On the spectral properties of an operator pencil

191

ISSN 0965-5425, Computational Mathematics and Mathematical Physics, 2007, Vol. 47, No. 2, pp. 191–199. © Pleiades Publishing, Ltd., 2007.Original Russian Text © Yu.V. Bychenkov, 2007, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2007, Vol. 47, No. 2, pp. 197–205.

On the Spectral Properties of an Operator Pencil

Yu. V. Bychenkov

Faculty of Mechanics and Mathematics, Moscow State University, Leninskie gory, Moscow, 119992 Russiae-mail: [email protected]

Received April 10, 2006; in final form, July 7, 2006

Abstract

—For a broad class of iterative algorithms for solving saddle point problems, the study of theconvergence and of the optimal properties can be reduced to an analysis of the eigenvalues of operatorpencils of a special form. A new approach to analyzing spectral properties of pencils of this kind is pro-posed that makes it possible to obtain sharp estimates for the convergence rate.

DOI:

10.1134/S0965542507020042

Keywords:

saddle point operator, operator pencil, spectral properties, estimate of the convergence rate

1. INTRODUCTION

The design of efficient algorithms for solving problems with saddle point operators is an important topicin modern numerical analysis. This is explained by the extensive use of this class of problems in applications(see survey [1]). Especially important are large-scale sparse saddle point problems, which are mainly solvedby iterative algorithms.

As evidenced by practice, an analysis of the convergence and especially the optimization of iterativealgorithms for solving saddle point problems are difficult and require that an individual approach be takenin each particular case. However, a typical step in analyses of many algorithms is to reduce the original prob-lem to an examination of the properties of a polynomial operator pencil

χ

(

λ

) =

P

(

λ

,

L

,

G

)

, where

λ ∈

�

and

L

and

G

are linear Hermitian operators in a finite-dimensional Hermitian space

U

.An inspection of the available algorithms (which is not undertaken in this paper) shows that, for a broad

class of problems, the analysis of the convergence can be reduced to the study of the spectrum

of the pencil

χ

(

λ

)

, where

(1.1)

It is assumed that

(1.2)

(1.3)

Hereafter, we use the following notation:

�

[

z

1

, …,

z

k

]

is the ring of polynomials in the variables

z

1

, …,

z

k

(

k

∈

�

) with the coefficients in

�

, and

σ

(

A

)

is the spectrum of the linear operator

A

(in other words, thespectrum of the pencil

A

–

λ

I

).Conventional approaches to the study of the spectral properties of pencil (1.1) make it possible to facil-

itate the analysis of the convergence of iterative algorithms for solving saddle point problems. For the par-ticular case of pencil (1.1), an approach of this kind was proposed in [2]. It is based on the use of operatorinequalities and allows one to derive good estimates for the convergence of certain algorithms (such as algo-rithms of the Arrow–Hurwitz type and algorithms with symmetric preconditioners). The shortcomings ofthis approach are that the class of the corresponding algorithms is too narrow and the estimates obtained areinsufficiently sharp (they may even have a wrong order of magnitude). Another and more general approachwas actually given by E.V. Chizhonkov (e.g., see [3]). It is based on the following additional assumption(which restricts the class of problems under analysis): the null space of

G

is invariant with respect to

L

. Itturned out that this approach made it possible to obtain sharp estimates for the convergence of many algo-rithms for solving saddle point problems.

σ χ( ) λ � kerχ λ( ) 0{ }≠∈{ }≡

χ λ( ) f λ L,( )g λ G,( ) h λ( )G, f � λ s,[ ], g � λ t,[ ], h � λ[ ].∈∈∈+=

L L*, σ L( ) δ1 δ2,[ ], δ1 δ2 �, δ1 δ2,≤∈,⊆=

G G* 0, σ G( ) 0{ } γ 1 γ 2,[ ], γ 1 γ 2 �, 0 γ 1 γ 2.≤<∈,∪⊆≥=

192

COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS

Vol. 47

No. 2

2007

BYCHENKOV

Below, we propose a new universal approach to the study of the spectral properties of pencil (1.1). Inparticular, this approach makes it possible to obtain simple and sharp estimates for the convergence rate ofmany iterative algorithms for solving saddle point problems.

