\titlea{Prolog}
This text was compiled to demonstrate the use of the Springer
plain-\TeX\ macropackages for one-column journals.

Parts of this ``article" were taken from different real articles, but
may have been changed to show a special feature of a macro.

\titlea{Notation}
Here are a few examples of how to use special fonts. Vectors are denoted
by boldface letters: $\vec V,\; \vec W$. Tensors are denoted by sans
serif letters: $\tens{A, B}$. If no tensors are needed, sans serif
letters may be reserved for other purposes. Vector spaces are denoted by
% N.B. It is also common for vector spaces to be denoted
% by roman letters. This may mislead the author. (EGD 27.10.91)
gothic letters: $\frak{G, H}$. Sets of functions are denoted by script
letters: $\cal{W}_i,\cal{F}$. Set of numbers are denoted by special
%letters: $\cal{W}_i,\cal{F}$. Sets of numbers are denoted by special
roman letters $\Bbb R, \Bbb C$.

You are of course (within limits) free to design your own notation but
sticking to conventions makes your article easier for others to read.

\titlea{Preliminaries}
The functions $f$ and $g$ of (1) and (2) fulfill the following
assumptions:
\medskip
\item{1.} $f: B_f \subset \bbbr^n \times \bbbr^n \times [a,b] \to
\bbbr^n$ \hfill\break
$f^\prime _x$, $f^\prime_y$ exist and are continous
\item{2.}ker$(f^\prime _y (y, x, t)) = N (t)\quad  \forall (y, x, t)
\in B_f$ \hfill\break
${\rm rank} (f^\prime _y (y, x, t)) = r$ \hfill\break
${\rm dim} (N (t)) = n - r$
\item{3.}$Q(t)$ denotes a projection onto $N(t)$ \hfill\break
$Q$ is smooth and $P(t) := I - Q (t)$
\item{4.} The matrix $G (y, x, t) := f^\prime _y (y, x, t) + f^\prime
_x (y, x, t) Q (t)$ is nonsingular \hfill\break
$\forall (y, x, t) \in B_f$\quad  (i.e. (1) is transferable)
\item{5.} $g: B_g \subset \bbbr^n \times \bbbr^n \to M \subset
\bbbr^n$ \hfill\break
$g^\prime _{x_a} , g^\prime _{x_b}$ exist and are continuous\hfill\break
${\rm im} (g^\prime _{x_a} , g^\prime _{x_b}) =: M$

\medskip
Now we give another example of a list with changed indentation.
\medskip
{\setitemindent{Shoot.}
\setitemitemindent{Jacob.}
\item{Shoot.}
Collocation methods for this type of equations are considered in [AMS]
and [D1]. Shooting and difference methods for linear, {\it 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.
\item{Diff.}
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.
\itemitem{Jacob.} We also deal with Jacobians, which means that we
explain the functions, advantages and inconveniences of calling them not
Jacobians.....
\itemitem{Nonl.} Nonlinear functions play an important role in
this connection. Please note that we always call them nonlinear whenever
there is no............
\par
}
\medskip

\titlea{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]
$$
a = t_0 < t_1 < \ldots < t_{m-1} < t_m = b
$$
the shooting equation reads
$$
\eqalign {
& g (z_0 , x (t_m; t_{m-1}, z_{m-1})) = 0 \cr
& P_i (z_i - x (t_i; t_{i-1}, z_{i-1})) = 0\; , \quad  i = 1, \ldots ,
m-1\; , \cr } \eqno (4.1)
$$
with $P_i := P (t_i)$.

\titleb{Disadvantages of the method}
The disadvantage of (4.1) is the singularity of the Jacobian. If we use
the representation of $z_i = P_i z_i + Q_i z_i =: u_i + v_i$, we obtain
the following system
$$
g (u_0 + v_0 , x (t_m, t_{m-1}, u_{m-1})) = 0 \eqno (4.2)
$$
$$
u_i - P_i x (t_i; t_{i-1}, u_{i-1}) = 0\; , \quad  i = 1, \ldots , m-1\;
. \eqno (4.3)
$$

\titleb{Specialization of $V$}
Now we specialize $V := \hat S^\prime $ in (2.3). Let $P_D$ be a
projector with ${\rm im} (P_D) = M$. If we demand (2.3) and
$$
\eqalign {
&VV^- = P_D \cr
&V^-V = P\; , \cr }
$$
the generalized inverse $V^-$ in uniquely determined. Using Lemma 2.1 we
%the generalized inverse $V^-$ is uniquely determined. Using Lemma 2.1 we
construct a regular matrix $K$ so that ${\rm im} (P_D) \oplus {\rm im}
(K^{-1} Q) = \bbbr^n$. This provides the possibility to add without loss
%(K^{-1} Q) = \bbbr^n$. This provides the possibility to add, without loss
of information, the Eqs.\ts (4.2) and (4.5) (after multiplying by
$K^{-1})$. The following shooting operator is created
$$
S (\xi ) := \left( \matrix {
S_1 (\xi):= \cases {g (u_0 + v_0, x (t_m; t_{m-1}, u_{m-1})) + K^{-1}
Q_0 u_0\hfill &\quad  (a)\cr
u_i - P_i x (t_i; t_{i-1} , u_{i-1})\;  i = 1, \ldots , m-1\hfill &
\quad  (b) \hfill\cr}
\hfill\cr
S_2 (\xi) := \cases {
Q_0 y_0 + P_0 v_0 \hfill & \quad  (c)\hfill\cr
f (y_0, u_0 + v_0, t_0) \hfill& \quad (d) \quad  ,\hfill\cr }
\hfill\cr}\right.
\eqno
(4.6)
$$
with $\xi := (u_0 , u_1, \ldots , u_{m-1} , y_0, v_0)^{\rm T}$.

