docuptex.tex 664 144 7 42503 6124605450 6472 %% %% SIAM Plain TeX macro documentation. %% Paul Duggan %% 9-15-94 \input siamptex.sty % The definitions are to provide a verbatim text environment \def\uncatcodespecials{\def\do##1{\catcode`##1=12 }\dospecials} \def\setupverbatim{\tt% \def\par{\leavevmode\endgraf}% \obeylines\uncatcodespecials\obeyspaces} {\obeyspaces\global\let =\ } \def\doverbatim#1{\def\next##1#1{##1\endgroup}\next} \def\verbatim{\begingroup\setupverbatim\doverbatim} \overfullrule=0pt \topmatter \vol{1} \no{1, pp.~000--000} \SIMAC \date{October 1994} \copyyear{1994} \code{000} \title Using SIAM's \TeX\ Macros\endtitle \shorttitle{USING SIAM'S \TeX\ MACROS} \recdate{*}{August 9, 1994. This work was supported by the Society for Industrial and Applied Mathematics} \author Paul Duggan\fnmark{$^{\dag}$}\endauthor \address{$^{\dag}$}{Society for Industrial and Applied Mathematics, Philadelphia, Pennsylvania ({\tt duggan@siam.org}). Questions, comments, or corrections to this document may be directed to that e-mail address} \abstract{Documentation is given for use of the SIAM \TeX\ macros. Instructions and suggestions for compliance with SIAM style standards are also included.} \subjclass\endsubjclass \keywords\endkeywords \endtopmatter \heading{1}{Introduction} This file is documentation for the SIAM \TeX\ macros and provides instruction for submission of files formatted in \TeX. To accommodate authors who electronically typeset their manuscripts, SIAM supports the use of \TeX. To ensure quality typesetting according to SIAM style standards, SIAM provides a \TeX\ macro style file. Using \TeX\ to format a manuscript should simplify the editorial process and lessen the author's proofreading burden. However, it is still necessary to proofread the galley proofs with care. Electronic files should not be submitted until the paper has been accepted, and then not until requested to do so by someone in the SIAM office. Once an article is slated for an issue, someone from the SIAM office will contact the author about any or all of the following: editorial and stylistic queries, supplying the source files (and any supplementary macros) for the properly formatted article, and handling figures. When submitting electronic files (electronic submissions) (to {\tt tex@siam.org}) write the journal, issue, and author's name in the subject line of the message. Authors are responsible for ensuring that the paper generated from the source files exactly matches the paper that was accepted for publication by the review editor. If it does not, information on how it differs should be indicated in the transmission of the file. When submitting a file, please be sure to include any additional macros (other than those provided by SIAM) that will be needed to run the paper. SIAM uses MS-DOS-based computers for \TeX\ processing. Therefore all filenames should be restricted to eight characters or less, plus a three character extension. Once the files are corrected here at SIAM, we will mail the revised proofs to be read against the original edited hardcopy manuscript. We are not set up to shuttle back and forth varying electronic versions of each paper, so we must rely on hard copy of the galleys. The author's proofreading is an important but easily overlooked step. Even if SIAM were not to introduce a single editorial change into your manuscript, there would still be a need to check, because electronic transmission can introduce errors. This distribution contains the following items: {\tt siamptex.sty}, the main macro package; this documentation file; and a sample file {\tt pexample.tex}. The sample file is representative of the standard way to apply the macros. The rest of this paper emphasizes some aspects of applying the macros, points out options and special cases, and describes the SIAM style standards. The SIAM macros make use of the small caps font, {\tt cmcsc}, which is not installed in some systems. This font along with other AMS-\TeX\ fonts can be retrieved from the American Mathematical Society via anonymous FTP to {\tt e-math.ams.com}. \heading{2}{Headings} The top matter of a journal is in a standard format. The macro and initial definitions should be included as follows. \verbatim: \input siamptex.sty \overfullrule=0pt : The \verbatim:\overfullrule: should be set wider than zero points while still being edited by the author to assist in locating lines that extend beyond the margins. The ``slugline,'' which contains the journal and copyright information, is created by special commands following the \verbatim:\topmatter:. \verbatim: \SIMAX \vol{0} \no{0, pp.