A well-posed shooting method for transferable DAE's

Numer. Math. 59, 815-829 (1991) Numerische Mathematik (~ Springer-Verlag 1991

A well-posed shooting method for transferable DAE's

Ren6 Lamour

Institut fiJr Angewandte Mathematik, Fachbereich Mathematik, Humboldt-Universit~it, PSF 1297, O-1086 Berlin, Federal Republic of Germany

Received September 15, 1989/Revised version received December 20, 1990

Summary. Consider a TPBVP for transferable nonlinear DAE's. In general the shooting equation has a singular Jacobian. A multiple shooting method which has a nonsingular Jacobian and also produces consistent initial values for the integration is presented. The estimation of the condition of the Jacobian shows the well-posedness of the method. Some illustrative examples are given

Mathematics Subject Classification (1991): 65L10

Consider the differential algebraic equation (DAE)

(1) f ( x ' ( t ) , x ( t ) , t ) = O, t E [a,b],

with the boundary condition

(2) g(x(a), x(b)) = O .

Problems of this kind arise in the description of multibody systems [F], semiconductor devices JAMS, A], electrical networks and other important fields.

The technique of shooting methods for explicit ODE's is well-known [D2] and lots of codes have been developed, too. However, in case of DAE's we are facing new difficulties.

[GM], [M] develop a general theory of shooting and difference methods for transferable BVP's in DAE's. The shooting equation consists of the boundary condition and the continuity conditions in the inner shooting points. For DAE's, this leads to a system of algebraic equations with a singular Jacobian. The numerical behaviour of this possibility was tested in [BN].

A further difficulty arises in particular for DAE's in shooting methods. The integration into every subinterval requires consistent initial values. The consistency of initial values for transferable DAE's is described by three sup- plementary equations [see (3.4) and (3.5)]. The combination of a shooting

816 Ren6 Lamour

equation with consistency conditions generates a shooting method with nonsingular Jacobian. [L] proposes a method with the double dimension as in case of ODE shooting methods.

Collocation methods for this type of equations are considered in [AMR] and [D1]. Shooting and difference methods for linear, solvable DAE's in the sense of [C], also with higher index, are treated in [CP] under the assumption that consistent initial values can be calculated and a stable integration method is available.

This paper aims at constructing an algorithm for solving a BVP in transferable nonlinear DAE's with nonsingular Jacobian and the same dimension as in the ODE case.

1. Preliminaries

The functions f and g of (1) and (2) fulfill the following assumptions:

A1.) f : / 3 / c F , n xF~ n x [a,b] ---~F.~ n f ' , fy exist and are continous

A2.) ker(f'y(y,x,t)) = N(t) V(y,x,t) E By rank(fy(y, x, t)) = r dim(N(t)) = n - r

A3.) Q(t) denotes a projection onto N(t) Q is smooth and P(t) := I - Q(t)

A4.) The matrix G(y, x, t) := fy(y, x, t) + f~x(Y, x, t)Q(t) is nonsingular V(y,x,t) E Bf (i.e. (1) is transferable)

A5.) g : Bg c ~;(n • ~ n --o M c ~n gtxa, gtxb exist and are continuous

im(g'a' g'b) = : M

Let x , be a solution of (1), (2). By X(t, s) we denote the fundamental matrix for which

a(t)" ' ' .= fy (x . (t), x . (t), t) ? !

(1.1) B(t) := fx(X,(t), x,(t), t) A(t)X'(t, s) + B(t)X(t, s) = 0 P (s)(X(s, s) - / ) = 0 .

In [GM] the representation

(1.2) X(t, s) = Ps(t) U(t, s)P (s),

is given for X, where U(t, s) denotes the solution of

U' = (P' Ps -- PG-1B)U (1.3)

U(s, s) = I ,

and Ps := I - QG-1B the so-called canonical projector. It is well known that (see [GM]) :

Theorem 1.1. The BVP (1), (2) is well-posed iff the shooting matrix

A well-posed shooting method for transferable DAE's

~' := fx, ,X(a,a) + g'bX(b,a) (1.4)

satisfies the condition

B"

817

ker(S') = N(a) , im(S') = M .

In the following let assumption ( B ) be fulfilled.

