A Relaxation of the Strangeness Index

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Technical report from Automatic Control at Linköpings universitet

A relaxation of the strangeness index

Henrik Tidefelt, Torkel Glad

Division of Automatic Control

E-mail: tidefelt@isy.liu.se, torkel@isy.liu.se

22nd February 2010

Report no.: LiTH-ISY-R-2932

Address:

Department of Electrical Engineering Linköpings universitet

SE-581 83 Linköping, Sweden

WWW: http://www.control.isy.liu.se

AUTOMATIC CONTROL REGLERTEKNIK LINKÖPINGS UNIVERSITET

Technical reports from the Automatic Control group in Linköping are available from http://www.control.isy.liu.se/publications.

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Abstract

A new index closely related to the strangeness index of a differential-algebraic equation is presented. Basic properties of the strangeness index are shown to be valid also for the new index. The definition of the new index is conceptually simpler than that of the strangeness index, hence making it potentially better suited for both practical applications and theoretical developments.

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1 Introduction

Kunkel and Mehrmann have developed a theory for analysis and numerical solution of differential-algebraic equations. The theory centers around the strangeness index, which differs from the differentiation index in that it does not consider the derivatives of the solution to be independent of the solution itself at each time instant. Instead, it takes the tangent space of the manifold of solutions into account, thereby reducing the number of dimensions in which the derivative has to be determined. The book Kunkel and Mehrmann (2006) covers the theory well and will be the predominant reference used in the paper.

The numerical solution procedure applies to general nonlinear differential-algebraic equations of higher in-dices, and is currently the only one we know of that can handle such problems, let be that it does not provide a sensitivity analysis. Our interest in this matter is mostly due to this capability.

The present paper presents highlights from the corresponding chapter in the first author’s thesis Tidefelt (2009).

2 Two definitions

In this section, two index definitions will be presented along with some basic properties of each. The one to be presented first is the strangeness index, found in Kunkel and Mehrmann (2006). The second, which is proposed as an alternative, is called thesimplified strangeness index. Both are based on the derivative array equations.

2.1 Derivative array equations and the strangeness index

As always when working with dae, it is crucial to be aware that the solutions are restricted to a manifold. In practice, one is interested in obtaining equations describing that manifold, and the way this is done in the present paper is by using the derivative array introduced in Campbell (1993).

Consider the dae

f ( x(t), x0(t), t )= 0! (1)

Assuming sufficient differentiability of f and of x, the idea is that the original dae is completed with derivatives of the equations with respect to time. This will introduce higher order derivatives of the solution, but the key idea is that, given values of x(t), it suffices to be able to determine x0

(t) in order to compute a numerical solution to the equations. That is, higher order derivatives such as x00

(t) may appear in the equations, but are not necessary to determine.

If the completion procedure is continued until the derivative array equations areone-full with respect to x0(t), the procedure has revealed the differentiation index of the dae. The meaning of one-full is defined in terms of the equation considered pointwise in time, so that a variable and its derivative become independent variables. We emphasize the independence by using the variable ˙x(t) instead of x0(t), where the dot is just an ornament, while the prime is an operator. The equations are then said to beone-full if they determine ˙x(t) uniquely within some open ball, given x(t) and t. An equivalent characterization can be made in terms of the Jacobian of the derivative array with respect to its differentiated variables; then the equations are one-full if and only if row operations can bring the Jacobian into block diagonal form, with a non-singular block in the block column corresponding to derivatives with respect to ˙x(t) (clearly, this shows that it is possible to solve for ˙x(t) without knowing the variables corresponding to higher order derivatives of x at time t).