~000--000} \date{April 1995} \copyyear{1995} \code{000} : Because authors will probably not know exact volume, number, date, or code, insert zeros in their place as above. SIAM will place the appropriate information in the commands in production; however, the commands must be used. \verbatim:\SIMAX:, \verbatim:\SIAP:, \verbatim:\SICOMP:, \verbatim:\SICON:, \verbatim:\SIDMA:, \verbatim:\SIMA:, \verbatim:\SIMAX:, \verbatim:\SINUM:, \verbatim:\SIOPT:, \verbatim:\SISC:, and \verbatim:\SIREV: are the commands provided to include the journal title in the heading. The title and author(s) of the paper are indicated by the \verbatim:\title\endtitle: and \verbatim:\author\endauthor: commands. Author support and address information is indicated as follows. \verbatim: \author A.~U. Thorone\fnmark{$^{\dag}$} \and A.~U. Thortwo\fnmark{$^{\ddag}$}\endauthor \address{$^{\dag}$}{Address of the first author and support} \address{$^{\ddag}$}{Address of the second author and support} : If more than two authors are included, each should be separated by a comma after the \verbatim:\fnmark{}: command. If more than one author shares common footnote information, then a common footnote and footnote mark should be used. The commands \verbatim:\shorttitle{}: and \verbatim:\shortauthor{}: are used to designate the running heads for the paper. The \verbatim:\abstract{}:, \verbatim:\keywords\endkeywords:, and \verbatim:\subjclass\endsubjclass:\break commands are used to include the abstract, key words, and AMS subject classification numbers, respectively. If there is to be only one subject classification number, the \verbatim:\subjclass: command should be preceded by \verbatim:\oneclass:. (If the AMS numbers are unknown, leave that information blank.) Authors are responsible for providing AMS numbers. They can be found in the Annual Index of Math Reviews or through {\tt e-Math} ({\tt telnet e-math.ams.com}; login and password are both {\tt e-math}). Complete the topmatter section by including \verbatim:\endtopmatter:. \heading{3}{Equations and mathematics} Equations and mathematics are handled by standard \TeX\ commands. SIAM style is for numbered equations to appear flush with the left margin. The \verbatim:\leqno: and \verbatim:\leqalignno{}: commands are used for this purpose. If any letters indicating subequations are to be used with the numbers, they should be set in roman type. Clear equation formatting using \TeX\ can be challenging. Aside from the regular \TeX\ documentation, authors will find Nicholas J. Higham's book {\it Handbook of Writing for the Mathematical Sciences\/} [1] useful for guidelines and tips on formatting with \TeX. The book covers many other topics related to article writing as well. Authors commonly make mistakes by using \verbatim:<:, \verbatim:>:, \verbatim:\mid:, and \verbatim:\parallel: as delimiters, instead of \verbatim:\langle:, \verbatim:\rangle:, \verbatim:|:, and \verbatim:\|:. The incorrect symbols have particular meanings distinct from the correct ones and should not be confused. \bigskip \eightpoint \halign{#\hfil\quad&&#\hfil\quad\cr &{\bf Wrong} &&&& {\bf Right}&\cr &\verbatim:: & $$ &&& \verbatim:\langle x, y\rangle: & $\langle x, y\rangle$ \cr &\verbatim:5 < \mid A \mid: & $5 < \mid A \mid$ &&& \verbatim:5 < |A|: & $5 < |A|$ \cr & \verbatim:6x = \parallel x: \cr &\verbatim: - 1\parallel_{i}: & $6x = \parallel x - 1\parallel_{i}$ &&& \verbatim:6x = \|x - 1\|_{i}: & $6x = \| x - 1\|_{i}$\cr} \tenpoint\rm \bigskip Another common author error is to put large (and even medium sized) matrices in-line with the text, rather than displaying them. This creates unattractive line spacing problems, and should be assiduously avoided. Text-sized matrices (like $({a \atop b} {b \atop c})$) might be used but anything much more complex than