖ =

1

(

j

=

1

,

2

. . . ,

p

).

(44)

We call the minimizer

{ˆ

xj

}

pj=1, the refined periodic Ritz vectors.

The following theorem shows that the refined periodic Ritz vectors converge when sin

θ

j

→

0 for all j.

{

(

Aj

,

Ej

)}

pj=1have spectral representations(32)and(33), where

‖

xj

‖ =

1 for all j. Let

(π

µ

, π

ν

) ≡ (∏

pj=1

µ

j

,

∏

p

j=1

ν

j

)

be a Ritz value pair with respect to the orthonormal bases

{

Uj

}

pj=1and let

{ˆ

xj

}

be the corresponding refined periodic Ritz

vectors. If sep

((π

_µ

, π

_ν

), {(

Lj

,

Kj

)}

p j=1

) >

0, then sin̸

(

_x j

, ˆ

xj

) ≤

‖

η‖

sep

((π

_µ

, π

_ν

), {(

Lj

,

Kj

)}

pj=1

)

(

j

=

1

,

2

, . . . ,

p

),

(45) where

η = [η

1

, . . . , η

p

]

⊤and

η

j

≡

ρ

j

+

2

|

µ

j

||

β

j

|

sin2θj−1₂

+

2

|

ν

j

||

α

j

|

sin2θ₂j cos

θ

j−1cos

θ

j

+

|

µ

j

|‖

Ej

‖

sin

θ

j cos

θ

j

+

|

ν

j

|‖

Aj

‖

sin

θ

j−1 cos

θ

j−1 (46)

with

ρ

j

≡ |

µ

j

β

j

−

α

j

ν

j

|

for all j.

Proof. Let xj

=

qj

+

q⊥j , where qj

=

UjUjHxjand q⊥j

=

(

I

−

UjUjH

)

xjfor all j. Then

‖

qj

‖ =

cos

θ

jand

‖

q⊥j

‖ =

sin

θ

j. Define the

normalized vectors

ˆ

qj

≡

qj

‖

qj

‖

=

qj cos

θ

j

,

j

=

1

,

2

, . . . ,

p

.

(47)

By(47), the residuals

ˆ

rjsatisfy

ˆ

rj

≡

µ

jEjq

ˆ

j

−

ν

jAjq

ˆ

j−1

=

µ

jEjqj cos

θ

j

−

ν

jAjqj−1 cos

θ

j−1

=

µ

jEj

(

xj

−

q⊥ j

)

cos

θ

j

−

ν

jAj

(

xj−1

−

q⊥ j−1

)

cos

θ

j−1

.

(48)

Denote the ith column of the identity matrix by ei. Pre-multiplying(48)by the unitary matrixY

ˆ

jH

≡ [

yj

,

Yj

]

Hand using(32)

and(33), we have, for all j,

ˆ

Y_jH

ˆ

rj

=

µ

j

β

je1 cos

θ

j

−

ν

j

α

je1 cos

θ

j−1

−

µ

j

ˆ

Y_jHEjq⊥j cos

θ

j

+

ν

j

ˆ

Y_jHAjq⊥j−1 cos

θ

j−1

=

µ

j

β

jcos

θ

j−1

−

ν

j

α

jcos

θ

j cos

θ

jcos

θ

j−1 e1

−

µ

jY

ˆ

jHEjq⊥j cos

θ

j

+

ν

j

ˆ

Y_jHAjq⊥j−1 cos

θ

j−1

.

(49)

Using the identity cos

θ =

1

−

2 sin2θ

2and taking norm of(49), we obtain

‖ˆ

rj

‖ ≤

η

jfor all j. By the minimization in(44),

we also have p

−

j=1

‖

µ

_jEjx

ˆ

j

−

ν

jAjx

ˆ

j−1

‖

2

≤

p

−

j=1

‖ˆ

rj

‖

2

≤ ‖

η‖

2

.

(50)

The inequalities(45)then follow fromTheorem 4.1and(50).