However, instead of requiring that the completed equations be one-full, it turns out that there are good reasons for using the weaker requirement that the equations display thestrangeness index instead. The definition of strangeness index will soon be considered in detail. It turns out that equations displaying the strangeness index determine x0

(t) uniquely if one takes into account the connection between x(t) and x0

(t) being imposed by the non-differential constraints which locally describe the solution manifold. Strangeness-free equations (strangeness index 0) are suitable for numerical integration. (Kunkel and Mehrmann, 1996)

In the sequel, it will be convenient to speak of properties which hold on non-empty open balls inside the set L_ν S 4 = t, x, ˙x, . . . , ˙xνS+1 _{: F} νS( x, ˙x, . . . , ˙x (νS+1)_{, t )}_{= 0}! (2)

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2.1 Definition (Strangeness index). Thestrangeness index νS at ( t0, x0) is defined as the smallest number (or ∞ if no such number exists) such that the derivative array equations1

FνS( t, x(t), x

0

(t), x0(2)(t), . . . , x0(νS+1)_{(t) )}_{= 0}! ₍₃₎

satisfy the following properties on L_ν S \ n t, x, ˙x, . . . , ˙xνS+1 _{: t ∈ B} t0(δ) ∧ x ∈ Bx0(δ) o | {z } bδ for some δ > 0.

• P1a–[2.1] There shall exist a constant number nasuch that the rank of MνS( t, x, ˙x, . . . , ˙x (νS+1)₎₌4 ∂F_νS( t, x, ˙x, ..., ˙x(νS+1)) ∂ ˙x . . . ∂F_νS( t, x, ˙x, ..., ˙x(νS+1)) ∂ ˙x(νS+1)

is pointwise equal to ( νS+ 1 ) nx−na, and shall exist a smooth matrix-valued function Z2with napointwise linearly independent columns such that ZT

2MνS = 0.

• P1b–[2.1] Let nd= nx−na, and let AνS = Z T 2NνSwhere NνS( t, x, ˙x, . . . , ˙x (νS+1)₎₌4 ∂FνS( t x, ˙x, . . . , ˙x (νS+1)₎ ∂x

Then the rank of AνS shall equal na, and there shall exist a smooth matrix-valued function X with nd

pointwise linearly independent columns such that AνSX = 0.

• P1c–[2.1] The rank of ∇2f X shall be full, and there shall exist a smooth matrix-valued function Z1with ndpointwise linearly independent columns such that Z1T∇2f X is non-singular.

Assume that the strangeness index is finite. Then the dimension of the solution manifold is nd = nx−na. P1b–[2.1] then states that it is possible to construct a local coordinate map x(t) = φ−1( xd(t), t ) with coordi-nates xd, determined by the partial differential equation

∇₁_φ−1_{( x}_d_{(t), t )}_{= X( x(t), t )}! ₍₄₎ where the columns of X are smooth functions and pointwise linearly independent. The columns of X can be selected as an orthonormal basis for the right null space of the matrix A, and the local coordinates xdare denoted thedynamic variables.

The last property, P1c–[2.1], is finally there to ensure that the time derivative of the local coordinates on the solution manifold are determined by the original equation (1). Replacing (1) by an equation with residual expressed only through the dynamic variables,

fd( xd, ˙xd, t ) 4 = fφ−1( xd, t ), ∇1φ −₁ ( xd, t ) ˙xd+ ∇2φ −₁ ( xd, t ), t (5) property P1c–[2.1] states that the Jacobian with respect to ˙xd,

∇₂_f_d_{( x}_d_{, ˙x}_d_{, t ) = ∇}₂_f_φ−1_{( x}_d_{, t ), ∇}₁_φ−1_{( x}_d_{, t ) ˙x}_d_{+ ∇}₂_φ−1_{( x}_d_{, t ), t}_X ₍₆₎ is full-rank. Since there are only ndderivatives to be determined, and there are nxequations, there are na more equations than unknowns. The property P1c–[2.1] also states that nd linear combinations, given by the columns of Z1, of the equations in (1) can be chosen smoothly and linearly independent (and hence or-thonormal), so that these linear combinations are sufficient to determine the time derivatives of the dynamic variables.