Theorem 1.1 ensures that

- x, is an isolated solution of (1), (2), - if we consider the IVP

(1.5) f(x '( t) , x(t), t) = 0 P ( s ) ( x ( s ) - z) = o

and denote the solution of (1.5) by x(t; s, z), then the IVP (1.5) has a unique solution at least in a neighbourhood

llP(s)(x.(s) - z)]l < 6

of a solution x..

Remark . The solution x depends only on the initial value P(s)z, i.e.

x ( t ; s , z ) = x ( t ; s , P ( s ) z ) .

Later, we shall also consider the linearization of (1), (2) in x.

A(t)x' (t) + B(t)x(t) = q(t)

Dlx(a) + D2x(b) = fl

D1 := fxa(X.(a),x.(b))

02 : = gtXb (x,(a), x,(b)) .

(1.6)

and

(1.7)

with

2. B a s i c t h e o r e m s

One characteristic of transferable DAE's

corresponds with the nonsingularity of ( Q

This is shown by multiplication of the matrices

is the nonsingularity of G. This

P ] whose inverse is given by B ' ]

p G - I QG -I ) "

818 Ren6 Lamour

) = ( Q + PPs 0 \ - A P G - 1 B + B P s (A+BQ)G - 1 /

with --APG-1B + BPs = --AG-1B + B- - BQG-1B = --B + B .

Lemma 2.1. Let the images of the two projectors P1, P2 c L(]R n) have equal dimensions, then there is a nonsingular matrix K so that

P2 = K-1P1K �9

Proof. Let dim(im(Pi)) =: r, then Pi has the representation

P i = Y i ( IrO O) Ti-1 ' i = 1 , 2 .

This assertion follows with

(2.2) K := T2T~ 1. []

We now consider a singular matrix V, and V- denotes a reflexive generalized inverse of V with

(2.3) V V - V = V , V - V V - = V - .

The V V - and V - V are projectors and dim(im(VV-)) = dim(im(V-V)).

Lemma 2.2. Let V be a singular matrix and V - a reflexive inverse of V with (2.3) and V V - = Po, V - V = P, where P and PD satisfy the conditions of Lemma 2.1. Then the matrix V + K-1Q is nonsingular and

(V + K-1Q) -1 = V- + QK ,

where K is defined in (2.2).

Proof.

(V + K-1Q)(V - + QK) = V V - + VQK + K-1QV - + K-1QK

= P D + O + O + Q D = I . []

3. The shooting method

The natural way to construct a shooting method for DAE's is described in [GM].

Using the subdivision of the interval [a,b]

a = t o < tl < . . . < t i n - 1 < tm =-b

the shooting equation reads

A well-posed shooting method for transferable DAE's 819

g(z0, X(tm ; tin-l, Zm-1)) = 0 (3.1)

Pi(zi -- X(t i ; t i_l ,Zi-1)) = O, i = 1 . . . . . m -- 1 ,

with Pi := P (ti). The disadvantage of (3.1) is the singularity of the Jacobian. If we use the

representation of zi = Pizi + Qizi =: ui + vi, we obtain the following system

(3.2) g(uo + vo, x(tm ; tin-l, u~- l ) ) = 0

(3.3) u i - Pix( t i ; t i - l ,ui-1) = O, i = 1 . . . . . m - 1,

and in order to determine v0 we add the equations

(3.4a) Qoyo + Povo = 0

(3.4b) f(Yo, uo + vo, to) = 0

(3.5) Qouo = O .

Now we specialize V := S' in (2.3). Let PD be a projector with im(Po) = M. If we demand (2.3) and

V V - ----'PD

V - V = P ,

the generalized inverse V - i s uniquely determined. Using Lemma 2.1 we construct a regular matrix K so that im(PD) @ im(K-1Q) = n;~ n. This provides the possibility to add without loss of information, the Eqs. (3.2) and (3.5) (after multiplying by K-l) . The following shooting operator is created

~ g(u0 + VO, X( tm; tm- l ,Ura-1) )+ K-1Qouo (a) SI(~) := i. u i - - P i x ( t i ; t i - l , U i - 1 ) i~- 1 . . . . . m - - 1 (b)

(3.6) S(~) Ooyo + Povo (c)

82(~) := f(yo, uo + vo, to) (d) ,

with ~ := (uo, ub . . . , um-b Yo, vo) T-

Theorem 3.1. le t the assumptions ( A ), ( B ) be fulfilled. Then the nonlinear equation

(3.7) S(r = 0

has a nonsingular Jacobian in a neighbourhood o f

= 4. : = ( u . 0 , . . . , u . m - b y*0, v .0 ) ,

which corresponds with x , .

