Adjoint pairs of differential-algebraic equations and Hamiltonian systems

ctableof thepairs ofharac-erlying

e

Applied Numerical Mathematics 53 (2005) 131–148www.elsevier.com/locate/apnum

Adjoint pairs of differential-algebraic equationsand Hamiltonian systems✩

Katalin Ballaa,∗, Vu Hoang Linhb

a Computer and Automation Research Institute, Hungarian Academy of Sciences, H-1518 Budapest P.O. Box 63, Hungaryb Faculty of Mathematics, Mechanics and Informatics, Vietnam National University, Hanoi, 334 Nguyen Trai Str., Vietnam

Abstract

We consider linear homogeneous differential-algebraic equationsA(Dx)′ + Bx = 0 and their adjoints−D∗(A∗x)′+B∗x = 0 with well-matched leading coefficients in parallel. Assuming that the equations are trawith index less than or equal to 2, we give a criterion ensuring the inherent ordinary differential equationspair to be adjoint each to other. We describe the basis pairs in the invariant subspaces that yield adjointessentially underlying ordinary differential equations. For a class of formally self-adjoint equations, we cterize the boundary conditions that lead to self-adjoint boundary value problems for the essentially undHamiltonian systems. 2004 IMACS. Published by Elsevier B.V. All rights reserved.

Keywords: Differential-algebraic equations; Adjoint pairs of differential-algebraic equations; Self-adjoint boundary valuproblems

1. Introduction

Recently, differential-algebraic equations (DAEs) of the form

A(Dx)′ + Bx = f, (1)

✩ This work was supported by OTKA (Hung. National Sci. Foundation) Grants # T043276, T031807.* Corresponding author.

E-mail addresses: [email protected] (K. Balla), [email protected] (V.H. Linh).

0168-9274/$30.00 2004 IMACS. Published by Elsevier B.V. All rights reserved.doi:10.1016/j.apnum.2004.08.015

132 K. Balla, V.H. Linh / Applied Numerical Mathematics 53 (2005) 131–148

ascon-

f ordi-ODEsmoge-n willhe onet may

5] thatealing

ationf adjointin [9].

all theto [4,5].

ODEs.th (1) andrmally)s, we

point

onsg wecients,

where A,D,B :C([a, b],L(Cm)), f :C([a, b],Cm), were introduced in [4] and their analysis w

launched. A motivation to considering this class of implicit equations was that under very mildditions the formal adjoint equation

−D∗(A∗x∗)′ + B∗x∗ = g, (2)

g :C([a, b],Cm), shares the basic properties with (1). In the paper, we intend to study two kinds o

nary differential equations (ODEs) that are derived from (1) and (2) in parallel and called inherent(INHODEs) and essentially underlying ODEs (EUODEs). Assuming that Eqs. (1) and (2) are honeous, we focus on the relationship within the INHODE pairs and EUODE pairs. Special attentiobe paid to a particular subclass of (formally) self-adjoint homogeneous equations different from tanalyzed in [1] and similar to one in [7,6,3]. For this class, we formulate boundary conditions thabe considered as ones giving to rise to a (formally) self-adjoint BVP.

The results of the paper rely strongly upon the analysis in [4,5]. We adopt the notions from [differ slightly from those in [4]. Some of the ideas used here were inspired by constructions in [1] dwith a special class of (formally) self-adjoint equations

iD∗(Dx)′ + Bx = 0, with B = B∗. (3)

Our general results if applied to (3) coincide with those proved in [1]. If specialized into the equexamined in [3], the statements here are complementary to those in [3]. Some special cases oDAE pairs with their specially chosen EUODEs appear in [8] and the results were implementedOur results cover these special cases, too.

The outline of the paper is as follows. For the convenience of the reader in Section 2 we recnotions and notations related to next sections. For an explanation of the relevance, we referThe basic results concerning the notions will be used here freely. Section 3 is addressed to INHSection 4 deals with the construction of the bases and pairs of bases in subspaces connected wi(2). Section 5 derives the EUODEs and the pairs of EUODEs. Section 6 defines a class of (foself-adjoint DAEs. In terms of the boundary conditions posed for these (formally) self-adjoint DAEformulate a condition sufficient to give a self-adjoint BVP for the EUODE. Finally, in Section 7 weout the practical importance of associated ODEs.

2. Preliminaries

In the paper, the source functionsf,g occurring in the general linear differential-algebraic equatiof the form (1) and (2) will not be involved into the analysis. Therefore, without further mentioninwill assume thatf = 0, g = 0, i.e., the equations are homogeneous. As to the remaining coeffidefined onI := [a, b], first we assume that the leading coefficientsA andD are “well-matched”, i.e.they fulfill as followsCondition C1: For eacht ∈ I,

kerA(t) ⊕ imD(t) = Cm, (4)

and there exist functionsa1, . . . , am−r , d1, . . . , dr ∈ C1(I,Cm) such that for allt ∈ [a, b],

kerA(t) = span{a (t), . . . , a (t)

}, imD(t) = span

{d (t), . . . , d (t)

}. (5)

K. Balla, V.H. Linh / Applied Numerical Mathematics 53 (2005) 131–148 133

s

With Eq. (1), the following chain is associated:

G0 := AD, B0 := B;For i = 0,1, Qi,Pi,Wi ∈ C(I,L(Cm)):

Q2i = Qi, W 2

i = Wi,

Ni := kerGi = imQi, Pi = I − Qi,

kerWi = imGi,

Gi+1 := Gi + BiQi, Bi+1 = BiPi,

Si := {z :I �→ Cm, Biz ∈ imGi}

= kerWiBi. (6)

Similarly, Eq. (2) gives rise to the chain

G∗0 := −D∗A∗, B∗0 := B∗;For i = 0,1, Q∗i , P∗i ,W∗i ∈ C(I,L(Cm)):

Q2∗i = Q∗i , W 2

∗i = W∗i ,

N∗i := kerG∗i = imQ∗i , P∗i = I − Q∗i ,

kerW∗i = imG∗i ,

G∗i+1 := G∗i + B∗iQ∗i , B∗i+1 = B∗iP∗i ,

S∗i := {z :I �→ Cm, B∗iz ∈ imG∗i}

= kerW∗iB∗i . (7)

To characterize DAEs we need a further assumption on smoothness:Condition C2:

dimD(t)S1(t) = const= :ρ and dimD(t)N1(t) = const= :ν, (8)

and there exist functionssD1 , . . . , sD

ρ , nD1 , . . . , nD

ν ∈ C1(I,Cm) such that for allt ∈ I,

D(t)S1(t) = span{sD

1 (t), . . . , sDρ (t)

}, D(t)N1(t) = span

{nD

1 (t), . . . , nDν (t)

}.