We now end this section with a lemma that will be useful later. 2.2 Lemma. If it is known that νS≥ν, and the matrixˆ

h

Nνˆ Mνˆ i

does not have full row rank, then νS= ∞. Proof: Let i ≥ ˆν. The upper part ofhNi Mi

i

equalshNνˆ Mνˆ 0 i

, sohNi Mi

i

cannot have full row rank. It follows that P1b–[2.1] cannot be satisfied for νS= i.

1_{The notation is defined such that x}0₍₁₎

= x0.

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2.2 The simplified strangeness index

By discretizing the derivatives (using a bdf method) in the original equation (1) (and scaling the equations by the step length), we get that the gradient of these equations with respect to x tends to ∇2f ( x, ˙x, t ) as the step length tends to zero. Hence, joining these equations with the full derivative array equations (where no derivatives are discretized) yields a set of equations which (locally) shall determine x uniquely. This leads to the following definition.

2.3 Definition (Simplified strangeness-index). Thesimplified strangeness index νq at ( t0, x0) is defined as the smallest number (or ∞ if no such number exists) such that the derivative array equations

Fνq( t, x(t), x

0

(t), x0(2)(t), . . . , x0(νq+1)_{(t) )}_{= 0}! ₍₇₎

satisfy the following property on L_ν q \ n t, x, ˙x, . . . , ˙xνq+1 _{: t ∈ B} t0(δ) ∧ x ∈ Bx0(δ) o | {z } bδ for some δ > 0. • P2–[2.3] Let Hνq 4 =            ∂f ( x, ˙x, t ) ∂ ˙x 0 . . . 0 ∂F_νq( t, x, ˙x, ..., ˙x(νq+1)) ∂x ∂F_νq( t, x, ˙x, ..., ˙x(νq+1)) ∂ ˙x . . . ∂F_νq( t, x, ˙x, ..., ˙x(νq+1)) ∂ ˙x(νq+1)            =        ∇₂_f ₀ Nνq Mνq       

where Nνq and Mνq are defined as in definition 2.1. Then it shall hold that

rank          I ₀ ∇₂_f ₀ Nνq Mνq          ! = rank"∇_N2f 0 νq Mνq #

That is, the basis vectors corresponding to x shall be in the span of the rows of Hνq, which may be

recog-nized as the property of Hνqbeing one-full.

The property P2–[2.3] can be interpreted as that there is no freedom in the x components of the solution to h f ( x, 1 h( x − q −₁ x ), t ) Fνq( t, x, ˙x, . . . , ˙x (νq+1)₎ ! ! = 0

since adding additional equations for the x variables alone does not decrease the solution space of the lin-earized equations. For theoretic considerations, however, the continuous-time interpretation provided by lemma 4.4 below is more relevant.

Of course, we must show what the simplified strangeness index is for the inevitable pendulum. Example 1

Let us consider the following familiar model of a pendulum.

f                                  ξ u y v λ                 ,                  ˙ ξ ˙ u ˙y ˙v ˙ λ                  , t                  4 =                  λ ξ − ˙u λ y − g − ˙v ξ2+ y2−₁ ˙ ξ − u ˙y − v                 

We consider initial conditions where the pendulum is in motion and neither x nor y is zero.

To check P2–[2.3] we look at the projection of a basis for the right null space of Hionto the space spanned by

the basis vectors corresponding to x, for i = 1, 2, . . . , νq. (The projection is implemented by just keeping the

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five first entries of the vectors.) The basis for the null space is computed usingMathematica , and for i = 0, 1, 2 the projected basis vectors are, in order,

                                 0 0 0 0 0                 ,                  0 0 0 0 −1 y                                                                      0 0 0 0 0                 ,                   0 0 0 0 ξ ˙y ξ− ˙ξ y                   ,                   0 0 0 0 −_y ˙y ξ− ˙ξ y                                                                       0 0 0 0 0                 ,                 0 0 0 0 0                 ,                 0 0 0 0 0                                 

Assuming that the symbolic null space computations are valid in some neighborhood of the initial conditions inside Li, it is seen that the λ component is undetermined for i = 0 and i = 1, and as all components are

determined for i = 2 we get νq = 2.