Since

(π

_µ

, π

_ν

)

converges to

(π

_α

, π

_β

)

as

θ

j

→

0 for j

=

1

,

2

, . . . ,

p, we have

sep

((π

_µ

, π

_ν

), {(

Lj

,

Kj

)}

p

j=1

) →

sep

((π

α

, π

β

), {(

Lj

,

Kj

)}

p j=1

),

which is a positive constant independent of the procedure, whenever

(π

_α

, π

_β

)

is a simple eigenvalue pair of

{

(

Aj

,

Ej

)}

p j=1. So

(10)

Remarks. In order to ensure the convergence of

ρ

_j, we should renormalize the complex ordered pairs

{

(α

_j

, β

_j

)}

p_j₌₁ and

{

(µ

_j

, ν

_j

)}

p_j₌₁inTheorem 5.1by periodic complex numbers of modulo one so that (i)







α

j

:= |

α

j

|

,

β

j

:= |

β

j

|

(

j

=

1

,

2

, . . . ,

p

−

1

),

α

p

:= |

α

p

|

e ι  p ∑ j=1 arg(αj)−arg(βj) 

,

β

p

:= |

β

p

|

,

whenever

π

_α

̸=

0 and

π

_β

̸=

0; (ii)

α

j

:= |

α

j

|

,

β

j

:= |

β

j

|

(

j

=

1

,

2

, . . . ,

p

),

whenever

π

_α

=

0 or

π

_β

=

0. A similar renormalization for

{

(µ

_j

, ν

_j

)}

p_j₌₁can also be carried out. With these new normalized ordered pairs

{

(α

_j

, β

_j

)}

p_j₌₁and

{

(µ

_j

, ν

_j

)}

p_j₌₁, byTheorem 3.1and the periodic Bauer–Fike theorem [5], we have

ρ

j

→

0 when

sin

θ

j

→

0 for all j. It follows fromTheorem 5.1that̸

(

xj

, ˆ

xj

) →

0; i.e., unlike the periodic Ritz vectors, the refined Ritz

vectors are guaranteed to converge.

(iii) Again, let

ϵ =

max_j=1,2,...,psin

θ

j. Then using Taylor expansions and ignoring higher order terms, we have

η

j

≤

ρ

j

+

(|µ

j

| ‖

Ej

‖ + |

γ

j

| ‖

Aj

‖

)ϵ.

(51)

We now propose a numerical procedure to compute the refined periodic Ritz vectors efficiently and reliably.

From(44), the set of refined periodic Ritz vectors can be computed via the following constrained minimization problem

min ˆ z f

(ˆ

z

) ≡

p

−

j=1

‖

µ

_jEjUjz

ˆ

j

−

ν

jAjUj−1z

ˆ

j−1

‖

2 subject to cj

(ˆ

z

) ≡ ˆ

zHjz

ˆ

j

−

1

=

0

(

j

=

1

,

2

. . . ,

p

),

(52)

wherez

ˆ

≡ [ ˆ

z₁⊤

, . . . , ˆ

z⊤_p

]

⊤

∈

_Ckp_{. Newton’s method can be applied to the Lagrangian function of the constrained}

optimiza-tion problem(52), with the periodic Ritz vectors utilized as the feasible initial iterate. An approximate solution to(52)will be acceptable in the sense ofTheorems 3.1,4.1and5.1if the associated residuals are reasonably small.

Remarks. (i) For periodicity p

=

1, the minimization problem(44)can be solved via the singular value decomposition (SVD). Indeed, as mentioned in [26,27], it is easily seen that the refined Ritz vectorx

ˆ

1

=

U1

ˆ

z1, wherez

ˆ

1is the right singular

vector of

(µ

1E1

−

ν

1A1

)

U1corresponding to its smallest singular value. Unfortunately, the refined periodic Ritz vectors

{ˆ

xj

}

p j=1

with periodicity p

≥

2 cannot be computed via(52)by any SVD-like algorithm since, instead of

‖[ ˆ

z⊤

1

, ˆ

z ⊤ 2

, . . . , ˆ

z ⊤ p

]

⊤

_{‖ =}

_1,

the constraints

‖ˆ

zj

‖ =

1

(

j

=

1

,

2

, . . . ,

p

)

have to be satisfied simultaneously.