Let conditions C1 and C2 be valid. Eq. (1) is said to beanindex-0 tractable DAE if

N0(t) = {0}, t ∈ I, (9)

anindex-1 tractable DAE if

N0(t) �= {0}, (10)

N0(t) ∩ S0(t) = {0}, t ∈ I, (11)

anindex-2 tractable DAE if

dimN0(t) ∩ S0(t) = const> 0, (12)

N (t) ∩ S (t) = {0}, t ∈ I. (13)

134 K. Balla, V.H. Linh / Applied Numerical Mathematics 53 (2005) 131–148

e

pecial

-then

r(2).

Denote the reflexive generalized inverses (RGIs) ofD andA∗ by D− andA∗− if defined by

D−D = P0, A∗−A∗ = P∗0,

DD− = R, A∗A∗− = R∗.While chain (6) depends on projectorsQ0,Q1, the index does not. If (1) has an indexµ, µ ∈ {0,1,2},then (2) is also tractable with indexµ and vice versa.

In index-1 case, projector functionQ0 (P0) is called canonical and it is denoted byQ0c (P0c), ifkerQ0 = S0 (imP0c = S0). In index-2 case, the subspaceN1 depends on the special choice ofQ0 (P0)while S1 does not. ProjectorQ1 is marked by if kerQ1 = S1 holds. All terms in chain (6) derived by thuse ofQ1 are marked by , too. In index-2 case, canonical projector functionQ0c ontoN0 is defined as

Q0c = Q0P1G−12 B + Q0Q1D

−(DQ1D

−)′D,

whereQ0 is an arbitrary projector function ontoN0; Q0c does not depend on the special choice ofQ0.All terms in chain (6) derived by the use ofQ0c are marked with subscriptc, too.Q1c = Q1 holds. Notethat if P1 = I is set, then we get the formula valid for index-1 case. Therefore, there is no need for snotation segregating the index-1 and index 2 canonical projectorsQ0c.

In index-2 case, decompositionD(t)S1(t) ⊕ D(t)N1(t) ⊕ kerA(t) = Cm induces projector func

tions DP1D−, DQ1D

−, I − R ∈ C1(I,L(Cm)) onto the subspaces in the decomposition alongother couple. Similarly, projectorsA∗P∗1A

∗−, A∗Q∗1A∗− and I − R∗ correspond to decompositio

A∗(t)S∗1(t) ⊕ A∗(t)N∗1(t) ⊕ kerD∗(t) = Cm and identity(DP1D

−)∗ = A∗P∗1A∗− holds.

For an arbitraryV ∈ C(I,L(Cm)), C1V denotes function space{v ∈ C(I,C

m): V v ∈ C1(I,Cm)}. With

Eq. (1) we associate operator

L :C1D → C1

DQ1G−12

, Lx := A(Dx)′ + Bx, x ∈ C1D,

and similarly, Eq. (2) is related to operator

L∗ :C1A∗ → C1

A∗Q∗1G−1∗2

, L∗x∗ := −D∗(A∗x∗)′ + B∗x∗, x∗ ∈ C1A∗ .

A function x ∈ C1D is called the solution if it satisfies (1) pointwise. For eacht ∈ I, solutions to (1) form

a linear subspaceSµ(t) ⊆ Cm,

Sµ(t) := imΠcanµ(t),

Πcan 0= I, Πcan1= P0c, Πcan2= P0cP1c = P0cP1 = UP0P1,

here

U = I − Q0P1G−12 B − Q0P1D

−(DQ1D

−)′D,

U is invertible,Q0 is an arbitrary projector function ontoN0. Solutionsx∗ ∈ C1A∗ and canonical projecto

functions for (2) are introduced in an analogous way with changes corresponding to the chain forThorough the paper we use the notational convention: IfH ∈ L(Cn) is invertible, thenH−∗ := (H−1)∗,

if H is not invertible thenH− is a (fixed) RGI,H−∗ := (H−)∗; in the latter case,H−∗ = H ∗− does nothold, in general. IfH : I → L(Cn), thenH−(t) := [H(t)]−, H ∗(t) := [H(t)]∗.

3. Inherent ODEs for adjoint pairs

We remind that the equations under consideration are homogeneous.

K. Balla, V.H. Linh / Applied Numerical Mathematics 53 (2005) 131–148 135

they areint pair

teness,

re con-

ies

be

3.1. Index-0 equations

Now, Condition C1 combined with (9) ensures that bothA andD are invertible. Letu := Dx, u∗ :=A∗x∗. Equations

u′ + A−1BD−1u = 0, (14)

−u′∗ + D∗−1B∗A∗−1u∗ = 0, (15)

are called inherent ODEs (INHODEs) associated with (1) and (2), respectively. On the one hand,equivalent to equations from which they are derived. On the other hand, (14) and (15) form an adjoof ODEs. Trivially,x ∈ C1

D (x∗ ∈ C1A∗ ) is a solution of (1) ((2)) if and only ifDx (A∗x∗) is a solution of (14)

((15)). Practically, we do not treat equations of index-0 anymore, we included them for compleonly.

3.2. Index-1 equations

In index-1 case the canonical projectors and the arbitrary ones belonging to pair (1) and (2) anected by the following proposition.

Lemma 1. Let (1) (and/or (2)) be an index-1 DAE and let Q0 and Q∗0 be arbitrary projector functionsonto N0 and N∗0, respectively. Then, P0c = −(D∗A∗G−1

∗1 )∗ and P∗0c = (ADG−11 )∗.