the example cited will not be easy to read and should be displayed. More information on the formatting of equations and aligned equations is found in Knuth [2]. Authors bear primary responsibility for formatting their equations within margins and in an aesthetically pleasing and informative manner. The SIAM macros include additional roman math words, or ``log-like" functions, to those provided in standard \TeX. The following commands are added: \verbatim:\const:, \verbatim:\diag:, \verbatim:\grad:, \verbatim:\Range:, \verbatim:\rank:, and \verbatim:\supp:. These commands produce the same word as the command name in math mode, in roman type. Groups of equations that are not directly related to each other should normally be centered independently. This may be done through the \TeX\ math command \verbatim:\displaylines{}:. Numbering independently centered equations can be difficult, so Seroul and Levy's [3] macro \verbatim:\ldisplaylinesno{}: has been included in {\tt siamptex.sty}. \verbatim:\ldisplaylinesno{}: works just like \verbatim:\leqalignno:, except no ampersand is used to align the equations, since they are to be centered. \heading{4}{Text formatting} Section and subsection headings are both included using the \verbatim:\heading{}{}: command, which requires two arguments. The first argument is for the number, and the second is the title of the section or subsection. No extra spacing should be placed between paragraphs. The \verbatim:\heading: command inserts the required spacing between sections. SIAM style does not normally make use of plain \TeX's \verbatim:\item: command. The \verbatim:\meti: command is preferred for lists of items beginning with, for instance, bullets ($\bullet$) or roman numerals (iv). The \verbatim:\meti: command retains normal paragraph shape but places all labels aligned flush right. For example: \verbatim: \meti{(i)} This is the first item. \meti{(ii)} This is the second item of the series. : produces \meti{(i)} This is the first item. \meti{(ii)} This is the second item of the series. \medskip The \verbatim:\meti: macro was adapted from Seroul and Levy [3]. \heading{4.1}{Punctuation} All standard punctuation and all numerals should be set in roman type (upright) even within italic text. The only exceptions are periods and commas. They may be set to match the surrounding text. References to sections should use the symbol \S, generated by \verbatim:\S:. (If the reference begins a sentence, the term ``Section'' should be spelled out in full.) Authors should not redefine \verbatim:\S:, say, to be a calligraphic S, because \verbatim:\S: must be reserved for use as the section symbol. Authors sometimes confuse the use of various types of dashes. Hyphens (\verbatim:-:, -) are used for some compound words (many such words should have no hyphen but must be run together, like ``nonzero,'' or split apart, like ``well defined.'' Minus signs (\verbatim:$-$:, $-$) should be used in math to represent subtraction or negative numbers. En dashes (\verbatim:--:, --) are used for ranges (like 3--5, June--August), or for joined names (like Runge--Kutta). Em dashes (\verbatim:---:, ---) are used to set off a clause---such as this one---from the rest of the sentence. \heading{4.2}{Theorems, lemmas, and proofs} Theorems, lemmas, propositions, and so forth, have macros included for correct formatting. Below is an example. \verbatim: \thm{Theorem 4.1} Sample theorem included for illustration. Numbers and parentheses, like equation $(3.2)$, should be set in roman type. Note that words in displayed equations, such as $$ x^2 = Y^2 \sin z^2 \hbox{ for all } x $$ will appear in italic type in a theorem, though normally they should appear in roman.\endthm : This sample produces Theorem 4.1 below. \thm{Theorem 4.1} Sample theorem included for illustration. Numbers and parentheses, like equation $(3.2)$, should be set in roman type. Note that words in displayed equations, such as $$ x^2 = Y^2 \sin z^2 \hbox{ for all } x $$ will appear in italic type in a theorem, though normally they should appear in roman.