2. BOUNDS FOR THE SPECTRUM OF OPERATOR PENCILS

In what follows, we assume that

χ

(

λ

)

is an operator pencil of type (1.1).

Theorem 1

(lower bound).

Let M

⊂

�

be a finite nonempty set with

card

M

≤

dim

U

.

Assume that, for any

λ

0

∈

M

,

there exist s

0

∈

[

δ

1

,

δ

2

]

and

t

0

∈ [γ1, γ2] such that f(λ0, s0) = 0 or f(λ0, s0)g(λ0, t0) + h(λ0)t0 = 0. Then,there exist linear operators L and G acting in U that satisfy (1.2) and (1.3) and are such that M ⊆ σ(χ).

Proof. Let {ei be an arbitrary orthonormal basis in U, where NU ≡ dimU. Let M = {λi , whereNM ≡ cardM > 0 and NM ≤ NU. By assumption, there exist si ∈ [δ1, δ2] and ti ∈ {0} ∪ [γ1, γ2] such thatf(λi, si)g(λi, ti) + h(λi)ti = 0 (i = 1, 2, …, NM).

For i = 1, 2, …, NU, define the operators L and G by the relations

It is easy to verify that the operators L and G thus defined satisfy assumptions (1.2) and (1.3). Moreover, fori = 1, 2, …, NM, we have

that is, λi ∈ σ(χ). The theorem is proved.The main implication of this result is that any upper bound for σ(χ) that is valid for the class of operators

(1.2), (1.3) contains the set of the functions λ(s, t) satisfying one of the equations

for various s ∈ [δ1, δ2] and t ∈ [γ1, γ2]. Below, we indicate the conditions under which a bound of this kindis sharp.

Note that the eigenvalues of χ(λ) satisfying the condition g(λ, 0) = 0 are most often ignored in the anal-ysis of the properties of algorithms for solving saddle point problems. Taking this into account, we state allthe subsequent assertions so as to drop these eigenvalues already at the stage at which the conditions areformulated. This, however, does not restrict the generality of our estimates, which, if required, can beextended by adding the solutions to g(λ, 0) = 0.

Theorem 2 (upper bound). Let assumptions (1.2) and (1.3) be fulfilled. Assume that λ0 ∈ σ(χ) and g(λ0,0) ≠ 0. Then, either there exists s0 ∈ [δ1, δ2] such that

or, in the opposite case,

Here, conv denotes the closed convex hull of a set in �.Proof. Under the conditions of the theorem, there exists a vector u ∈ U\{0} such that χ(λ0)u = 0. Denote

by S the linear operator f(λ0, L). If f(λ0, s0) = 0 for a certain s0 ∈ [δ1, δ2], then the theorem is proved.Assume that f(λ0, s) ≠ 0 for any s ∈ [δ1, δ2]. Then, we have

(2.1)

Hence, 0 ∉ σ(S); that is, S is an invertible operator. The assumption made above also implies that Gu ≠ 0.Indeed, if Gu = 0, then

By assumption, g(λ0, 0) ≠ 0; therefore, u ∈ kerS, which leads to a contradiction.

}i 1=NU }i 1=

NM

Lei

siei, if i NM,<sNM

ei, if i NM,≥⎩⎨⎧

Gei

tiei, if i NM,<tNM

ei, if i NM.≥⎩⎨⎧

= =

χ λi( )ei f λi si,( )g λi ti,( ) h λi( )ti+( )ei 0;= =

f λ s,( ) 0,=

f λ s,( )g λ t,( ) h λ( )t+ 0=

f λ0 s0,( ) 0,=

conv t 1– g λ0 t,( ) t γ 1 γ 2,[ ]∈{ } conv f λ0 s,( ) 1– h λ0( ) s δ1 δ2,[ ]∈{ } � 0.+