Proo f . The Jacobian of (3.7) has the representation

820 Ren6 Lamour

(3.8) S'(~)=

I - - 1 ! [ g x . + K Qo g~ X (t . , t ._ l) [--P1 X(t l , to) I

-P2 X(t2, t0 I

- P . - 1 X ( t . - 1 , t,,-2) I

0 g~,~ \

1 o f" : r f ;

We consider the equation

(3.9)

(3.13)

Newton's method

s ' ( r = o

with ~/= (zo . . . . . zm-b qo, w0) r. The second up to the m-th equation of (3.9) is

(3.10) PiX(ti, ti-l)Zi-1 --zi = 0 , i = 1 . . . . . m-- 1 .

This leads to

(3.11) Pro-iX(tin-i, to)zo - Zm--i = O .

From the last two equation of (3.9) and (2.1) we know that (qo) ,o; . (0) (3.12) wo = ~, f'y f'x --f'xzo = - a o G - l f~xzo J"

The first equation of (3.9) gives

I I (g~xa + K-1Qo)zo + gxbX(tm, tm-1)Zm-1 4- gx WO = O.

With (3.11) and (3.12) we have the following result

(glxaX(to, to) q- gPxb X(tm, tO))ZO + K-1Qozo = 0 , i.e.

(iS' + K-1Qo)zo = O.

With Lemma 2.2 we conclude z0 = 0, and (3.10) implies that zi = 0 for i = 1 . . . . . m - 1, and (3.12) that q0 = 0 and w0 = 0. []

With Theorem 3.1 it is possible so solve (3.7) by Newton's method. We do so by using the structure of S' (see (3.8)). With

St = ( J1 J 2 ) J3 J4

j(i) A~i+l = -S (~ i) (3.14) r := ~i + 2A~i+l

with e.g. IIJ (0 - S'(r i) II -< a is of the form


(J~ jJ24)(~i ) (;12) Ay = -

With the nonsingularity of J4 (see (2.1)) this leads to

(3.15) J4 Av = -($2 + J3Au)

(J1 - S2s~ l S3)Au = - (S~ -- S # 4 1 S 2 )

or, in detail,

-1"2 X ( t 2 , t2) I = �9 . . . . . ' ' ' " . . . . . . .

" " " . . . . . . . " " . . . . . . . . .

--Pro-1 X(tm-1, t i n - z ) " I \ A U r a _ l /

(3.16)

/ /g (Uo + v0, x (tin; tin- 1, urn- 1)) + K - 1 Qo Uo - g'~, (P~ Vo + Qo G- lf(yo, Uo + Vo, to))\ - - lU. - -P1 X ( t l ; tO, UO) )

\urn-l--Pro-1 x(t,~- l ; t in- 2, /"/m- 2)

and

( A y ) ( Q o y o + P o v o ) (3.17) J4 Av = - f(yo, uo + vo, to) + f'~Auo "

Remark. The value w := (Psvo + QoG-l f(yo, uo + vo, to)) at the right-hand side of (3.16) is the solution of the linear system

(3.18, j 4 ( ~ ) = ( Qoyo-k-Povo ) k,f(YO, uo + vO, t o ) / "

This leads to the following algorithm to compute the iteration ~i: o o o vo) 0 - initial value 4 ~ := (u 0 . . . . . urn-i, Yo,

1 - i : = 0 2 - compute u i+l with (3.16) 3 - compute y~+l,v~+l with (3.17) using A u i+l := u i+l - - u i

4 - i : = i + 1 5 - TI? accuracy not reached THEN GOTO 2 ELSE STOP

4. The condition of the shooting method

We have to estimate the matrices S' and S t -1 . Now we consider the linearized Eqs. (1.6), (1.7). For convenience, we introduce a further variable Urn with

(4.1) Um - - P m x ( t m ; tm-1 , urn - l ) = O .

822 Ren+ Lamour

Using A(tm)Xt (tm) + B(tm)X(tm) = q(tm) Qmx'(tm) = 0

and (2.1) we have the representation

(4.2) X(tm ; t m - l , Urn--l) = X(tm) : Ps,mUm + QmG-l q(tm) �9

The additional equation (4.1) does not change the structure of (3.13) signifi- cantly. For the matrix

t Ol P~,o 02 Ps,m) Pa X(tl, to) P1

(4.3) M.'= --P2X(t2,tl) P2 �9 . . . . .