3 Relations

In this section, the two indices νSand νqwill be shown to be closely related. This is done by means of a matrix decomposition developed for this purpose. We first show the matrix decomposition, and then interpret the two definitions in terms of this decomposition.

3.1 Lemma. The matrix

h N Mi

where N ∈ Rk×l, M ∈ Rk×k, rank M = k − a, a ≥ 1, can be decomposed as

h N Mi=hQ1,1 Q1,2 i " 0 0 Σ 0 A 0 0 0 #                     QT 3,1 0 QT 3,2 0 Σ−1QT 1,1N QT2,1 0 QT 2,2                    

In this decomposition, the left matrix is unitary, as are the diagonal blocks of the right matrix. The matrix Σ is a diagonal matrix of the non-zero singular values of M. The matrix A is square.

3.2 Theorem. Definition 2.1 and definition 2.3 satisfy the relation νS≥νq.

Proof: Suppose that the strangeness index is νS and finite, as the infinite case is trivial. Let the matrices N and M in lemma 3.1 correspond to MνSand NνSas in definition 2.1.

First, let us consider νSin view of this decomposition. The left null space of M is spanned by Q1,2, and making these linear combinations of N results in

QT 1,2N = h A 0 0 0i              QT 3,1 QT 3,2 Σ−1QT 1,1N 0              =hA 0i"Q T 3,1 QT 3,2 #

where A has full rank due to P1b–[2.1]. This matrix determines the tangent space of the non-differential constraints as being its null space, spanned by the independent columns of Q3,2. Hence, we can parameterize x as x = Q3,2xd.

Turning to νq, we follow the constructive interpretation of P2–[2.3] in section 2.2. The right null space of h

N Miis spanned by the second and fourth rows of the right factor in the decomposition; h N Mi x y ! ! = 0 ⇐⇒ ∃_z₁_{, z}₂_: x y ! ="Q3,2 0 0 Q2,2 # z1 z2 ! (8)

Extracting the part of this equation which only involves x, we find that it can be parameterized in z1alone, and since the columns of Q3,2are independent, we can use z1as dynamic variables; x = Q3,2xd.

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Since the strangeness index is νS, ∇2f Q3,2has full column rank according to P1c–[2.1]. Hence, "∇₂_f ₀ N M # x y ! ! = 0 ⇐⇒ ∃_z₂_: x y ! = 0 Q2,2z2 ! (9)

which is exactly the condition captured by P2–[2.3]. Since νq is the smallest index such that this condition is satisfied, it is no greater than νS.

3.3 Theorem. Definition 2.1 and definition 2.3 satisfy the relation νS ∈ n

νq, ∞ o

, with νS = νq if and only if νq= ∞ or the following property holds.

• P1–[3.3] The matrixhNνq Mνq

i

has full row rank on the set Lνq∩bδin definition 2.3. That is,

rankhNνq Mνq

i

= ( νq+ 1 ) nx (10)

Proof: In view of theorem 3.2, the statement is trivial in the case νq = ∞. Hence, consider νq < ∞, in which case it shall to be shown that νS = νq when P1–[3.3] holds, and νS = ∞ otherwise. The latter case follows immediately from lemma 2.2, so it remains to consider the case when P1–[3.3] holds.

Due to theorem 3.2 it suffices to show νS≤νq. Let the matrices N and M in lemma 3.1 correspond to Mνqand

Nνqas in definition 2.3. The rank condition (10) implies that A in lemma 3.1 is non-singular.

Consider (8) and (9). Since adding the equation ∇2f x= 0 is sufficient to conclude x = 0 given x = Q! 3,2z1, it is seen that ∇2f Q3,2z1 = 0 must imply z! 1= 0. This is only true if ∇2f Q3,2 has full column rank, which shows that P1c–[2.1] holds. Since νSis the smallest index such that this condition is satisfied, it is no greater than νq.