σ S( ) f λ0 s,( ) s σ L( )∈{ } f λ0 s,( ) s δ1 δ2,[ ]∈{ }.⊆=

0 χ λ0( )u f λ0 L,( )g λ0 G,( )u Sg λ0 0,( )u.= = =

COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS Vol. 47 No. 2 2007

ON THE SPECTRAL PROPERTIES OF AN OPERATOR PENCIL 193

We write u in the form u0 + u1 , where u0 ∈ H ≡ kerG and u1 ∈ K ≡ (kerG)⊥ = imG. The original eigen-value problem is rewritten as

Take the scalar products of both sides of this equality with Gu and then divide them by (Gu, Gu) ≠ 0. Inthese calculations, we use the equalities Gu = Gu1 and (u, Gu) = (u1, Gu1). This yields

(2.2)

The operator F ≡ G|K : K K is self-adjoint in K and σ(F) ⊆ [γ1, γ2]. Hence, F–1g(λ0, F) is a normal operatorand σ(g(λ0, F)) = g(λ0, σ(F)). Thus, we have

(2.3)

Since L is self-adjoint, S = f(λ0, L) is a normal operator; hence, S–1 is also normal. It follows that

(2.4)

Relations (2.2), (2.3), and (2.4) immediately imply the assertion of the theorem.In many problems, the spectrum of pencil (1.1) is a priori known to be real (see, e.g., [2, 4, 5]). In such

a case, one can use the following implication of Theorem 2.Corollary 1. Let the conditions of Theorem 2 be fulfilled. Assume that the polynomials f(λ, s) and g(λ,

t) have real coefficients. If λ0 ∈ �, then there exist s0 ∈ [δ1, δ2] and t0 ∈ [γ1, γ2] such that f(λ0, s0) = 0 orf(λ0, s0)g(λ0, t0) + h(λ0)t0 = 0.

Proof. Assume that f(λ0, s) ≠ 0 for any s ∈ [δ1, δ2]. Then, by Theorem 2, we have the inclusion

By assumption, the polynomial P(s) ≡ f(λ0, s) has real coefficients. Hence, the image under P of the segment[δ1, δ2] is a segment in �\{0}. Since h(λ0) is independent of s, the set

is a segment in �; thus, this set is convex and closed. In a similar way, we show that {t–1g(λ0, t) | t ∈ [γ1, γ2]}is also a convex closed set.

Thus, it holds that

Therefore, there exist t0 ∈ [γ1, γ2], s0 ∈ [δ1, δ2] such that

which is equivalent to the relation

The corollary is proved.Note that the bound given in this corollary is identical to the lower bound obtained in Theorem 1. Hence,

this is a sharp bound.

3. BOUNDS FOR THE SPECTRAL RADIUS

The minimization of the spectral radius of the transition operator underlies the asymptotic optimizationof stationary algorithms. For an algorithm of this kind, the size of the spectral radius of the transition oper-

g λ0 G,( )u S 1– h λ0( )Gu+ 0.=

g λ0 G,( )u1 Gu1,( )Gu1 Gu1,( )

-------------------------------------------S 1– h λ0( )Gu1 Gu1,( )

Gu1 Gu1,( )------------------------------------------------+ 0.=

g λ0 G,( )u1 Gu1,( )Gu1 Gu1,( )

-------------------------------------------F 1– g λ0 F,( )Fu1 Fu1,( )

Fu1 Fu1,( )------------------------------------------------------- conv t 1– g λ0 t,( ) t γ 1 γ 2,[ ]∈{ }.⊆=

S 1– h λ0( )Gu1 Gu1,( )Gu1 Gu1,( )