Proof. The representationQ0c = G∗−1∗1 Q∗

∗0G∗∗1 is a simple consequence of the trivial identit

kerQ0G−11 B = kerW0B andQ∗

∗0G1 = G∗∗1Q0. Indeed,G∗−1

∗1 Q∗∗0G

∗∗1 = G∗−1

∗1 Q∗∗0B = Q0G

−11 B. We im-

mediately getP0c = I − G∗−1∗1 Q∗

∗0G∗∗1 = G∗−1

∗1 P ∗∗0G

∗∗1 = −G∗−1

∗1 AD. The second representation canchecked in a similar way. �

The lemma results in identitiesQ∗∗0cBP0c = 0 andQ∗

0cB∗P∗0c = 0, which, in turn, yieldBP0c =

P ∗∗0cBP0c = (P ∗

0cB∗P∗0c)

∗ = (B∗P∗0c)∗ = P ∗

∗0cB. Therefore, (1) can be rewritten as

0 = A(Dx)′ + BP0cx + BQ0cx = A(Dx)′ + P ∗∗0cBP0cx + Q∗

∗0cBQ0cx. (16)

Take the projections onto imQ∗∗0c and kerQ∗

∗0c and use RGIsA∗−c ,D−

c : P∗0c = A∗−c A∗, P0c = D−

c D andA∗−∗

c A = R = DD−c , R ∈ C1(I,L(Cm)):

Q∗∗0cBQ0cx = 0, (17)

(Dx)′ − R′Dx + A∗−∗c BD−

c Dx = 0. (18)

A particular case of the above lemma is representationQ∗0c = G−∗1c Q∗

0cG∗1c. Thus, Q∗

∗0c(BQ0c) =(G−∗

1c Q∗0cG

∗1c)

∗G1cQ0c = G1cQ0c, and therefore,

Q0cx = 0,

is an equivalent to (17). Setu = DP0cx = Dx and get the INHODE for (1) from (18):

u′ − R′u + A∗−∗c BD−

c u = 0. (19)

The next “invariance” gives importance to the INHODE:

136 K. Balla, V.H. Linh / Applied Numerical Mathematics 53 (2005) 131–148

es

on theoriginaluely

ker

-

(I1) If u(t0) ∈ imD(t0) holds for an arbitraryt0, thenu ∈ imD.

Indeed,A∗−c (I −R∗) = 0 and (19) involve that functionw := (I −R)u satisfies the ODEw′ = (I −R)′w,

therefore,w(t0) = 0 for an arbitraryt0 yieldsw = 0.The similar procedure applied to (2) with the same RGIs yields the decomposition

Q∗0cx∗ = 0,

−(A∗x∗

)′ + R∗ ′A∗x∗ + D−∗c B∗A∗−

c A∗x∗ = 0, (20)

while settingu∗ = A∗P∗0cx∗ = A∗x∗ results in the inherent ODE for (2)

−u′∗ + R∗′u∗ + D−∗

c B∗A∗−c u∗ = 0. (21)

Now, the invariance of the inherent ODE reads as

(I1∗) If u∗(t0) ∈ imA∗(t0) holds for an arbitraryt0, thenu∗ ∈ imA∗.

Let us return to arbitrary projector functionsQ0 andQ∗0. Due to the properties of RGIs, the identiti

A∗−c = P∗0cA

∗−c R∗ = P∗0cA

∗−c R∗R∗ = P∗0cA

∗−c

(A∗A∗−)(

DD−)∗

= P∗0c

(A∗−

c A∗)A∗−(DD−)∗ = P∗0cP∗0cA

∗−(DD−)∗ = P∗0cA

∗−(DD−)∗

,

D−c = P0cD

−c R = P0cD

−c DD− = P 2

0cD− = P0cD

−,

are valid. Therefore,

A∗−∗c BD−

c = DD−A∗−∗P ∗∗0cBP0cD

− = DD−A∗−∗P ∗∗0cBD− = DD−A∗−∗G1P0G

−11 BD−

= DD−A∗−∗ADG−11 BD− = DD−RDG−1

1 BD− = DG−11 BD−,

and the inherent equation turns to be identical to that in [4]. Similarly,

D−∗c B∗A∗−

c = A∗G−1∗1 B∗A∗−.

The new forms (19) and (21) of INHODEs show transparently that these ODEs depend onlygeometric characteristics of the problem: Eqs. (19) and (21) contain terms dependent on thedata, only: projector functionsR,P0c,P∗0c are defined uniquely by them and RGIs are induced uniqby the latter.

Furthermore, the following statement becomes transparent.

Theorem 2. In the index-1 case, the adjoint of the inherent ODE (19) of the DAE (1) coincides with theinherent ODE (21) of the adjoint DAE (2) if and only if neither imD(t) nor kerA(t) depend on t .

Proof. Let imD and imA∗ be constant subspaces. Then, so are their orthogonal complements,D∗and kerA. Then,R′ = 0 and vice versa. �3.3. Index-2 equations

In index-2 case, we use again canonical projectorsQ0c, Q1, Q∗0c, Q∗1, Πcan2, Π∗can 2and the decompositions relying upon them. In the analysis we also utilize the identityQ = Q , i.e., Q = Q P

K. Balla, V.H. Linh / Applied Numerical Mathematics 53 (2005) 131–148 137

,st

(23) is

rary

ndent

and its counterpart marked by∗. We take the projections of the equations onto imG2cΠcan 2G−12c =

imADP1G−12c and imG2c(I − Πcan2)G

−12c = im G2c(Q0c + P0cQ1)G

−12c . In the solutionx partΠcan2x =

P0cP1x is segregated. TermsDP1x = DP0cP1x = DP1D−c Dx andDQ1x = DQ1D

−c Dx are differen-

tiable for x ∈ C1D and DQ0cP1x = 0, so we may split their sumDx. Note that due to condition C1

kerAD = kerD holds. RelationG−12c A = P1D

−c also will be useful. After partial differentiation, the fir

projection simplifies to(DP1x

)′ − (DP1D

−c

)′DP1x − (

DP1D−c

)′DQ1x + DP1G

−12c BD−

c DP1D−c DP1x = 0. (22)

We undertake the second projection to scaling byG−12c that yields

0 = Q0cP1D−c (Dx)′ + (

Q0c + P0cQ1)G−1

2c Bx

= −Q0cQ1D−c (Dx)′ + Q0c

(P1 + Q1

)G−1

2c Bx + P0cQ1x

= −Q0cQ1D−c (Dx)′ − Q0cQ1D

−c

(DQ1D

−c

)′Dx

+ (Q0cQ1D

−c

(DQ1D

−c

)′D + Q0cP1G

−12c B

)x + Q1x

= −Q0cQ1D−c

(DQ1x

)′ + Q0cx + Q1x. (23)

A consequence of (23) is that the projections of the right side also vanish. We obtain that in fact,equivalent to a system of algebraic equations:

P0cQ1x = 0, (24)

Q0cx + Q1x = 0. (25)

The hidden constraint now sits inside the system, it becomes transparent if one returns to an arbitQ0.On the other hand, if (24) is taken into account, (22) yields the INHODE foru := DP1x

u′ − (DP1D

−c

)′u + DP1G

−12c BD−

c DP1D−c u = 0. (26)

This is the same equation we got in [4]. Indeed, the multiplierDP1D−c of u in the last term could be

inserted in [4]. Subscriptc was not present there. However, it was shown that the equation is indepeof the choice ofQ0.