4 Uniqueness and existence of solutions

The present section gives a result corresponding to what Kunkel and Mehrmann (2006, theorem 4.13) states for the strangeness index. As the difference between the two index definitions is basically a matter of whether P1–[3.3] is required or not, the main ideas in Kunkel and Mehrmann (2006) apply here as well.

4.1 Lemma. If the simplified strangeness index νqis finite, there exist matrix functions Z1, Z2, X, similar to those in definition 2.1. They are all smooth with pointwise linearly independent columns, satisfying

ZT

2Mνq = 0 and columns of Z2span left null space of Mνq (11a)

ZT

2NνqX = 0 and columns of X span right null space of Z T

2Nνq (11b)

ZT

1∇2f X is non-singular (11c)

Proof: Using the decomposition of lemma 3.1, we may take Z2 B Q1,2 and X = Q3,2. As in the proof of theorem 3.3, (8) and (9) then imply that ∇2f X has full column rank, and the existence of Z1follows.

Multiplying the relations in (11) by smooth pointwise non-singular matrix functions shows that the matrix functions Z1, Z2, X are not unique, but they can be replaced by any smooth matrices with columns spanning the same linear spaces. For numerical purposes, the smooth Gram-Schmidt orthonormalization procedure may be used to obtain matrices with good numerical properties, while the theoretical argument of the present section benefits from another choice, to be derived next.

Select the non-singular constant matrix P =hPd Pa i

such that ZT

2NνqPais non-singular in a neighborhood of

the initial conditions, and make a change of the un-dotted variables in Lνq according to

x =hPd Pa i x_d xa ! (12) 5

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The following notation will turn out to be convenient later (note that Nνaqis non-singular) Nνdq 4 = ZT 2NνqPd N a νq 4 = ZT 2NνqPa (13)

The next result corresponds to Kunkel and Mehrmann (2006, corollary 4.10) for the strangeness index. 4.2 Lemma. There exists a smooth function R such that

xa= R( xd, t ) (14)

inside Lνq, in a neighborhood of the initial conditions.

Proof: In Lνq it holds that Fνq= 0 and Z T

2Mνq= 0, and it follows that

∂ZT 2Fνq ∂ ˙x(1+) = Z T 2 ∂Fνq ∂ ˙x(1+)+ ∂ZT 2 ∂ ˙x(1+)Fνq = 0

Hence, the construction of Z2is such that Z2TFνqonly depends on t and x, and the change of variables (12) was

selected so that the part of the Jacobian corresponding to xais non-singular. It follows that xacan be expressed locally as a function of xdand t.

We now introduce the function φ−1to describe the local parameterization of x using the coordinates xdand t, x = φ−1( xd, t ) 4 = P xd R( xd, t ) ! (15) and the next lemma shows an important coupling between φ−1and lemma 4.1.

4.3 Lemma. The matrix X in lemma 4.1 can be chosen in the form ˆ X = P " I ∇₁_{R( x}_d_{, t )} # = ∇1φ −₁ ( xd, t ) (16)

Proof: Clearly, the columns are linearly independent and smooth. By verifying that the matrix is in the right null space of ZT

2Nνq we will show that its column spans the same linear space as X. It will then follow that X

and ˆX are related by a relation in the form

ˆ X = X W

for some smooth non-singular matrix function W . Using the form X W then shows that (11c) is also satisfied. Hence, it remains to show that ˆX is in the right null space of ZT

2Nνq.