------------------------------------------------ conv f λ0 s,( ) 1– h λ0( ) s δ1 δ2,[ ]∈{ }.⊆

conv t 1– g λ0 t,( ) t γ 1 γ 2,[ ]∈{ } conv f λ0 s,( ) 1– h λ0( ) s δ1 δ2,[ ]∈{ } � 0.+

f λ0 s,( ) 1– h λ0( ) s δ1 δ2,[ ]∈{ } P s( ) 1– h λ0( ) s δ1 δ2,[ ]∈{ }=

t 1– g λ0 t,( ) t γ 1 γ 2,[ ]∈{ } f λ0 s,( ) 1– h λ0( ) s δ1 δ2,[ ]∈{ } � 0.+

t01– g λ0 t0,( ) f λ0 s0,( ) 1– h λ0( )+ 0,=

f λ0 s0,( )g λ0 t0,( ) h λ0( )t0+ 0.=

194


BYCHENKOV

ator is directly related to the quantity

(3.1)

Bounds for ρ(χ) can be derived from the spectral bounds given in Theorem 2. It makes sense, however, toobtain more compact and convenient bounds imposing additional limitations on the form of pencil (1.1).

Let P ∈ �[z1, …, zn] (n ∈ �). Denote by P the degree of the polynomial ∈ �[zj] for the

fixed values zi = (i ≠ j).

Theorem 3. Let assumptions (1.2) and (1.3) be fulfilled. Assume additionally that degs f ≤ 1, degtg ≤ 1,and the quantity

is independent of s1, s2, s3 ∈ [δ1, δ2], and t ∈ [γ1, γ2]. Then, we have

where the maximum is taken over all λ(s, t) satisfying one of the equations

(3.2)

Proof. We first observe that the condition degtg ≤ 1 implies the equality

for any λ ∈ �. Let λ ∈ σ(χ) and g(λ, 0) ≠ 0. Using the observation just made, we derive from Theorem 2the existence of s ∈ [δ1, δ2] and t ∈ [γ1, γ2] such that f(λ, s) = 0 or

where N(λ) ≡ conv{f(λ, s)–1h(λ) | s ∈ [δ1, δ2], f(λ, s) ≠ 0}. Denote by M the set of λ ∈ � that satisfy thissystem for some s ∈ [δ1, δ2] and t ∈ [γ1, γ2]. It is obvious that ρ(χ) ≤ supM|λ|.

We fix λ ∈ � and assume that f(λ, s) ≠ 0 for all s ∈ [δ1, δ2]. Then, w(z) = f(λ, z)–1h(λ) is a linear fractionalfunction whose poles do not belong to [δ1, δ2]. Therefore, the image of [δ1, δ2] under w is either a line seg-ment or an arc of a circle with the endpoints w(δ1) = f(λ, δ1)–1h(λ) and w(δ2) = f(λ, δ2)–1h(λ). Thus, any pointz ∈ N(λ) can be represented in the form

where s1, s2 ∈ [δ1, δ2], s1 ≤ s2 , α ∈ [0, 1], and s3 ≡ (1 – α)s1 + αs2 .This representation implies that λ ∈ M if and only if there exist s2 ∈ [δ1, δ2], s1 ≤ s2, s3 ∈ [s1, s2], and t ∈

[γ1, γ2] such that

Since the left-hand side of this equation is continuous, M is a closed set. By the assumption of the theorem,the quantity degλ( f(λ, s1)f(λ, s2)g(λ, t) + f(λ, s3)h(λ)t) > 0 is independent of s1, s2, s3, and t. Since the rootsof a polynomial are continuous functions of its coefficients, M is a bounded set, which means that

where ∂M is the boundary of M.Assume that λ ∈ ∂M is such that f(λ, s) ≠ 0 for all s ∈ [δ1, δ2]. We show that λ satisfies the inclusion