Now, we check thatDP1G−12c BD−

c DP1D−c = DP1D

−c A∗−∗

c BD−c DP1D

−c .

DP1G−12c BD−

c DP1D−c − DP1D

−c A∗−∗

c BD−c DP1D

−c

= DP1D−c

(D − A∗−∗

c G2c

)G−1

2c BD−c DP1D

−c

= −DP1D−c A∗−∗

c

[BQ0c

(P1 + Q1

)G−1

2c BD−c DP1D

−c + BP0cQ1G

−12c BD−

c DP1D−c

]= −DP1D

−c A∗−∗

c

[BQ0cP1G

−12c BD−

c DP1D−c + BQ0cQ1P0cP1D

−c + BP0cQ1P0cP1D

−c

]= −DP1D

−c A∗−∗

c B(Q0c − Q0cQ1D

−c

(DQ1D

−c

)′D

)P0cP1D

−c

= −DP1D−c A∗−∗

c BQ0cQ1D−c

(DQ1D

−c

)′DP0cP1D

−c

= +DP1D−c A∗−∗

c ADQ1D−c

(DQ1D

−c

)′DP0cP1D

−c

= +DP1Q1D−c

(DQ1D

−c

)′DP0cP1D

−c = 0.

Thus, the transparent form of (26) is

u′ − (DP D−)′

u + (DP D−)

A∗−∗BD−(DP D−)

u = 0. (27)

138 K. Balla, V.H. Linh / Applied Numerical Mathematics 53 (2005) 131–148

nDE

ment-

n

erent

he ad-ll. The

We can treat the adjoint equation (2) in a quite similar way. The final result foru∗ = A∗P∗1x∗ is

−u′∗ + (

A∗P∗1A∗−c

)′u∗ + (

A∗P∗1A∗−c

)D−∗

c B∗A∗−c

(A∗P∗1A

∗−c

)u∗ = 0, (28)

while the constrains are

P∗0cQ∗1x∗ = 0, Q∗0cx∗ + Q∗1x∗ = 0. (29)

Recall thatDP1D−c = DP1D

− andA∗P∗1A∗−c = A∗P∗1A

∗−. Eq. (27) has the “invariance” property:

(I2) If u(t0) ∈ imD(t0)P1(t0) is valid for an arbitraryt0, thenu ∈ imDP1,

while for (28) the next claim holds:

(I2∗) If u∗(t0) ∈ imA∗(t0)P∗1(t0) is valid for an arbitraryt0, thenu∗ ∈ imA∗P∗1.

The proof is similar to the index-1 case, projectorR is replaced byDP1D−.

Relation (DP1G−12c BD−

c DP1D−c )∗ = A∗P∗1A

∗−c D−∗

c B∗A∗−c A∗P∗1A

∗−c connecting the last terms i

(27) and (28) calls for comparing the adjoint of the INHODE (27) of the DAE (1) with the INHO(28) of the adjoint DAE (2). The next Theorem gives the trivial answer.

Theorem 3. In the index-2 case the adjoint of the inherent ODE (27) of the DAE (1) coincides with theinherent ODE (28) of the adjoint DAE (2) if and only if projector DP1D

− is constant.

Proof. We again recall that(A∗P∗1A∗−)∗ = DP1D

−. �We formulate this result in terms of the basic subspaces.

Theorem 4. In the index-2 case the adjoint of the inherent ODE (27) of the DAE (1) coincides with theinherent ODE (28)of the adjoint DAE (2) if and only if neither the subspace D(t)S1(t) nor the subspaceA∗(t)S∗1(t) depend on t .

Proof. Let D(t)S1(t) and A∗(t)S∗1(t) be constant subspaces. Then, the orthogonal compleA∗(t)N∗1(t) ⊕ kerD∗(t) of the first one does not depend ont either. It means that neither the image nor the kernel of projectorA∗(t)P∗1(t)A

∗−(t) depend ont , i.e.,A∗(t)P∗1(t)A∗−(t) ≡ const. Then,

D(t)P1(t)D−(t) is constant, as well.

In the opposite direction: Let projector functionDP1D− and together with it, projector functio

A∗P∗1A∗− be constant. Then, their images,DS1 andA∗S∗1 are constant. �

Remark 5. In the proof we referred to identitiesA∗S∗1 = (DN1 ⊕ kerA)⊥ and DS1 = (A∗N∗1 ⊕kerD∗)⊥. They allow to formulate the above results in terms of one equation, either (1) or (2):

In the index-2 case, the adjoint of the inherent ODE (27) of the DAE (1) coincides with the inhODE (28) of the adjoint DAE (2) if and only if the subspace pairsD(t)S1(t), D(t)N1(t) ⊕ kerA(t) orA∗(t)S1(t), A∗(t)N1(t) ⊕ kerD∗(t) are constant.

Remark 6. There is another way to show the connection between the last terms in INHODEs of tjoint pairs of DAEs. We provide it since the method used here is applicable in later material, as we

K. Balla, V.H. Linh / Applied Numerical Mathematics 53 (2005) 131–148 139

tructions upon

es of

oher

sformanges

case,atrix

ce

method is based on an appropriate decomposition of the coefficient matrices and the explicit consof matrix chains for the adjoint pair (1), (2). In fact, the technique used here as well as in [2] reliethe pointwise reduction to a matrix pair equivalent to the original pair(AD,B).

Let us fix t = t0 and omit the argument. We begin with a decomposition resulting block matricspecial form

G0 = AD = U

I 0 0

0 0 0

0 0 0

V, B0 = B = U

B11 B12 0

B21 0 0

0 0 B33

V,

whereU andV are nonsingular. The blocksizes in the decompositions ofG0 andB0 correspond each tother. Let the diagonal blocks be of dimensionn1, n2, n3, respectively. One always may achieve eitn3 = 0 (no third block-row and no block-column appear) orB33 be nonsingular. The special casen2 = 0indicates that the matrix pair has index 1. (The decomposition trivially exists: first one should tranG0 to the indicated form, say, by a forward-backward Gauss elimination with row and column chif necessary and then by a next transformation that makes the appropriate blocks inB0 to vanish anddoes not affect the pattern ofG0). For brevity, in the next considerations, we assume the generaln2 �= 0, n3 �= 0 (andB33 is nonsingular). The special cases may be treated in a similar way. A mchain corresponding to the scheme in (6) can be constructed explicitly: Set

Q0 = V −1

0 0 0

0 I 0

0 0 I

V, thenP0 = V −1

I 0 0

0 0 0

0 0 0

V.