--PmX(tm, tm-m) Pm /

we find, in [LM], the representation

(4.4) M = MI~

with

(4.5)

/Dx P~ o+K-1Qo

M I : = ( -'U(tl,to)

\ I

-- U(t2, tl) I - . . " . .

-U(t~, t~ O

O2 P~ "1 I ' /

(4.6)

Let

(4.7)

~3 := diag (Po, P1 . . . . . Pro) �9

SQ : = ~S' + K-1Qo

= D1X(to, to) + D2X(tm, to) + K-tQo ;

then the matrix M1 is nonsingular with the inverse

(4.8) U(to, to)S~ -1 G(to, tl) . . . . . . . G(to, tin) I

M I I = U(tm, io)S~ 1 G(:tm, tl) G(tm, tmi ]

where G(t,s) denotes Green's function (see [LOR]). With ~o := diag(S'S'-, P1 ..... Pro), M- := M/-I~ is the generalized inverse of M with M - M = ~, M M - = ~o (see [LM]) and with

(4.9) R := diag (K, I . . . . . I)

we obtain


Now, for the matrix Jl - J 2 J 4 1 J 3 of (3.15) we have the representation

(4.10) 31 - J2J41J3 = M --~ .~-1~ (~ := [ _ ~) .

Lemma 2.2 now becomes

(4,11) (Jl - J2J41j3) -1 = M - + ~ R .

(4.10) provides the representation

(4.12) J1 = M + ~-I!i~ + J2d~lJ3 .

We denote IIAII~ := max {llA(t)ll , t E [a,b]},

where [I.II is an arbitrary matrix norm. [LM] introduces some suitable constants for estimating the condition of St:

~cl := max {H U(t, t0)S~ 1 ]1 ,

K2 := (b - a) sup { tl G(t, s)[I , ~3 := 11PG-11100

(4.13a) /s := IIQG-llloo

7s := [Ie, ll~

:= IIP II~ x := max{max { IlX(t, t j)II ,

J The estimation needs some further constants

(4.13b)

b) c+d)

e)

t ~ [a, b]}

t ,s e [a,b]}

t e [tj, t j+ t ] } } .

~Ca := [[AIIoo

Xb : : IIB]loo k : : max{llK[Ioo, 1}

k - := max{IlK -1[Io~, 1}.

Using the block row sum norm as matrix norm, the following esti-

f PG-~B ) J21J3 = ~, QG-1B. �9 []

Lemma 4.1. mations hold a) Let max { H D1 II, lID2 II} = 1, i.e. the boundary conditions are scaled, then

JIM + R - I ~ I I ~ < max {y(x + 1),27s} + k - ( l + ~) b) [IM- + ~Rl[~o < ~:I7 + ~:2m/(b - a) + k(1 + y) c) llJ411oo <- max{1 + 27,x, + xb} d) IIJ3lloo < IlJ41[oo e) IIJalJ311oo < max {x3,~:a}Xb.

Proof . a) Using the triangular inequality and the structure of M and !R-1Q the

estimation follows immediately. See [LM] Corollary 5.12. Follows immediately from the structure of J4 and J3.

824 Ren+ Lamour

Using the structure of S', (3.12) and Lemma 4.1d we obtain

(4.14) IIS'll~ < max {ItM + R-I~[ [~ + [IJ2J41j311~ + llJzll~, 2 tlJ411oo} �9

The inverse of S' is

St-1 = ( (J1 -- J2J41J3) -1 --(Jl - J3J41J3)-l J2J41 ) --J41J3(J1 -- J2J41J3) -1 J4-1(I + J3(J1 - J2J41J3)-lJ2J41) ,] "

(4.15)

Using (4.15), (4.11) and the structure of J2 and J3 we obtain

(4.16) I[ S'-11t -< ILM- + ~Rl[oo(1 + IlJ2J41 [[oo)max {1, IlJ41J3loe} + IIJ411fo~. This yields

Theorem 4.2.

condoo(S') _< (max {I[M + R-l~lloo + IIJflulJ3[Ioo + ilJ2[Ioo, 2 [IJ411o~})

(4.17) x (ItM- + aStll~(1 + tlJ2J41-ll~)max {1, [IJ41J3[l~} + JlJ4111~)

Proof. (4.14) + (4.16). []

If the boundary conditions act on the differentiable components only, i.e. g" does not depend on v0, the condition J2 = 0 holds.