ρ χ( ) λ .λ σ χ( )∈

g λ 0,( ) 0≠

sup=

degz jP

zi zi0

i j≠,=

zi0

degλ f λ s1,( ) f λ s2,( )g λ t,( ) f λ s3,( )h λ( )t+( ) 0>

ρ χ( ) λ s t,( ){ },s δ1 δ2,[ ]∈

t γ 1 γ 2,[ ]∈

max≤

f λ s,( ) 0,=

f λ s,( )g λ t,( ) h λ( )t+ 0,=

f λ δ1,( ) f λ δ2,( )g λ t,( ) f λ s,( )h λ( )t+ 0.=

conv t 1– g λ t,( ) t γ 1 γ 2,[ ]∈{ } t 1– g λ t,( ) t γ 1 γ 2,[ ]∈{ }=

0 t 1– g λ t,( ) N λ( ),+∈

z α/ f λ s1,( ) 1 α–( )/ f λ s2,( )+( )h λ( )f λ s3,( )h λ( )

f λ s1,( ) f λ s2,( )--------------------------------------,= =

f λ s1,( ) f λ s2,( )g λ t,( ) f λ s3,( )h λ( )t+ 0.=

λM

sup λ ,∂M

max=

0 t 1– g λ t,( ) ∂N λ( )+∈



for some t ∈ [γ1, γ2]. If the contrary is true, then there exist t ∈ [γ1, γ2] and z ∈ ]N(λ)[ such that

Since f(µ, s)–1h(µ) is continuous in a certain neighborhood of the set {λ} × [δ1, δ2], there exist ε0 , ε1 > 0such that (z) ⊂ ]N(µ)[ for any µ ∈ (λ) (hereafter, Va(b) = {z ∈ ∈ � | |z – b| < a}). Choose ε2: 0 < ε2 ≤ε0 so that

for all µ ∈ (λ). Therefore, the relation

holds for all µ ∈ (λ). This means that µ ∈ M and, hence, (λ) ⊂ M, which contradicts the assumptionλ ∈ ∂M.

Thus, the condition λ ∈ ∂M implies that f(λ, s) = 0 for some s ∈ [δ1, δ2] or 0 ∈ t–1g(λ, t) + ∂N(λ) forsome t ∈ [γ1, γ2]. Now, the assertion of the theorem follows from the representation

and the inequality

In certain practically important situations, the bound given in Theorem 2 can be improved.Corollary 2. Let assumptions (1.2) and (1.3) be fulfilled. Moreover, assume the following:(1) degs f ≤ 1, degtg ≤ 1;(2) f, g, and h are polynomials with real coefficients;(3) degλf = 1, degλg = 1, and degλh ≤ 1 for s ∈ [δ1, δ2] and t ∈ [γ1, γ2];(4) f(0, s)h(0) is independent of s.Then, it holds that

where the maximum is taken over all λ(s, t) satisfying one of the equations

(3.3)

Proof. Note that all the conditions in Theorem 3 are fulfilled. Therefore, we have

(3.4)

where the maximum is taken over all λ(s, t) satisfying one of the equations (3.2). Define

where the maximum is taken over all λ(s, t) satisfying one of the equations (3.3). It is obvious that 0 ≤ q1 ≤q0. Let us show that, under the conditions made above, q0 = q1 . The assertion of the theorem clearly holdsif q0 = 0 or δ1 = δ2 .

Assume that q0 > 0 and δ1 < δ2. We fix the scalars t0 ∈ [γ1, γ2] and s0 ∈ [δ1, δ2], which supply the maxi-

mum to the function in (3.4), and write the third equation in (3.2) in the form R( λ, α0) = 0, where

0 t 1– g λ t,( ) z.+=

V ε1V ε0

t 1– g µ t,( ) t 1– g λ t,( )– ε1<

V ε2

t 1– g µ t,( )– V ε1t 1– g λ t,( )–( )∈ V ε1

z( ) ]N µ( )[,⊂=

V ε2V ε2

∂N λ( ) f λ s,( ) 1– h λ( ) s δ1 δ2,[ ]∈{ } f λ s,( ) f λ δ1,( ) 1– f λ δ2,( ) 1– h λ( ) s δ1 δ2,[ ]∈{ }∪=