\endthm The \verbatim:\cor:, \verbatim:\dfn:, \verbatim:\lem:, and \verbatim:\prop: commands all work similarly. Named theorems should be designated with the title in roman type, enclosed in parentheses. \verbatim: \thm{Theorem 3.2 {\rm (sample theorem with title)}} : Proofs are illustrated in the following example: \verbatim: \prf{Proof} The body of the proof. \qquad\endproof : If the proof ends with a displayed equation, the \verbatim:\endproof: box \endproof\ should appear two ems (\verbatim:\qquad:) from the end of the equation on line with it horizontally. \heading{5}{Figures and tables} Figures and tables are best handled in \TeX\ by putting them within a \verbatim:\topinsert \endinsert: or \verbatim:\midinsert \endinsert: environment. The appropriate amount of space should be left for the figure, and the caption should be formatted to be centered or as a paragraph if more than one line. Text should be italic, eight-point type, with the words ``Fig.~\#'' in small caps. A sample follows. \verbatim: \midinsert \vskip 22pc \centerline{\eightpoint{\smc Fig.~5.1}. \it Italic caption text.} \endinsert : SIAM tables should be formatted in eight-point type, with enough space left between entries and surrounding lines so that they do not touch. Take particular care with super- and subscript characters. See Knuth [2] or Seroul and Levy [3] for more information on the formatting of tables in \TeX. Table captions are similar to figure captions, but the word ``Table'' and the number appear on a separate line from the caption text. SIAM supports the use of {\tt psfig} for including {\smc PostScript} figures. All {\smc Post\-Script} figures should be sent in separate files. See the {\tt psfig} documentation (from wherever you acquired {\tt psfig}) for more details on the use of this style option. It is a good idea to submit high-quality hardcopy of all {\smc Post\-Script} figures just in case there is difficulty in the reproduction of the figure. Figures produced by other non-\TeX\ methods should be included as high-quality hardcopy when the manuscript is submitted. \heading{6}{Bibliographies} References are handled using the \verbatim:\Refs: command. All names are to be keyed initial upper case cap and small caps. Only the first and middle initials, followed by the last name, are to be used. Last names should never be listed first. Some representative sample entries are illustrated below: \verbatim| \Refs \ref 1\\ {\smc A.~U Thorone}, {\it Title of paper with lower case letters}, SIAM J. Abbrev. Correctly, 2 (1992), pp.~000--000.\endref \ref 2\\ \sameauthor, % generates a 3-em rule {\it Title of paper appearing in book}, in Book Title: With All Initial Caps, Publisher, Location, 1992.\endref \ref 3\\ {\smc W. Riter}, {\it Title of another paper appearing in a book}, in The Book Title, E.~D. One, E.~D. Two, and A.~N. Othereditor, eds., Publisher, Location, 1992, pp.~000--000.\endref \ref 4\\ {\smc A.~U. Thorone, A.~U. Thortwo, and A.~U. Thorthree}, {\it Title of Book{\rm III:} Note Initial Caps}, Publisher, Location, pp.~000--000, 1994.\endref \ref 5\\ {\smc A. Notherauth}, {\it Title of paper that's not yet published}, SIAM J. Abbrev. Correctly, to appear.\endref | Other types of references fall into the same general pattern. See the sample file or any SIAM journal for other examples. Authors must correctly format their bibliography to be considered as having used the macros correctly. An incorrectly formatted bibliography is not only time-consuming for SIAM to process but it is possible that errors may be introduced by keyboarders/copy editors. As an alternative to the above style of reference, an alphanumeric code may be used in place of the number (e.g., [AUTh90]). The same \verbatim:\Refs: and \verbatim:\ref: commands are used, but the command \verbatim:\resetrefindent{}: must be used before the \verbatim:\Refs: command, with the widest expected alphanumeric code as an argument. Another alternative is no number, simply the authors' names and the year of publication following in parentheses. The rest of the format is identical. To get an entry with no number in brackets before it, use the \verbatim:\xref: command. This method is acceptable but not encouraged. \heading{7}{Conclusion} Many other style suggestions and tips could be given to help authors but are beyond the scope of this document. Simple mistakes can be avoided by increasing your familiarity with how \TeX\ functions. The books referred to throughout this document are also useful to the author who wants clear, beautiful typography with minimal mistakes. \Refs \ref 1\\ {\smc N.