% linvmat.dem % LaTeX package LJour1 1.0: demo file for Inventiones mathematicae % (c) Springer-Verlag HD %---------------------------------------------------------------------- % % customization \documentstyle[bibay]{pljour1} \journalname{Inventiones mathematicae} % State name of journal \newcommand{\DXDYCZ}[3]{\left( \frac{ \partial #1 }{ \partial #2 } \right)_{#3}} % end of customization % \begin{document} % \title{ Optimality relationships for $p$-cyclic SOR\thanks{Research supported in part by the US Air Force under grant no. AFOSR-88-0285 and the National Science Foundation under grant no. DMS-85-0285.}\fnmsep\thanks{In memory of J.L. Verdier}} \subtitle{A demonstration text} \author{Daniel J. Pierce\inst{1} \and Apostolos Jadjidimos\inst{2}\fnmsep\thanks{{\it Present address:\/} Department of Computer Science, Purdue University, West Lafayette, IN 47907, USA.} \and Robert J. Plemmons\inst{3}} \mail{R. Plemmons} \titlerunning{Optimality relationships for $p$-cyclic SOR} \authorrunning{D. J. Pierce et al.} \institute{Boeing Computer Service, P.O. Box 24346, MS 7L-21, Seattle, WA 98124-0346, USA \and Department of Mathematics, University of Ioannina, GR-45 1210 Ionnanina, Greece \and Department of Computer Science and Mathematics, North Carolina State University, Raleigh, NC 27695-8205, USA} \date{Oblatum 20-I-1989 \& 3-VIII-1991} \maketitle \begin{abstract} The optimality question for block $p$-cyclic matrix into a block $q$-cyclic form, $q < p$, results in asymptotically faster SOR convergence for the same amount of work per iteration. As a consequence block 2-cyclic SOR is optimal under these conditions. \end{abstract} \section{Introduction} This text was compiled to demonstrate the use of the Springer \LaTeX\ macropackages {\em LJour1\/} for one-column journals. Please refer to \cite{leslie} for general information on coding \LaTeX{} and to the \cite{springer} for information concerning the Springer layout. Parts of this ``article" were taken from different real articles, but may have been changed to show a special feature of a macro. \section{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 may be denoted by gothic letters: $\frak{G, H}$. Sets of functions are denoted by script 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. \section{Preliminaries} Let us state a few well known results and demonstrate how to typeset lists. The functions $f$ and $g$ of (1) and (2) fulfill the following assumptions: \begin{enumerate} \item $f: B_f \subset {\Bbb R}^n \times {\Bbb R}^n \times [a,b] \to {\Bbb R}^n$ \\ $f^\prime _x$, $f^\prime_y$ exist and are continous \item ker$(f^\prime _y (y, x, t)) = N (t)\quad \forall (y, x, t) \in B_f$ \\ ${\rm rank} (f^\prime _y (y, x, t)) = r$ \\ ${\rm dim} (N (t)) = n - r$ \item $Q(t)$ denotes a projection onto $N(t)$ \\ $Q$ is smooth and $P(t) := I - Q (t)$ \item The matrix $G (y, x, t) := f^\prime _y (y, x, t) + f^\prime _x (y, x, t) Q (t)$ is nonsingular \\ $\forall (y, x, t) \in B_f$\quad (i.e. (1) is transferable) \item $g: B_g \subset {\Bbb R}^n \times {\Bbb R}^n \to M \subset {\Bbb R}^n$ \\ $g^\prime _{x_a} , g^\prime _{x_b}$ exist and are continuous\\ ${\rm im} (g^\prime _{x_a} , g^\prime _{x_b}) =: M$ \end{enumerate} Now we give another example of a list with changed indentation. \begin{description}[Shoot.] \item[Shoot.] Collocation methods for this type of equations are considered in \cite{yser} and \cite{wendl}. Shooting and difference methods for linear, {\it solvable} DAE's in the sense of [9], also with higher index, are treated in [8] 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. \begin{description}[Jacob.] \item[Jacob.] We also deal with Jacobians, which means that we explain the functions, advantages and inconveniences of calling them not Jacobians..... \item[Nonl.] Nonlinear functions play an important role in this connection. Please note that we always call them nonlinear whenever there is no............ \end{description} \end{description} \section{The shooting method} The natural way to construct a shooting method for DAE's is described by \cite{yser}. The physical meaning of $ \sigma_0 $ and $K$ is clearly visible in the equations above. $\sigma_0$ represents a frequency of the order one per free-fall time. $K$ is proportional to the ratio of the free-fall time and the cooling time. Substituting into Baker's criteria, using thermodynamic identities and definitions of thermodynamic quantities, \begin{displaymath} \Gamma_1 = \DXDYCZ{\ln P}{\ln \rho}{S} \, , \; \chi^{}_\rho = \DXDYCZ{\ln P}{\ln \rho}{T} \, , \; \kappa^{}_{P} = \DXDYCZ{\ln \kappa}{\ln P}{T} \end{displaymath} \begin{displaymath} \nabla_{\rm ad} = \DXDYCZ{\ln T}{\ln P}{S} \, , \; \chi^{}_T = \DXDYCZ{\ln P}{\ln T}{\rho} \, , \; \kappa^{}_{T} = \DXDYCZ{\ln \kappa}{\ln T}{T} \end{displaymath} one obtains, after some pages of algebra, the conditions for {\em stability} given below: \begin{eqnarray} \frac{\pi^2}{8} \frac{1}{\tau_{\rm ff}^2} ( 3 \Gamma_1 - 4 ) & > & 0 \label{ZSDynSta} \\ \frac{\pi^2}{\tau_{\rm co} \tau_{\rm ff}^2} \Gamma_1 \nabla_{\rm ad} \left[ \frac{ 1- 3/4 \chi^{}_\rho }{ \chi^{}_T } ( \kappa^{}_T - 4 ) + \kappa^{}_P + 1 \right] & > & 0 \label{ZSSecSta} \\ \frac{\pi^2}{4} \frac{3}{\tau_{ \rm co } \tau_{ \rm ff }^2 } \Gamma_1^2 \, \nabla_{\rm ad} \left[ 4 \nabla_{\rm ad} - ( \nabla_{\rm ad} \kappa^{}_T + \kappa^{}_P ) - \frac{4}{3 \Gamma_1} \right] & > & 0 \label{ZSVibSta} \end{eqnarray} For a physical discussion of the stability criteria see \cite{tetz} or \cite{yser}. \subsection{Disadvantages of the method} The disadvantage of Eq. (\ref{ZSVibSta}) 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 \begin{eqnarray} g (u_0 + v_0 , x (t_m, t_{m-1}, u_{m-1}))& = & 0 \label{dis}\\ u_i - P_i x (t_i; t_{i-1}, u_{i-1}) & = & 0\;, \quad i = 1, \ldots , m-1\;. \label{das} \end{eqnarray} \subsection{Specialization of $V$} Now we specialize $V := \hat S^\prime $ in. Let $P_D$ be a projector with ${\rm im} (P_D) = M$. If we demand Eq. (\ref{das}) and \begin{eqnarray*} VV^- &=& P_D \\ V^-V &=& P\; , \end{eqnarray*} % the generalized inverse $V^-$ in uniquely determined. Using Lemma 1 we construct a regular matrix $K$ so that ${\rm im} (P_D) \oplus {\rm im} (K^{-1} Q) = {\Bbb R}^n$. This provides the possibility to add without loss $(K^{-1} Q) = {\Bbb R}^n$. This provides the possibility to add, without loss of information, the Eqs.\ts (\ref{dis}) and (\ref{six}) (after multiplying by $K^{-1})$. The following shooting operator is created \begin{equation} \quad S (\xi ) := \left\{ \begin{array}{ll} S_1 (\xi):= & \left\{ \begin{array} {ll} g (u_0 + v_0, x (t_m; t_{m-1}, u_{m-1})) + K^{-1} Q_0 u_0 &\quad (a)\\ u_i - P_i x (t_i; t_{i-1} , u_{i-1})\; i = 1, \ldots , m-1 & \quad(b) \end{array} \right. \\ S_2 (\xi) := & \left\{ \begin{array} {ll} Q_0 y_0 + P_0 v_0 & \quad (c)\\ f(y_0, u_0 + v_0, t_0) & \quad (d) \quad , \end{array} \right. \end{array} \right.