ρ χ( ) λλ M∈sup≤ λ

λ ∂M∈max .=

ρ χ( ) λ s t,( ){ },s δ1 δ2,[ ]∈

t γ 1 γ 2,[ ]∈

max≤

f λ s,( ) 0,=

f λ s,( )g λ t,( ) h λ( )t+ 0.=

ρ χ( ) q0 λ s t,( ){ },s δ1 δ2,[ ]∈

t γ 1 γ 2,[ ]∈

max=≤

q1 λ s t,( ){ },s δ1 δ2,[ ]∈

t γ 1 γ 2,[ ]∈

max=

q01–

α0 s0 δ1–( )/ δ2 δ1–( ) 0 1,[ ],∈=

196


BYCHENKOV

It is obvious that degµR = 3 for any α ∈ [0, 1] and all the roots of the equation R(µ, α0) = 0 belong to the

closed unit disc , where D = {µ ∈ � | |µ| < 1}; moreover, there exist roots lying on the boundary of thisdisc.

Suppose that q1 < q0 . If α ∈ {0, 1}, then the equation R(µ, α) = 0 splits into the two equations

which correspond to the equations in (3.3). It follows that, for α ∈ {0, 1}, the moduli of the roots of R(µ,α) = 0 do not exceed q1/q0 < 1; hence, these roots belong to D.

According to the conditions of this corollary, the polynomial R(µ, α) has real coefficients and R(µ, α) =(1 – α)R(µ, 0) + αR(µ, 1). Consider R(µ, α) as a polynomial in µ with the coefficients depending on α. Then,condition (4) and the fact that the leading coefficient of R (that is, the coefficient of µ3) is the same as theleading coefficient of f(q0µ, δ1)f(q0µ, δ2)g(q0µ, t0) imply that the leading coefficient and the constant term inR are independent of α. Hence, R(µ, α) can be represented as

where a, d ∈ �, a ≠ 0, bα ≡ (1 – α)b0 + αb1, cα ≡ (1 – α)c0 + αc1, and b0, b1, c0, c1 ∈ �.

To estimate the roots of R(µ, α) = 0 for α ∈ [0, 1], we use the Schur–Cohn criterion (see [6]): all theroots of the polynomial P(z) = a0zn + … + an ∈ �[z] lie in D if and only if |a0| > |an| and all the roots of

P*(z) = z–1( P(z) – anzn ) also lie in D.

The roots of R(µ, 0) lie in D; hence, |a| > |d| and

Since the roots of R*(µ, 0) and R*(µ, 1) belong to D, we have

for any α ∈ [0, 1] and

The only root of R**(µ, α) has the form

This is a real linear fractional function of α whose denominator does not vanish for α ∈ [0, 1]. Thus, µα isa monotone function of α on the interval [0, 1]; moreover, the roots of R**(µ, 0) and R**(µ, 1) lie in D. Thisimplies the inequality

By the Schur–Cohn criterion, all the roots of R(µ, α0) = 0 belong to D, which contradicts the fact that thereare roots on the boundary of D. Thus, our original assumption is false, which means that q0 = q1 . The cor-ollary is proved.

Note that, in view of Theorem 1, the bound derived in this corollary cannot be weakened and, hence, issharp.

R µ α,( ) f q0µ δ1,( ) f q0µ δ2,( )g q0µ t0,( ) f q0µ 1 α–( )δ1 αδ2+,( )h q0µ( )t0.+≡

D

f q0µ 1 α–( )δ1 αδ2+,( ) 0,=

f q0µ αδ1 1 α–( )δ2+,( )g q0µ t0,( ) h q0µ( )t0+ 0,=

R µ α,( ) aµ3 bαµ2 cαµ d ,+ + +=

a0 P 1/z( )

R* µ α,( ) a2 d2–( )µ2 abα dcα–( )µ acα dbα–( ).+ +=

acα dbα– max ac0 db0– ac1 db1–,{ } a2 d2–<≤

R** µ α,( ) a2 d2–( ) acα dbα–( )–[ ] a2 d2–( ) acα dbα–( )+[ ]µ abα dcα–( )+{ }.=