\beglemma{4.1.} Let $V$ be a singular matrix and $V^-$ a reflexive
inverse
of $V$ with (2.3) and $VV^- = P_D$, $V^-V = P$, where $P$ and $P_D$
satisfy the conditions of Lemma 2.1. Then the matrix $V + K^{-1} Q$ is
nonsingular and
$$ (V + K^{-1} Q) ^{-1} = V^- + QK\; , $$
where $K$ is defined in (2.2).
\endlemma
\begproof.          % (Why is there a dot at the end? EGD 27.10.91)
$$\eqalign {
(V + K^{-1}Q)(V^- + QK) & = VV^- + VQK + K^{-1}QV^- + K^{-1} QK \cr
& = P_D + 0 + 0 + Q_D = I\; . \qed \cr }
$$
\endproof

\begremark.  The value $w := (P_s v_0 + Q_0 G^{-1} f (y_0, u_0 + v_0,
t_0))$ at the right-hand side of (4.16) is the solution of the linear
system
$$
J_4 \pmatrix {\eta \cr w \cr } = \pmatrix{ Q_0 y_0 + P_0 v_0 \cr f (y_0,
u_0 + v_0, t_0) \cr } .\eqno (4.18)
$$

\begdoublefig 4 cm
\figure{1}{The doping profile $C (t)$ has the same structure as $N_-$}
\figure{2}{}
\endfig

This leads to the following algorithm to compute the iteration $\xi^i$:
\medskip
{\setitemindent{5 ---}
\item{0 -- } initial value $\xi^0 := (u_0^0 , \ldots , u^0_{m-1} , y_0^0
, v_0^0)$
\item{1 -- } $i:= 0$
\item{2 -- } compute $u^{i+1}$ with (3.16)
\item{3 -- } compute $y^{i+1}_0, v_0^{i+1}$ with (3.17) using $\Delta
u^{i+1} := u^{i+1} - u^i$
\item{4 -- }$i:= i + 1$
\item{5 -- }{\tt IF} accuracy not reached {\tt THEN GOTO 2 ELSE STOP}
\par}
\endremark

\begtheorem{4.1.} Let the assumptions (A), (B) be fulfilled. Then the
non-linear equation
$$
S (\xi) = 0
$$
has a nonsingular Jacobian in a neighbourhood of
$$
\xi = \xi_\star := (u_{\star 0}, \ldots , u_{\star m-1} , y_{\star 0},
v_{\star 0})\; ,
$$
which corresponds with $x_\star$.
\endtheorem

\titlea{Implementation}
If listing of a program is desired, this is possible too:
\medskip
\begverbatim
void get_two_kbd_chars()
{
     extern char KEYBOARD;
     char c0, c1;

     c0 = KEYBOARD;
     c1 = KEYBOARD;
}
\endverbatim
\medskip
(Code taken from the book: {\it C, A software engineering approach} by
P.A. Darnell and P.E. Margolis, Springer Verlag 1988)
%
\titlea{Solutions}
We solve this problem with the relative accuracy of integration $1d-4$.
Results are given in Table 6.1.
\begtabfull
\tabcap{6.1}{Results using the shooting method}
{}
\halign{%\strut
#\quad \hfill &
#\quad &
\hfill # \hfill \quad  &
\hfill # \hfill \quad  &
\hfill # \hfill \quad  &
\hfill # \hfill \quad  &
\hfill # \hfill \quad  \cr
\noalign{\smallskip\hrule\smallskip}
{\bf Accuracy = $1d-4$} \cr
\noalign{\smallskip\hrule\smallskip}
Number of shooting intervals & 1 & 2 & 4 & 5 & 10 \cr
Number of Newton iterations & 2 & 3 & 5 & 4 & 3 \cr
Reached defect of $nl$-system & 1.4--5 & 6.2--5 & 4.2--4 & 1.2--4 &
9.6--5 \cr
Number of $f$-calls & 212 & 380 & 656 & 710 & 910 \cr
Number of Jacobians $J^{(i)} $ & 1 & 1 & 1 & 1  & 1 \cr
\noalign{\smallskip\hrule\medskip}
%{\bf Accuracy = $1 d-6$} \cr
%\noalign{\smallskip\hrule\smallskip}
%Number of shooting intervals & 1 & 2 & 4 & 5 & 10 \cr
%Number of Newton iterations & 2 & 3 & 3 & 3 & 3 \cr
%Reached defect of $nl$-system & 2.6--6 & 1.8--7 & 2.7--8 & 3.5--8 &
%2.4--8 \cr
%Number of $f$-calls & 418 & 768 & 1208&1397 & 2208\cr
%Number of Jacobians $J^{(i)} $ & 1 & 1 & 1 & 1  & 1 \cr
%\noalign{\smallskip\hrule}
}
\endtab

Text
text text text text text text text text text text text text text
text text text text text text text text text text text text text.
%
\acknow{I wish to thank Prof. Dr. Roswitha M\"arz for many helpful
discussions.}

\begref{References}{[AMR]}
\noindent%
As this demo file is meant for several journals for which different
reference systems apply, we do not give references here. Please refer to
the documentation for examples for all three systems.
\endref
\bye