Using (14) and allowing also the dotted variables ˙x(1+)_{to depend on x}

din (suppressing arguments) ∂ZT 2Fνq ∂xd = 0 it follows that ∂ZT 2 ∂xd Fνq+ Z T 2 ∂Fνq ∂xd +       ∂ZT 2 ∂xa Fνq+ Z T 2 ∂Fνq ∂xa       ∂xa ∂xd +       ∂ZT 2 ∂ ˙x(1+)Fνq+ Z T 2 ∂Fνq ∂ ˙x(1+)       ∂ ˙x(1+) ∂xd ! = 0 Here, Fνq= 0 and Z T 2 ∂F_νq ∂ ˙x(1+) = Z T 2Mνq= 0 implies that ZT 2 ∂Fνq ∂xd + ZT 2 ∂Fνq ∂xa ∇₁_{R = Z}T 2∇2Fνq h Pd Pa i " I ∇₁_R # = ZT 2NνqXˆ ! = 0

Back in section 2.2 it was indicated that we would be able to show that a finite simplified strangeness index implies local uniqueness of solutions. With lemma 3.1 at our disposal this statement can now be shown rather easily.

4.4 Lemma. If the simplified strangeness index is finite and x is a solution to the dae for some initial condi-tions in Lνq∩bδ, then the solution x is locally unique.

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Proof: Using the parameterization of x given by (15), it suffices to show that the coordinates x_dare uniquely defined. By the smoothness assumptions and the analytic implicit function theorem, Hörmander (1966), show-ing that x0_d(t) is uniquely determined given xd(t) and t will be sufficient, since then the corresponding ode will have a right hand side which is continuously differentiable, and hence locally Lipschitz on any compact set. One may then complete the argument by applying a basic local uniqueness theorem for ode, such as Codding-ton and Levinson (1985, theorem 2.2)).

Reusing (5) for the current context, x0_d(t) is seen to be uniquely determined if ∇2fd( xd, ˙xd, t ) is non-singular (in some neighborhood Lνq∩bδ of the initial conditions). Identifying (6) in (11c), lemma 4.3 completes the

proof.

With ˆX according to (16) it follows that ZT 2NνqX = Nˆ d νq+ N a νq∇1R ! = 0 (17)

using the notation (13). Before stating the main theorem of the section we derive one more equation. Using (14) and allowing also the dotted variables ˙x(1+)to depend on t in (suppressing arguments)

ZT 2 ∂Fνq ∂t ! = 0 it follows that ZT 2 ∇1Fνq+ ∇2Fνq∇2φ −₁ + Mνq ∂ ˙x(1+) ∂t ! = ZT 2 ∇₁_F_ν q+ ∇2Fνq∇2φ −₁ _! = 0 (18)

4.5 Theorem. Consider a sufficiently smooth dae (1), repeated here,

f ( x(t), x0(t), t )= 0! (1)

with finite simplified strangeness index νq and where the un-dotted variables in Lνq form a manifold of

di-mension nd. If the set where P2–[2.3] holds is the projection of a similar bδ+∩ Lνq+1, and P2–[2.3] also holds

on bδ+∩ Lνq+1with the same dimension nd, then there is a unique solution to (1) for any initial conditions in

bδ+∩ Lνq+1.

Proof: Considering how Fνq+1is obtained form Fνq, it is seen that the equality Fνq+1 = 0 can be written

∇₁_F_ν

q+ ∇2Fνq ˙x + ∇3+Fνq ˙x

(2+)_{= 0} Multiplying by ZT

2from the left and identifying the expressions for Nνqand Mνq, one obtains

ZT 2 ∇₁_F_ν q+ ∇2Fνq ˙x = 0 Using (18) and the change of variables (compare (12))

˙x =hPd Pa i ˙x_d

˙xa !

(19) leads to (using the notation introduced in (13))

hNd νq N a νq i −_P−1∇₂_φ−1₊ ˙xd ˙xa ! ! = 0 Using (15) and (17) yields

−_Na νq∇2R( xd, t ) − N a νq∇1R( xd, t ) ˙xd+ N a νq ˙xa= 0

and since Nνaqis non-singular, it must hold that

˙x = P ˙xd ˙xa ! = P _∇ I 1R( xd, t ) ! ˙xd+ P _∇ 0 2R( xd, t ) ! = ∇1φ −₁ ( xd, t ) ˙xd+ ∇2φ −₁ ( xd, t )