µαabα dcα–

a2 d2–( ) acα dbα–( )+-------------------------------------------------------.=

µα max µ0 µ1,{ } 1.<≤



4. EXAMPLES OF THE USE OF THE NEW BOUNDS

We demonstrate how the results obtained above can be applied to the class of symmetric saddle pointproblems

(4.1)

where A = A* > 0 is a square NU-by-NU matrix and B is a rectangular NU-by-NP matrix of full rank. Twoadditional matrices are often introduced in the design of algorithms for solving problems of this kind,namely, a square NU-by-NU matrix Q = Q* > 0 and a square NP-by-NP matrix C = C* > 0. The matrices Qand C are used to precondition A and the Schur complement B*A–1B, respectively. A priori informationabout the algorithm can be given by the constants δ, ∆, γ, and Γ in the inequalities

4.1. Symmetric Preconditioning

The symmetric preconditioning is used to improve the convergence of projective methods for solvinglinear equations such as the conjugate gradient method or the Lanczos algorithm. The structure of the spec-trum of the preconditioned operator is important for the convergence of these methods. Consider, forinstance, the block preconditioner

where α, β > 0. Since P = P* and L – P > 0 for α < δ, we conclude that P–1L has a real spectrum for α ≤ δ.It is easy to verify that, with the exception of λ = 0, this spectrum is identical to the spectrum of the pencil

Here, L ≡ A–1/2QA–1/2 and G ≡ A–1/2BC–1B*A–1/2. Let us write χ(λ) in form (1.1). To this end, define

With δ1 = ∆–1, δ2 = δ–1, γ1 = γ, and γ2 = Γ, assumptions (1.2) and (1.3) and all the conditions in Corollary 1are fulfilled. Therefore, the spectrum of the preconditioned operator P–1L belongs to

where λ1(s, t) satisfies the equation αλs – 1 = 0 (that is, λ1 = (αs)–1) and (s, t) are the roots of the equation

Thus, changing the variable s s–1, we obtain the following estimate for α ≤ δ:

Note that this estimate is equivalent to the one obtained in [5]. The latter estimate is, however, derived underthe additional assumption that kerG is invariant with respect to L.

Lz A B

B* 0⎝ ⎠⎜ ⎟⎛ ⎞ u

p⎝ ⎠⎜ ⎟⎛ ⎞

≡ f

g⎝ ⎠⎜ ⎟⎛ ⎞

,=

0 δQ A ∆Q, 0 γC B*A 1– B ΓC.≤ ≤<≤ ≤<

P αQ B

B* βC–⎝ ⎠⎜ ⎟⎛ ⎞

,≡

χ λ( ) λ2 αβL G+( ) λ βI 2G+( )– G.+=

f λ s,( ) β αλs 1–( ), g λ t,( ) λ, h λ( ) λ 1–( )2.≡ ≡ ≡

Λ λ1 s t,( ) λ21 s t,( ) λ2

2 s t,( ), ,{ },s δ1 δ2,[ ]∈

t γ 1 γ 2,[ ]∈

∪=

λ22 3,

λ2 αβs t+( ) λ β 2t+( )– t+ 0.=

σ P 1– L( ) Λ⊆ δ/α ∆/α,[ ] β 2t β2 4tβ 1 α/s–( )+±+2 αβ/s t+( )

--------------------------------------------------------------------s δ ∆,[ ]∈t γ Γ,[ ]∈

∪⎝ ⎠⎜ ⎟⎜ ⎟⎛ ⎞

.∪=

198


BYCHENKOV

4.2. Stationary Algorithms

The generalized Arrow–Hurwitz algorithm

(4.2)

is the classical example of a stationary algorithm for solving saddle point problems. In (4.2), (uk, pk)Ú is thekth approximation (k ∈ �); (u0, p0)Ú is a fixed initial approximation; and α > 0, τ > 0 are fixed iterationparameters. A definitive result concerning the optimal convergence of this algorithm was obtained in [7].