(ii) The Newton optimization of(52)is straightforward, but can be expensive since EjUjand AjUj−1

(

j

=

1

,

2

, . . . ,

p

)

,

though already available when forming the periodic Rayleigh–Ritz pairs

{

(

M_j

,

N_j

)}

p_j₌₁, are n

×

k. However, it is possible to

reduce the optimization problem to a much smaller one when the Rayleigh–Ritz method is applied to certain special periodic Krylov subspaces. In Section6, we will consider a periodic Arnoldi process that generates periodic orthonormal bases of the periodic Krylov subspaces. Based on it, we propose the refined periodic Arnoldi method and show that Newton optimization is particularly efficient.

6. Refined periodic Ritz vectors from a periodic Arnoldi process

Recall the eigen-equations in(1):

β

jAjxj−1

=

α

jEjxj

(

j

=

1

,

2

, . . . ,

p

).

Without loss of generality, Ejcan be assumed to be nonsingular, as a shift can always be applied to the periodic eigenvalue

problem. To apply the Arnoldi process for matrix products [17] to our periodic matrix pairs, we may consider two different products

Pl

≡

(

Ep−1Ap

)(

Ep−−11Ap−1

) · · · (

E1−1A1

),

Pr

≡

(

ApE−p−11

)(

Ap−1Ep−−12

) · · · (

A1Ep−1

).

First, construct A

∈

_Rnp×np_{from C in}₍₃₎_{by substituting}

_α

j

=

0

, β

j

= −

1

(

j

=

1

,

2

, . . . ,

p

)

. Similarly, denote by E the

matrix constructed with

α

j

=

1

, β

j

=

0

(

j

=

1

,

2

, . . . ,

p

)

in(3). Denote C in(3)slightly differently as C

(

A

,

E

;

α, β)

, with

α = [α

1

, . . . , α

p

]

⊤and

β = [β

1

, . . . , β

p

]

⊤. From the equivalence of(1)and(2)to(4)and(5), respectively, we can

see clearly that C

(

_A

,

_E

;

α, β)

defines the periodic eigenvalue problem under consideration. An appropriate shift

σ

can be applied to C

(

A

,

E

;

α, β)

, producing an equivalent periodic eigenvalue problem defined by C

(

A

,

E

−

σ

A

;

α, β)

. Obviously,

(11)

an eigenvalue

(π

_α

, π

_β

) = (∏

p_j=1

α

j

, ∏

p

j=1

β

j

)

for C

(

A

,

E

;

α, β)

is transformed to

( ˜π

α

, ˜π

β

) = ∏

pj=1

α

j

, ∏

p

j=1

(β

j

+

σ α

j

)

for

C

(

_A

,

_E

−

σ

_A

;

α, β)

, with identical eigenvectors and

β

j

+

σα

j

̸=

0 for all j.

Utilizing Pl, the Arnoldi process is applied to the equivalent eigenvalue equations after inverting Ej:

β

jE

−1

j Ajxj−1

=

α

jxj

(

j

=

1

,

2

, . . . ,

p

),

resulting in the refinement of the corresponding periodic Ritz vectors as summarized in(52). Alternatively with Pr, we consider another set of equivalent eigenvalue equations:

β

jAjEj−−11

(

Ej−1xj−1

) = α

j

(

Ejxj

)

⇒

β

_jAj

(

Ej−1xj−1

) = α

j

(

Ejxj

),

(53)

whereAj

≡

AjEj−−11

(

j

=

1

,

2

, . . . ,

p

)

and Pr

=

∏

1

k=pAk.