We may choose

W0 = U

0 0 0

0 I 0

0 0 I

U−1.

An elementary computation yields

G1 = G0 + B0Q0 = U

I B12 0

0 0 0

0 0 B33

V, B1 = B0P0 = U

B11 0 0

B21 0 0

0 0 0

V,

N1 = kerG1 =z =

z1

z2

z3

∈ C

m, z1 + B12z2 = z3 = 0, wherez = V z

.

S1 =z =

z1

z2

z3

∈ C

m, B21z1 = 0, wherez = V z

.

By definition in Section 2, if the index exists and it equals 2 thenN1 ∩ S1 = {0}. One can verify directlythat this is the case if and only if detB21B12 �= 0. Let this assumption hold. For brevity, let us introduthe notations

C = B (B B )−1B , M = (B B )−1B .

140 K. Balla, V.H. Linh / Applied Numerical Mathematics 53 (2005) 131–148

unt

If we choose

Q1 = V −1

C 0 0

−M 0 0

0 0 0

V,

thenQ1 is the projector ontoN1 alongS1. We compute

P1 = V −1

I − C 0 0

M I 0

0 0 I

V, G2 = G1 + B1Q1 = U

I + B11C B12 0

B21 0 0

0 0 B33

V.

Elementary calculations give us

G−12 B = V −1

(I − C)B11(I − C) + C 0 0

M[B11(I − C) − I ] I 0

0 0 I

V.

Let us denote the expressionDP1G−12 BD− by H . SinceD andR were not decomposed, we leaveD and

D− in the formula forH . However, the structure ofH still becomes transparent if we take into accothat nowP1 = P1 and, therefore,DP1 = DP0P1 = DP0P1P0 = DP1P0, i.e.,

H = DP1P0G−12 BD− = DV −1

(I − C)B11(I − C) 0 0

0 0 0

0 0 0

V D−.

Now, let us take the adjoint toH . If we make use of the definitionsA∗A∗− = D−∗D∗ = R∗, D∗R∗ =D∗,R∗A∗ = A∗ and the decomposition ofD∗A∗ = (AD)∗, we get

H ∗ = D−∗V ∗ (I − C∗)B∗

11(I − C∗) 0 0

0 0 0

0 0 0

V −∗D∗A∗A∗−

= D−∗V ∗ (I − C∗)B∗

11(I − C∗) 0 0

0 0 0

0 0 0

U ∗A∗−. (30)

Let H∗ = A∗P∗1G−1∗2 B∗A∗−. Similarly to (30), we obtain

H∗ = A∗U−∗ (I − C∗)B∗

11(I − C∗) 0 0

0 0 0

0 0 0

U ∗A∗−

= D−∗D∗A∗U−∗ (I − C∗)B∗

11(I − C∗) 0 0

0 0 0

0 0 0

U ∗A∗− = H ∗. (31)

It remains to recall thatH andH∗ were shown to be independent of the choice ofP0 andP∗0 [4] andH = HDP D− andH = H A∗P A∗−. By this the proof is completed.

K. Balla, V.H. Linh / Applied Numerical Mathematics 53 (2005) 131–148 141

es. Sup-by the

sedn

ntoc-

4. Invariant subspaces and their bases

This section deals with the existence and the construction of smooth bases in certain subspacpose that for eacht , E(t) ⊆ C

n is a subspace of the same dimension and assume that it is definedvalues of a continuously differentiable projector functionP , that is,E(t) = imP(t), t ∈ I. The existenceof a continuously differentiable basis function forE was proven in [10]. Based on the idea propoin [1], we describe a construction that results in a basis functionT satisfying a prescribed normalizatiocondition.

First, letk denote dimE , k � n andR an arbitrary continuously differentiable projector function oE(t), i.e., imR(t) = imP(t) = E(t), t ∈ I. Dissimilarly to [1], hereR is not necessarily an orthoprojetor.

Let Q ∈ L(Cn),S ∈ L(Ck) and rankQ � k = rankS . Assume that either bothQ andS are Hermitianor both of them are skew-Hermitian.

Additionally, we assume thatv ∈ E(t) impliesQ∗v ∈ E∗(t) := imR∗(t). Obviously, ifR is an ortho-projector, i.e.,R = R∗, the simplest matrix pair that satisfies the assumptions isQ = In, S = Ik, whereI�, � ∈ N, is the identity matrix of dimension�.

Let T ∈ C1(I,L(Ck,Cn)) be the solution to the initial value problem

T ′ = R′T , T (t0) = T0, (32)

where the columns ofT0 form a basis inE(t0) = imR(t0) = imP(t0) for some fixedt0 ∈ I.

Lemma 7. If T ∈ C1(I,L(Ck,Cn)) is the solution to the initial value problem (32), then, for each t

imT (t) = E(t). Moreover, if

T ∗0 QT0 = S, (33)

holds, then

T (t)∗QT (t) = S, (34)

is valid for all t ∈ I.

Proof. For the first part of the Lemma defineV = RT − T . Then,

V ′ = R′T +RT ′ − T ′ = RR′T .

SinceR2 = R, we haveRR′ = R′ −R′R. Hence,

V ′ = (R′ −R′R

)T = R′T −R′(V + T ) = −R′V.

SinceV(t0) = 0, we haveV(t) = 0 at anyt , i.e., the columns ofT belong toE .Since the columns ofT (t0) are linearly independent, the linear independence of the columns ofT (t)

for eacht follows from the elementary ODE theory. This completes the proof of the first part.For the second part, assume that (33) holds with some appropriate pair(Q,S) and for eacht , v ∈ E(t)

impliesQ∗v ∈ E∗(t) = imR∗. We derivedRT = T , it involvesRT ′ = 0. Then,

T ∗QT ′ = (Q∗T

)∗T ′ = (R∗Q∗T

)∗T ′ = T ∗QRT ′ = 0. (35)

Finally, whenQ is Hermitian, we have(T ∗QT

)′ = T ∗QT ′ + (T ∗QT ′)∗ = 0.

142 K. Balla, V.H. Linh / Applied Numerical Mathematics 53 (2005) 131–148

c-r-es (33).ction

ex-1al”

n

ODEiquely,

WhenQ is skew-Hermitian,(T ∗QT

)′ = T ∗QT ′ − (T ∗QT ′)∗ = 0.