With the exception of λ = 1, the spectrum of the transition operator T in (4.2) is identical to the spectrumof the pencil

(see [7]). Here, L ≡ A–1/2QA–1/2 and G ≡ A–1/2BC–1B*A–1/2. Let us write χ(λ) in form (1.1). To this end, define

With δ1 = ∆–1, δ2 = δ–1, γ1 = γ, and γ2 = Γ, assumptions (1.2) and (1.3) and all the conditions in Corollary 2are fulfilled. Therefore, we have

where λ1 is a root of the equation f(λ, s) = 0 (that is, λ1 = 1 – τ/s) and are the roots of the quadraticequation

Finally, changing the variable s s–1, we have the following bound:

Note that this bound is much simpler and more compact than the bound derived in [7]; however, it is notbetter than the latter bound.

5. CONCLUSIONS

In this paper, an approach to the analysis of the spectral characteristics of an operator pencil of specialform (1.1) under assumptions (1.2) and (1.3) was developed. A general upper bound was obtained for thespectrum of the pencil χ(λ) (see Theorem 2). Sharp estimates for the spectrum (Corollary 1) and the spectralradius (Corollary 2) were derived in practically important cases. These results can be used to analyze newalgorithms for solving saddle point problems and to refine the characteristics of the existing algorithms.

ACKNOWLEDGMENTS

This work was supported by the Russian Foundation for Basic Research, project no. 05-01-00511.

REFERENCES

1. M. Benzi, G. H. Golub, and J. Liesen, “Numerical Solution of Saddle Point Problems,” Acta Numer. 14, 1–137(2005).

Quk 1+ uk–

τ--------------------- Auk B pk+ + f ,=

αCpk 1+ pk–

τ----------------------– B*uk 1++ g,=

χ λ( ) λ2L λ 2L τI– τ2/αG–( )– L τI–( ),+=

f λ s,( ) 1 λ–( )s τ– , g λ t,( ) 1 λ– , h λ( ) λτ2/α.≡ ≡ ≡

ρ T( ) λλ σ T( )∈max≡ λ

λ σ χ( )\ 1{ }∈max ρ χ( ) λ1 λ2

1 2,,{ },s δ1 δ2,[ ]∈

t γ 1 γ 2,[ ]∈

max≤= =

λ21 2,

λ2s λ 2s τ– τ2t/α–( )– s τ–+ 0.=

ρ T( ) 1 τs– 1 τθ– θ2 ts/α–±,{ }, θ s 1 τt/α+( )/2.≡s δ ∆,[ ]∈t γ Γ,[ ]∈

max≤



2. W. Zulehner, “Analysis of Iterative Methods for Saddle Point Problems: A Unified Approach,” Math. Comput. 71(238), 479–505 (2002).

3. E. V. Chizhonkov, “On a Generalized Relaxation Method for Linear Problems with a Saddle Operator,” Mat. Mod-elir. 13 (12), 107–114 (2001).

4. E. V. Chizhonkov, “Improving the Convergence of the Lanczos Method in Solving Algebraic Saddle Point Prob-lems,” Zh. Vychisl. Mat. Mat. Fiz. 42, 504–513 (2002) [Comput. Math. Math. Phys. 42, 483–491 (2002)].

5. M. V. Gorelova and E. V. Chizhonkov, “On the Solution of Saddle Point Problems by the Methods Based on ModelSaddle Operators on the Upper Layer,” Izv. Vyssh. Uchebn. Zaved., Mat., No. 8, 19–27 (2003).

6. P. Henrici, Applied and Computational Complex Analysis, Vol. 1 (Wiley, New York, 1988).

7. Yu. V. Bychenkov, “Optimization of One Class of Nonsymmetrizable Algorithms for Saddle Point Problems,” Rus.J. Numer. Analys. Math. Modeling 17 (6), 521–546 (2002).

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On the spectral properties of an operator pencil