\label{six} \end{equation} % with $\xi := (u_0 , u_1, \ldots , u_{m-1} , y_0, v_0)^{\rm T}$. \begin{lemma} Let $V$ be a singular matrix and $V^-$ a reflexive inverse of $V$ with Sect. (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 Sect. (2.2). \end{lemma} \begin{proof} \begin{eqnarray*} (V + K^{-1}Q)(V^- + QK) & = & VV^- + VQK + K^{-1}QV^- + K^{-1} QK \\ & = & P_D + 0 + 0 + Q_D = I\; . \quad\qed \end{eqnarray*} \end{proof} \begin{remark} 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 Eq. (16) is the solution of the linear system \begin{equation} J_4 \left(\begin{array}{c} \eta \\ w \end{array} \right) = \left(\begin{array}{c} Q_0 y_0 + P_0 v_0 \\ f (y_0, u_0 + v_0, t_0) \end{array} \right) \end{equation} \end{remark} \begin{figure}\picplace {4 cm} \firstcaption{The doping profile $C (t)$ has the same structure as $N_-$} \secondcaption{The doping profile of $C (z)$} \end{figure} This leads to the following algorithm to compute the iteration $\xi^i$: \begin{description}[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} \end{description} \begin{theorem} 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 $x_\star$. \end{theorem} \section{Implementation} If listing of a program is desired, this is possible too \cite{darnell} \begin{verbatim} void get_two_kbd_chars() { extern char KEYBOARD; char c0, c1; c0 = KEYBOARD; c1 = KEYBOARD; } \end{verbatim} \section{Solutions} We solve this problem with the relative accuracy of integration $1d-4$. The experimental tests of the Standard Model and thereby of the unification of the weak and electromagnetic interactions have reached a new level of accuracy. The results are given in Table \ref{KapSou}. \begin{table} \caption{Opacity sources}\label{KapSou} \centering \begin{tabular}{ll} \hline\noalign{\smallskip} Source & T/[K] \\ \noalign{\smallskip} \hline \noalign{\smallskip} Yorke 1979, Yorke 1980a & $\leq 1700^{\rm a}$ \\ Kr\"ugel 1971 & $1700 \leq T \leq 5000$ \\ Cox and Stewart 1969 & $5000 \leq $ \\ \noalign{\smallskip}\hline\noalign{\smallskip} $^{\rm a}$ This is a footnote. \end{tabular} \end{table} \begin{acknowledgement}I wish to thank Prof. Dr. Roswitha M\"arz for many helpful discussions.\end{acknowledgement} \begin{thebibliography}[9]{References} % Note that space for square brackets is added to the width of the label % specified in the [] argument. If you don't use []s in your % bibliography, specify a narrower label or omit the specification % altogether. In this case \parindent is used. \bibitem{1.}{darnell}{[1]} Darnell, P.A., Margolis, P.E.: C, A software engineering approach. Berlin Heidelberg New York: Springer-Verlag, 1988 \bibitem{2.}{leslie}{[2]} Lamport, L.: \LaTeX: A document preparation system. Addison-Wesley Publishing Company, Inc., 1986 \bibitem{3.}{seroul}{[3]} Seroul, R., Levy, S.: A beginner's book of \TeX{}. New York Berlin Heidelberg: Springer, 1989 \bibitem{4.}{springer}{[4]} LJour1: Springer's \LaTeX{} style file for journals with one-column layout. Heidelberg: Springer-Verlag, 1993 \bibitem{5.}{stroud}{[5]} Strout, A.H.: Approximate calculation of multiple integrals. Englewood Cliffs, N.J.: Prentice Hall, 1971 \bibitem{6.}{tetz}{[6]} Tetzlaff, A.: Stability in the Common Market. (To appear) \bibitem{7.}{wendl}{[7]} Wendland, W.L.: Strongly elliptic boundary integral equations. In: A. Iserles, M. Powell, (eds.) The state of the art in numerical analysis. Oxford: Clarendon Press, 1987, pp. 511--561 \bibitem{8.}{yser}{[8]} Yserentant, H.: A remark on the numerical computation of improper integrals. Computing {\bf 30}, 179--183 (1983) \medskip\noindent \bibitem{Please}{}{}refer to a recent issue of the journal for further examples on how to format references. \end{thebibliography} \end{document}