Theorem 4.3. Let J2 = O, then there are constants 2M and 2j with

l < 2 m , 2 j < max{HM+R-~[I~176176 such that min {IIM + !~/-1~11oo, 2 llJ4lloo}

(4.18) cond~(S') < 2Mc cond~(M + R-Is + 2j 2condoo(J4)

and c = max {1, IlJ41J31[~}. Proof. The estimation of Theorem 4.2 and J2 = 0 imply

cond~(S') < (max {tIM + R-ls 2 IIJ41[o~})

• + ~StIIoomax {1, tlJ4JJ31too} + lIJu ~ Lo) �9

Let

and

s : = m a x { I lM+R- l~ t l~176 1} 2 IlJ411o~ ' '

2 [IJ4[{oo 1} 2M := max [IM+ R-Xs ' '

then

max { [IM + R - I ~ [ I ~ , 2 IIJ4 lion} = 2M I[M + ~1-1~[1~ = 22s IlJ4 lloo. []


R e m a r k . a) 1 =AM < 2j if IIM +~-1~11~ > 211J411~

1 = 2j < AM if IIM + ~ - l ~ l l ~ < 2llJall~. b) By scaling (1.6), the estimations

IlZll~-< 1, Ilnl[oo-< 1

are valid, i.e. 2M and 2j are of moderate size.

Let us consider two special cases of Theorem 4.3. The pure ODE-case can be characterized by Q(t) - O. Where J2 --- 0, -/3 -- 0, ~ --- 0, y = 1, 7s = 1 are valid, too. Then, (4.18) provides

condoo(S') < 2 M condo(M) + 2j2 condoo(J4)

and condo(M) < (x + 1)(K1 + x2m/(b - a)) .

This represents the classical estimation of the ODE-shooting matrix (see [LOR]) plus the condition of the matrix J4, which describes now the transformation of the implicit representation of the ODE to an explicit one.

The pure linear algebraic system case can be characterized by Q --- I, i.e., there is no differentiable part of x (r = 0) and therefore no boundary condition g. It follows that J1 = 0, J2 ~ 0, J3 =- 0 and J4 = diag(I, B). The estimation (4.18) gives

(4.19) cond~(S') < 1 + 2 max(l, lIB 11~) * max (1, lIB -11100) �9

The essential part of (4.19) is the well-known condition of linear systems.

5. Remarks on the implementation

(3.15) has been the basis for implementing this shooting method.

5.1 Projectors

We select the projector Q (P := I - Q) with im(Q) = ker(A) in the following way. Let

A = URpTc

with Pc - permutation matrix of columns, U - orthogonal matrix

R1 - nonsingular and r -- rank(A), then

o -Ri-IR: Q 0 ]n-r . ] p T . :~ ec

The UR-decomposition is performed by a Householder method with column pivoting.

826 Ren+ Lamour

We implement the boundary condition in such a way that the first r components of g are used. This means that the projector Po is given by

P D = ( IrO 00) and K -1 = (R1 In-rR2 ) p T .

Therefore, the term o o )p?

K - 1 Q = 0 In-r

of (3.15) is very simple and nothing has to be calculated additionally.

5.2 Nonlinear equations

The nonlinear algebraic equations are solved by a damped Newton-like method, where the cyclic structure of (3.16) is taken into consideration (this is the same structure as in case of ODE's!). We use the code NLZYK2, which is a modification of N L Z Y K in [HLW].

5.3 Integration method

As an integration method we use a modified BDF code based on the code I N T G R of Skelboe [S].

5.4 Initial values

In addition to the selection of proper start values for the nonlinear problems in DAE's we also have the problem of the consistency of initial values of integration. Therefore, to every inner shooting point ti (i = 1 . . . . . m - 1) we add two further equations

Qiyi --}- Pivi : o (5.1)

f(Yi, Ui -]- Vi, ti) = O .

The structure of (5.1) makes it possible to carry out one Newton step with the small system (5.1) after one step with (3.16) to determine the iteration values for Yi and vi. The solution of (5.1) is a set of consistent initial values for the integration (y~, ui + v~), i = 1 . . . . . m - 1. This means that this implementation results not only in shooting values ui, i = 0 . . . . . m - 1, but also in a set of consistent initial values for every shooting point.