Therefore, in both cases

T (t)∗QT (t) ≡ T (0)∗QT (0) = S. �In the next section, we may apply Lemma 7 in its simplest form by settingR = R in the index-1 case

andR= DP1D− in the index-2 case. Or, in the index-1 case we may set

R= RR+ and (Q,S) = (Im, Ir),

while in the index-2 case

R= DP1D−(

DP1D−)+

and (Q,S) = (Im, Iρ).

Here,+ denotes the Moore–Penrose generalized inverse. Clearly,R defined as above is an orthoprojetor. Proceeding from an arbitrary basis in imD(t0) or imD(t0)P1(t0), respectively, the Gram–Schmidt othonormalization procedure provides us the required orthonormal initial value, i.e., one that satisfiConsequently, the solution of the initial value problem (32) yields a continuously differentiable funthe values of which form an orthonormal basis in imR(t) = imD(t) or D(t)S1(t) = imD(t)P1(t)D

−(t),respectively for eacht ∈ I.

5. Essentially underlying ODEs for adjoint pairs

In this section, we work with the adjoint pair of DAE-s of index-2, only. The result for the indcase follows directly, by settingP1 = P∗1 = Im. We recall the INHODEs (27) and (28) for the originequation (1) and its adjoint (2). We also refer to the properties (I2) and (I2∗) of the “invariant subspacesimDP1D

− and imA∗P∗1A∗− and recall that dim imDP1D

− = dim imA∗P∗1A∗− = ρ.

Let u ∈ imDP1D− be a solution of (27) andT ∈ C1(I,L(Cρ,C

m)) an arbitrary basis function iimDP1D

−. Then there existsv :I → Cρ such thatu = T v. Moreover,v ∈ C1(I,C

ρ), since

v = Iρv = (T ∗T

)−1T ∗T v = [(

T ∗T)−1

T ∗]T v = [(T ∗T

)−1T ∗]u,

and both terms in the latter product are differentiable. Partial differentiation and (27) yields

v′ = [(T ∗T

)−1T ∗]′

u + [(T ∗T

)−1T ∗]u′

= [(T ∗T

)−1T ∗]′

T v + [(T ∗T

)−1T ∗][(DP1D

−)′ − (DP1D

−)A∗−∗

c BD−c

(DP1D

−)]T v.

It is worth noting that(T ∗T )−1T ∗ = T +. SinceDP1D−T = T , we may simplify to get

v′ + T +(DP1D

−)(T ′ + A∗−∗

c BD−c T

)v = 0, (36)

Eq. (36) is called the essentially underlying ODE (EUODE) for Eq. (1). It is a lower dimensionalthan (27) is. However, unlike (1), it is not uniquely defined since the basis function is not defined uneither.

In a similar way, we could derive the EUODE for (2): Setu∗ = T∗v∗ whereu∗ ∈ imA∗P∗1A∗− is the

solution of (28) andT∗ is a smooth basis function in imA∗P∗1A∗−. The result is

−v′∗ + T +

∗(A∗P∗1A

∗−)(−T ′∗ + D−∗

c B∗A∗−c T∗

)v∗ = 0. (37)

K. Balla, V.H. Linh / Applied Numerical Mathematics 53 (2005) 131–148 143

ows in

dnctions

f

heir

Corollary 8 (of Theorems 2 and 3). Let DP1D− be constant. Then, there exist constant basis functions

T and T∗ in DS1 and A∗S∗1, respectively, and with them, the EUODE pair takes the form

v′ + T +(DP1D

−)A∗−∗

c BD−c T v = 0, (38)

−v′∗ + T +

∗(A∗P∗1A

∗−)D−∗

c B∗A∗−c T∗v∗ = 0. (39)

Now our aim is to find appropriate basis pairs allowing us to see the connection between the flgeneral case. That is we look for specific pairs of EUODEs for pair (1), (2).

Lemma 9. Let T ∈ C1(I,L(Cρ,Cm)) be an arbitrary function such that

imT (t) = imD(t)P1(t)D−(t), t ∈ I,

and E ∈ L(Cρ) be invertible.There exists a unique function T∗ ∈ C1(I,L(Cρ,C

m)) such that

imT∗(t) = imA∗(t)P∗1(t)A∗−(t), t ∈ I,

and

T ∗∗ (t)T (t) ≡ E, t ∈ I.

Remark 10. The existence of a functionT ∈ C1(I,L(Cρ,Cm)) with the required property is ensure

by Lemma 7. However, the construction of Lemma 7 does not exhaust the set of all basis fuapplicable in Lemma 9.

Proof. Take an arbitrary smooth matrix functionS such that the columns ofS(t) constitute a basis okerD(t)P1(t)D

−(t). SinceDP1D− is a projector function, we have

imD(t)P1(t)D−(t) ⊕ kerD(t)P1(t)D

−(t) = Cm, t ∈ I.

Hence, the columns ofT (t) andS(t) form a basis ofCm. Therefore, there exists a unique solutionT∗ tothe system of equations{

T ∗∗ T = E,

T ∗∗ S = 0.(40)

This solution is obviously smooth. The second equation means that the columns ofT∗(t) belong to(kerD(t)P1(t)D

−(t))⊥ = im

(D(t)P1(t)D

−(t))∗ = imA∗(t)P∗1(t)A

∗−(t).

On the other hand, sinceE is nonsingular, the first equation implies that

rankT∗(t) = rankT (t) = dim(imD(t)P1(t)D

−(t))

= dim(imA∗(t)P∗1(t)A

∗−(t)) = ρ.

Consequently, the columns ofT∗(t) form a basis of imA∗(t)P∗1(t)A∗−(t). The solutionT∗ of (40) de-

pends purely onT (and E). Indeed, if S �= S, and imS = kerD(t)P1(t)D−(t), then S = Sη, η(t) ∈

L(Cρ), η(t) is nonsingular. The corresponding system is equivalent to (40).�Obviously, a lemma similar to Lemma 9 is valid ifT andT∗ and the relevant subspaces change t

role.