With the k-step periodic Arnoldi process for

{

Aj

}

p j=1, we have A1Up

=

U1H1

, . . . ,

AjUj−1

=

UjHj

, . . . ,

ApUp−1

=

UpHp

+

hk+1,ku p k+1e ⊤ k

,

(54)

where H1

, . . . ,

Hp−1

∈

Ck×kare upper triangular and Hp

∈

Ck×kis upper Hessenberg. Denote



Hp

=



_H p h_k+_1,ke⊤k



, we have ApUp−1

= [

Up

|

upk+1

]

Hp

.

It is easy to show that Hj

=

MjNj−1

(

j

=

1

,

2

, . . . ,

p

)

, with Mjand Njas defined in(20).

Without loss of generality, assume

ν

j

=

1

(∀

j

)

. We then selectx

ˆ

j

(

j

=

1

,

2

, . . . ,

p

)

from

min ˆ xj p

−

j=1

‖

Ajx

ˆ

j−1

−

µ

jEjx

ˆ

j

‖

2 subject to

ˆ

xj

∈

span

(

Ej−1Uj

), ‖ˆ

xj

‖ =

1

(

j

=

1

,

2

. . . ,

p

).

(55)

It is easy to show that the refinement in(55)is equivalent to the one in(44)with the periodic Arnoldi process providing

{

Uj

}

, when

{

Ej−1Uj

}

are orthogonalized. There is no reason in doing so because of the saving in reusing computed quantities

from the periodic Arnoldi process, as shown below.

From(55), the set of refined periodic Ritz vectors can be computed via the following constrained minimization problem

min ˆ z f

(ˆ

z

) ≡

p

−

j=1

‖

Aj

(

Ej−−11Uj−1

)ˆ

zj−1

−

µ

jEj

(

Ej−1Uj

)ˆ

zj

‖

2 subject to

‖

(

E_j−1Uj

)ˆ

zj

‖ =

1

(

j

=

1

,

2

. . . ,

p

),

(56) wherez

ˆ

≡ [ ˆ

z₁⊤

, . . . , ˆ

z⊤ p

]

⊤

_∈

Ckp. By(54), we have min ˆ z f

(ˆ

z

) ≡

p

−

j=1

‖

Aj

(

Ej−−11Uj−1

)ˆ

zj−1

−

µ

jEj

(

Ej−1Uj

)ˆ

zj

‖

2

⇒

min ˆ z f

(ˆ

z

) ≡

p−1

−

j=1

‖

Uj

(

Hjz

ˆ

j−1

−

µ

jz

ˆ

j

)‖

2

+



[

Uj

|

uk+1

]





Hpz

ˆ

p−1

−

µ

p

[ ˆ

zp 0

]



2

⇒

min ˆ z f

(ˆ

z

) ≡

p−1

−

j=1

‖

Hjz

ˆ

j−1

−

µ

j

ˆ

zj

‖

2

+







Hpz

ˆ

p−1

−

µ

p

[ ˆ

zp 0

]



2

.

(57)

Then we consider the Lagrangian function of the constrained optimization problem(56):

L

(ˆ

z

, λ) =

f

(ˆ

z

) +

p

−

j=1

λ

j

(ˆ

zjHBjz

ˆ

j

−

1

),

(58) where

λ = [λ

1

, . . . , λ

p

]

⊤

,

Bj

≡

(

Ej−1Uj

)

⊤

(

Ej−1Uj

)

.

The derivatives of L

(ˆ

z

, λ)

are:

f1

≡

∂

L

∂ ˆ

z1

=

2

(µ

2₁z

ˆ

1

−

µ

1H1z

ˆ

p

+

H2⊤H2z

ˆ

1

−

µ

2H2⊤z

ˆ

2

+

λ

1B1z

ˆ

1

),

fj

≡

∂

L

∂ ˆ

zj

=

2

(µ

2_jz

ˆ

j

−

µ

jHjz

ˆ

j−1

+

Hj⊤+1Hj+1z

ˆ

j

−

µ

j+1Hj⊤+1z

ˆ

j+1

+

λ

jBjz

ˆ

j

) (

j

=

2

, . . . ,

p

−

2

),