6. Examples

In this section we present some numerical results using this method. The examples are given in [CP] and [A]. The code is written in FORTRAN77 and all the computations are performed in double precision on an Atari 1040ST computer.


The first test p rob lem on [0, 1] was given by

(i -t'2) (i 1 - - t x' + 0 0

with the b o u n d a r y cond i t ions

The exact so lu t ion is

(0) - 1 ( t - 1) x = 0 0 1 sin t

xt (0) = 1

x 2 ( I ) - - x3(1) = e .

xl = e x p ( - t ) + texp( t )

x2 = exp(t) + t sin(t)

x3 = sin(t) .

We solve this p rob lem with the relat ive accuracy o f in tegra t ion l d - 4 and l d - 6 (this implies a defect accuracy o f Newton ' s m e t h o d o f 5d -4 and 5d-6) . The s tar t values o f x, y and v were zero. The interval [0, 1] was equid is tan t ly subdivided,

Accuracy-- ld -4

Number of shooting intervals 1 2 4 5 10 Number of Newton iterations 2 3 5 4 3 Reached defect of hi-system 1.4-5 6.2-5 4.2-4 1.2-4 9.6~5 Number of f-calls 212 380 656 710 910 Number of Jacobians j(i) 1 1 1 1 1

Accuracy -- ld -6

Number of shooting intervals 1 2 4 5 10 Number of Newton iterations 2 3 3 3 3 Reached defect of nl-system 2.(~6 1.8 7 2.7-8 3.5-8 2.4-8 Number of f-calls 418 768 1208 1397 2208 Number of Jacobians j(i) 1 1 1 1 1

The h igher accuracy o f in tegra t ion genera tes a "be t t e r " a p p r o x i m a t i o n o f the J acob ian so tha t the convergence is faster (number o f Newton i te ra t ions in the cases o f rn = 4 and m = 5).

The increase o f the n u m b e r o f f - ca l l s is caused by the length of the in tegra t ion intervals. A smal ler in terval genera tes a smal ler in tegra t ion order i.e. smal ler stepsizes.

The second example is a mode l o f a s teady s tate semiconduc to r device descr ibed in [A] in the version o f [CP].

0 = N _ - C( t )

J ' + = 0

f _ = 0

J+ = (N'_ - - N+~v')

N ' J _ = ( + - N _ r

with b o u n d a r y condi t ions

828 Ren6 Lamour

N+ = v / C 2 + 464 , t = - - 1 , + 1 ,

1/) = Ipb i + V(t + 1)/2

and using the functions and parameters

tPbi(t) = ln((N+(t) + N_(t) ) /2) - ln(62)

C(t) = 0.5 + arctan (2t)/r~

f i = l d - - 4 , V = I .

C describes the doping profile and we are interested in the symmetric dxode case. This problem was investigated as a singularly disturbed ODE-sys tem in [VS] and a simplified version was solved by shoot ing methods in [Mai]. We use the parameter 2 as an imbedding parameter f rom 1 to 100. We solve this problem with relative accuracy o f integrat ion l d - 6 (this implies a defect accuracy o f Newton ' s me thod o f 5d - 6). By imbedding, we obtain the start values for single shoot ing

J'+ = 0 J+ = 6.6 -- 3

J ' = 0 J _ = 6 . 6 1 - - 3

N ' = 6 . 6 - - 3 N + = 3 . 3 5 - - 3 +

N ' = 6 . 6 1 - - 3 N _ = 3 . 3 5 - - 3

~v' = 0 ~ = 12.7

(Note that consistent initial values are not necessary.) For ). = 100, after 6 Newton iterations we obtain the solution

J+ -- J_ - 6.294507 - 3 .

3~Q70523-3 J

- - 0 .99681701

i -I 0 +1 -I +1

Fig. 1. The doping profile C(t) has the same structure as N_

12,,52,3

0

Fig. 2.

17.418

Acknowledgement. I wish to thank Prof. Dr. Roswitha M~irz for many helpful discussions.


References

[A]

[AMR]

[AMS]

[BN]

[Cl

[CP]

[DI]

[D2]

[F]

[GM]

[HEW]

[LI

[LMI

[LOR]

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A well-posed shooting method for transferable DAE's