144 K. Balla, V.H. Linh / Applied Numerical Mathematics 53 (2005) 131–148

be

to

is the

Now, let us apply Lemma 9 and for the givenT find the associatedT∗ such that

T ∗∗ (t)T (t) ≡ T ∗(t)T∗(t) ≡ Iρ (41)

hold. We show that with this choice ofT∗, Eqs. (36) and (37) are adjoint each to other.First, we check thatT∗ = A∗P∗1A

∗−T (T ∗T )−1.Indeed, letV := A∗P∗1A

∗−T (T ∗T )−1. Obviously,V = T∗η with someη. Further,

η = T ∗T∗η = T ∗V = T ∗A∗P∗1A∗−T

(T ∗T

)−1

= (DP1D

−T)∗

T(T ∗T

)−1 = T ∗T(T ∗T

)−1 = Iρ.

Analogously, one can verify thatT = DP1D−T∗(T ∗∗ T∗)−1 holds. Thus, the pair (36) and (37) may

rewritten as follows

v′ + T ∗∗ T ′v + T ∗

∗(A∗−∗

c BD−c

)T v = 0, (42)

−v′∗ − T ∗T ′

∗v∗ + T ∗(D−∗c B∗A∗−

c

)T∗v∗ = 0. (43)

Finally, note that(T ∗∗ T ′)∗ = −T ∗T ′∗ sinceT ∗T∗ = Iρ. In order to formulate the result, it remainsrecall that we could proceed from a fixedT∗ and use its counterpartT uniquely defined by (41).

Theorem 11. Let (T ,T∗) be a continuously differentiable basis pair satisfying (41). Then, the adjoint ofthe EUODE (36) of the DAE (1) coincides with the EUODE (37) of the adjoint DAE (2).

Example 12. Consider the adjoint pair of index-2 DAE-s of upper Hessenberg form{x ′

1 + B11x1 + B12x2 = 0,

B21x1 = 0,(44)

and {−x ′∗1 + B∗

11x∗1 + B∗21x∗2 = 0,

B∗12x∗1 = 0.

(45)

Here, one may set

A = D =(

I 0

0 0

), B =

(B11 B12

B21 0

).

Matrix chains for (44) and its adjoint (45) can be constructed similarly to those in Remark 6. (Thiscase withn3 = 0.) The index-2 property of (44)–(45) is equivalent to the condition detB21B12 �= 0. Weobtain the INHODE for (44)

u′ −(

(I − C)′ 0

0 0

)u +

((I − C)B11 0

0 0

)u = 0, (46)

whereC = B12(B21B12)−1B21 and

DP1D− =

(I − C 0

0 0

), u =

((I − C)x1

0

).

Choose an arbitrary smooth basisT = (T0

)in imDP1, T ∈ im(I − C), and the correspondingT∗ = (

T∗0

)in imA∗P∗1, T∗ ∈ im(I − C∗). The EUODE of (44) is of the form

v′ + (T ∗T ′ + T ∗B T

)v = 0, (47)

K. Balla, V.H. Linh / Applied Numerical Mathematics 53 (2005) 131–148 145

e basisewhat:

-e

t

e

wherev is defined by(I − C)x1 = T v.Similarly, we have the EUODE of (45)

−v′∗ + (−T ∗T ′

∗ + T ∗B∗11T∗

)v∗ = 0, (48)

wherev∗ is defined by(I − C∗)x∗1 = T∗v∗.Eqs. (47) and (48) illustrate the claim in Theorem 11. Furthermore, one can observe that

im(I − C) = kerB21, ker(I − C) = imB12,

im(I − C∗) = kerB∗12, ker(I − C∗) = imB∗

21.

That is, the EUODE pairs in [8] are particular cases of (47) and (48). Here, we do not require thT to be orthonormalized. Moreover, the smoothness condition of the coefficients is relaxed somfunctionsB12 andB21 are not necessarily continuously differentiable, but functionC, only.

6. Self-adjoint DAEs and self-adjoint BVPs

In this section, we consider the special DAE

DTJ (Dx)′ + Bx = 0, (49)

with real coefficientsD,J,B, the superscriptT denotes transposition. HereJ is a constant skewsymmetric matrix, the leading pair(DTJ,D) is well matched,B is symmetric. If not confusing, thargumentt is omitted.

Without loss of generality, we may assume that eitherJ = J2n, m = 2n or J = diag(J2n,0m−2n),1� n, 2n < m, where

J2n =(

0 −In

In 0

),

and 0m−2n is a zero matrix of dimensionm − 2n.In the second case, the condition C1 gives

D =(

D1 D2

0 0

), D1(t) ∈ L

(R

2n), D2(t) ∈ L

(R

m−2n,R2n

).

Indeed, letJ be the given arbitrary skew-symmetric constant matrix and the pair(D TJ , D) be well-matched. It follows from [12] that there exists a decompositionJ = U TJ U with nonsingular constanU . Then, dim kerJ = m − 2n and forD := UD, kerDTJ ⊇ kerJ , that is, imD ∩ kerJ = {0} holds, i.e.,in, the second caseD(t) has the lastm − 2n rows filled with zeros, only and dim imD � 2n.

With the notation of the previous sections, hereA = DTJ . SinceAT = (DTJ )T = −JD, we immedi-ately haveC1

D = C1AT and for eachx ∈ C1

D, DTJ (Dx)′ + Bx and−DT((DTJ )Tx)′ + BTx coincide.We show now, that imDP1 is of even dimension, i.e.,ρ = imDP1 = 2l, with somel � n.Note that one may choose projectorsQ0 andQ∗0 to be equal in chains (6) and (7). Then,P∗1 = P1.

This leads to

ATP∗1AT− = −JDP1D

−J. (50)

Let T be an arbitrary smooth basis functionT ∈ C1(I,L(Rρ,Rm)) in DS1. Due to Lemma 9, ther

exists a unique basis functionT ∈ C1(I,L(Rρ,Rm)) in ATS such thatT T(t)T (t) = I , while (50)

146 K. Balla, V.H. Linh / Applied Numerical Mathematics 53 (2005) 131–148

d

ult in

gives imT∗ ⊆ imJ T . Since dim imT∗ = dim imT , there exists an invertible functionω such thatT∗ =J T ω. Thus,ωTT TJ TT = Iρ , or, equivalently,T TJ T = ω, with a differentiable, skew-symmetric aninvertibleω. If the rank of a skew-symmetric real matrix is odd, then it is singular [12]. Thus rankω(t)

is even,ρ = 2l with somel � n.It follows from [12] and the eigenstructure ofJ , that there exists a nonsingularU0 ∈ L(R2l)

such thatT T(0)J T (0) = UT0 J2lU0. Then, T (t) = T (t)U−1

0 is also a differentiable basis inDS1, withT T(0)J T (0) = J2l .