Since f ( x, ˙x, t )= 0 holds by definition on L! νq, it follows that

f ( φ−1( xd, t ), ∇1φ −₁ ( xd, t ) ˙xd+ ∇2φ −₁ ( xd, t ), t ) = 0 7

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where ˙xdis uniquely determined given xdand t by (11c) with ∇1φ −₁

= ˆX in place of X. Hence, the dae

f ( φ−1( xd(t), t ), ∇1φ −₁ ( xd(t), t ) x 0 d(t) + ∇2φ −₁ ( xd(t), t ), t ) = 0 has a (locally unique) solution and the trajectory generated by

x(t) = φ−1( xd(t), t ) is a solution to the original dae (1).

5 Conclusions and future work

In our view, a simpler way of computing the strangeness index has been proposed. While the original def-inition follows a three step procedure, the proposed defdef-inition has just one step. Once the index has been determined according to the new definition, it is known that the original definition leads to the same or an infinite index, and there is a simple test that distinguishes the two cases. The new index definition is also appealing due to its immediate interpretation from a numerical integration perspective.

Analogues of central results for the original strangeness index have been derived for the simplified strangeness index. In particular, it has been shown that a finite simplified strangeness index implies that if a solution exists, it will be unique, and existence of a solution can be established by checking the property that defines the index for two successive values of the index parameter.

An important aspect of the analysis of the strangeness index provided in Kunkel and Mehrmann (2006, chap-ter 4) is that the strangeness index is shown to be invariant under some transformations of the equations which are known to yield equivalent formulations of the same problem. It is an important topic for future research to find out whether the simplified strangeness index is also invariant under these transformations. Another interesting topic for future work is to seek examples where νq , νSin order to get a better understanding of this exceptional case.

6 Acknowledgment

The authors would like to acknowledge Ulf Jönsson at the Royal Institute of Technology, Sweden, for strength-ening theorem 3.3.

References

Stephen L. Campbell. Least squares completions for nonlinear differential algebraic equations. Numerische Mathematik, 65(1):77–94, December 1993.

Earl A. Coddington and Norman Levinson.Theory of ordinary differential equations. Robert E. Krieger Publish-ing Company, Inc., third edition, 1985.

Lars Hörmander. An introduction to complex analysis in several variables. The University Series in Higher Mathematics. D. Van Nostrand, Princeton, New Jersey, 1966.

Peter Kunkel and Volker Mehrmann. A new class of discretization methods for the solution of linear differential-algebraic equations with variable coefficients. SIAM Journal on Numerical Analysis, 33(5):2941– 1961, October 1996.

Peter Kunkel and Volker Mehrmann. Regular solutions of nonlinear differential-algebraic equations. Nu-merische Mathematik, 79(4):581–600, June 1998.

Peter Kunkel and Volker Mehrmann.Differential-algebraic equations, analysis and numerical solution. European Mathematical Society, 2006.

Henrik Tidefelt.Differential-algebraic equations and matrix-valued singular perturbation. PhD thesis, Linköping University, 2009.

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Avdelning, Institution Division, Department

Division of Automatic Control Department of Electrical Engineering

Datum Date 2010-02-22 Språk Language _{Svenska/Swedish} _{Engelska/English} Rapporttyp Report category _{Licentiatavhandling} _{Examensarbete} _C-uppsats _D-uppsats _{Övrig rapport}

URL för elektronisk version

http://www.control.isy.liu.se

ISBN — ISRN

—

Serietitel och serienummer Title of series, numbering

ISSN 1400-3902

LiTH-ISY-R-2932

Titel Title

A relaxation of the strangeness index

Författare Author

Henrik Tidefelt, Torkel Glad

Sammanfattning Abstract

A new index closely related to the strangeness index of a differential-algebraic equation is pre-sented. Basic properties of the strangeness index are shown to be valid also for the new index. The definition of the new index is conceptually simpler than that of the strangeness index, hence making it potentially better suited for both practical applications and theoretical developments.

Nyckelord