Let T be a smooth basis inDS1 constructed by the help of the IVP (32) withR= DP1D− and setting

Q = J,S = J2l in (33). Clearly, the assumptions of Lemma 7 are fulfilled. Hence, we haveT TJT ′ = 0(see the proof of Lemma 7) andT T(t)JT (t) ≡ const= T T(0)JT (0) = J2l .

With a basisT indicated above, we derive the EUODE (36) for (49). In order to have the resa transparent form, we shall obtain the same equation in a different form. Recall again thatAT−T

c =−JD−T

c . We could representv asv = Iρv = (T TJT )−1(T TJT )v = J−12l T TJu. Therefore,

J2lv′ = (

T TJu)′ = T T′Ju + T TJu′

= T T′JT v + T TJ[(

DP1D−)′

T + (DP1D

−)JD−T

c BD−c

(DP1D

−)T

]v

= {(T TJT ′)T + T TJ

[(DP1D

−)T

]′ − T TJ(DP1D

−)T ′

+ T TJ(DP1D

−)JD−T

c BD−c

(DP1D

−)T

}v. (51)

HereT TJT ′ = 0. Due to(DP1D−)T = T , the second term also vanishes. Since imJT ⊆ ATS∗1, we

have

T TJ(DP1D

−)T ′ = [−(

ATP1AT−)

JT]T

T ′ = (−JT )TT ′ = T TJT ′ = 0.

In order to simplify the last term, additionally we take into account that imT = DS1 ⊆ imD, thus thelastm − 2n rows inT (t) vanish. ThereforeJ 2T = −ImT = −T .

T TJDP1D−JD−T

c BD−c

(DP1D

−)T v

= [−(ATP1A

T−)JT

]TJD−T

c BD−c T v

= T TJ 2D−Tc BD−

c T v = −T TD−Tc BD−

c T v.

The final result is

J2lv′ + Fv = 0, F = T TD−T

c BD−c T = F T. (52)

This equation is self-adjoint.Now, we look for boundary conditions for (49) that yield a self-adjoint BVP for (52).

Theorem 13. Let Ka,Kb ∈ L(Rm) be given. The boundary condition

KTa D(a)x(a) + KT

b D(b)x(b) = 0, (53)

for Eq. (1) induces a self-adjoint boundary condition for (52) if and only if the conditions

KTa JKa = KT

b JKb, (54)

imKa ⊂ JD(a)S1(a), imKb ⊂ JD(b)S1(b), (55)

rank(KT

a ,KTb

) = dimDS1, (56)

hold.

K. Balla, V.H. Linh / Applied Numerical Mathematics 53 (2005) 131–148 147

self-

showses orditionsmoreof the

tion toE are

ered.decou-thod)

problem

case,

initial

cidesy the

Proof. Let Ka,Kb satisfy the assumptions. It means that

imKa ⊂ JD(a)S1(a) = imJT (a) = imJ TT (a)J2l .

Therefore, there existsM ∈ L(R2l) such that

Ka = J TT (a)J2lM, (57)

whereT is the basis functions as above. Similarly, there existsN ∈ L(R2l) such that

Kb = J TT (b)J2lN. (58)

If x is a solution of (1), thenDx = T v wherev satisfies (52). Condition (53) turns into

MTv(a) + NTv(b) = 0, (59)

while the relation (54) takes the form

MTJ2lM = NTJ2lN. (60)

The condition (56) means that rank(KTa Ka +KT

b Kb) = 2l. In terms ofM,N we obtain that, i.e.,(MT,NT)

is of full rank 2l. We recall [11, Theorem 3.2, Chapter 11], that the boundary condition (59) yields aadjoint boundary value problem for (52) if and only if (60) is valid and(MT,NT) is of full rank.

In the opposite direction: If (52) and (60) form a self-adjoint BVP, then one can defineKa,Kb by (57)and (58). They satisfy the assumptions.�

7. Numerical consequences

The decoupling of the linear DAE into the INHODE and the algebraic constraints in Section 3that the dynamic properties of DAE are defined by the INHODE. Any type of conditions, initial valuboundary conditions may be imposed only on the components involved into the INHODE. The conshould allow to exist a solution within the invariant subspace of the INHODE. Theoretically, atransparent picture of the freedom in imposing conditions arises if one turns to the descriptiondynamics in minimal coordinates that is to an EUODE. Practically, however, neither the reducan EUODE nor that to the INHODE is desirable. Nevertheless, the stability properties of the DAinherited from the INHODE [13,14], or, respectively, from an EUODE [8,9].

In [13,14], numerical integration of DAEs (IVPs) with well matched leading terms is considA particular case is the linear equation (1). One of the questions concerned there is whether thepling of the DAE and the discretization (by a stiffly accurate Runge–Kutta method or a BDF mecommute. In other words, the question is whether the numerical method properly reproduces thedynamics. It is stated [13, Proposition 3] that in the index-1 case property imD = const is sufficient forthat. Therefore, DAEs of index-1 with this property are called numerically qualified. In the index-2a sufficient condition isDS1 = const andDN1 = const [14, Lemma 7].

In [5], two-point boundary value problems for index-1 and index-2 DAEs (1) are reduced tovalue problems for DAE (2). Clearly, an index-1 DAE (2) is numerically qualified if imA∗ = const. Ifcompared with Theorem 2, we conclude that provided the INHODE of the index-1 DAE (1) coinwith the adjoint to the INHODE of the DAE (2), solutions of both the IVP and the BVP (solved bhelp of the adjoint equation) are properly reproduced numerically and vice versa.

148 K. Balla, V.H. Linh / Applied Numerical Mathematics 53 (2005) 131–148

ulta-

d then,mbiningif and

entialsl. Mat.

ass ofl. Mat.

timalät II der

ndun-

.ontrol

nstitute

t DAE

tware., J. Com-

pis

(2003)

(2003)

Computation in Example 12 shows that for an index-2 Hessenberg system,DS1 = const andDN1 =const hold if and only if kerB21 = const and imB12 = const. These two subspaces are constant simneously with their orthogonal complements. It means thatA∗S∗1 = const andA∗N∗1 = const. Thus, onone hand, an index-2 Hessenberg system is numerically qualified if and only if so is its adjoint, anthe numerical boundary value problem also well reproduces the dynamics. On the other hand, cothis with Theorem 4, we can state that an index-2 Hessenberg DAE (1) is numerically qualifiedonly if its INHODE coincides with the adjoint to the INHODE of the DAE (2).

Acknowledgement

The authors thank the referees offering suggestions for the improvement of the paper.

References

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