~J. Higham}, {\it Handbook of Writing for the Mathematical Sciences}, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1993.\endref \ref 2\\ {\smc D.~E. Knuth}, {\it The \TeX book}, Addison Wesley, Reading, MA, 1986.\endref \ref 3\\ {\smc R. Seroul and S. Levy}, {\it A Beginner's Book of {\rm \TeX}}, Springer-Verlag, Berlin, New York, 1991.\endref \bye out this document are also useful to the author who wants clear, beautiful typography with minimal mistakes. \Refs \ref 1\\ {\smc N.~J. Higham}, {\it Handbook of Writing for the Mathematpexample.tex 664 144 7 47320 6124605440 6453 % Sample file for SIAM's plain TeX macro package. % 9-14-94 Paul Duggan \input siamptex.sty % author defined macros included for illustrative purposes only. % symbols for real numbers, complex, ... (\Bbb font from AMS-TeX % fonts v2.x also usable) \def\fR{{\bf R}} \def\fC{{\bf C}} \def\fK{{\bf K}} % misc. operators \def\Span {\mathop{\hbox{\rm span}}\nolimits} \def\Range{\mathop{\hbox{\rm Range}}\nolimits} \def\Det {\mathop{\hbox{\rm det}}} \def\Re {\mathop{\hbox{\rm Re}}} \def\Im {\mathop{\hbox{\rm Im}}} \def\Deg {\mathop{\hbox{\rm deg}}} % misc. \def\Kr{\hbox{\bf K}} \def\K { { K}} \def\sT{\hbox{$\cal T$}} \def\sB{\hbox{$\cal B$}} \def\bmatrix#1{\left[ \matrix{#1} \right]} % Each of the following commands have to be filled in with % something. If the data is unknown, the arguments can be % left blank. \topmatter \journal{SIAM J. E{\smc XAMPLE} F{\smc ILES}} \vol{1} \no{1, pp.~000--000} \date{October 1994} \copyyear{1994} \code{000} \title SAMPLE FILE FOR SIAM PLAIN \TeX\ MACRO PACKAGE\endtitle \shorttitle{SIAM MACRO EXAMPLE} \recdate{*}{October 1, 1994; accepted by the editors Month, x, xxxx. This work was supported by the Society for Industrial and Applied Mathematics, Philadelphia, Pennsylvania} \author Paul Duggan\fnmark{$^{\dag}$} \and Various A.~U. Thors\fnmark{$^{\ddag}$}\endauthor \address{$^{\dag}$}{Composition Department, Society for Industrial and Applied Mathematics, 3600 University City Science Center, Philadelphia, Pennsylvania, 19104-2688 ({\tt duggan@siam.org})} \address{$^{\ddag}$}{Various affiliations, supported by various foundation grants} \abstract{An example of SIAM \TeX\ macros is presented. Various aspects of composing manuscripts for SIAM's journals are illustrated with actual examples from accepted manuscripts. SIAM's stylistic standards are adhered to throughout, and illustrated.} \keywords polynomials, SI model\endkeywords \subjclass 33H40, 35C01\endsubjclass % if there is only one AMS subject number, the % command \oneclass should precede the \subjclass command. \endtopmatter \heading{1}{Introduction and examples} This paper presents a sample file for the use of SIAM's \TeX\ macro package. It illustrates the features of the macro package, using actual examples culled from various papers published in SIAM's journals. This sample will provide examples of how to use the macros to generate standard elements of journal papers, e.g., equations, theorems, or figures. This paper also serves as an exmple of SIAM's stylistic preferences for the formatting of such elements as bibliographic references, displayed equations, and aligned equations, among others. Some special circumstances are not dealt with this the sample file; for that information, please see the associated documentation file. {\it Note}. This paper is not to be read in any form for content. The conglomeration of equations, lemmas, and other text elements were put together solely for typographic illustrative purposes. For theoretical reasons, it is desirable to find characterizations of the conditions of breakdown of the algorithms that are based on the key {\it spaces} $\Kr_n(r^{(0)},A)$ and $\Kr_n(\tilde r^{(0)},A^*)$ rather than the {\it formulas} for the algorithms. In particular, we will characterize breakdown of the three Lanczos algorithms in terms of the {\it moment matrices} $\K_n(\tilde r^{(0)},A^*)^*\K_n(r^{(0)},A)$ and $\K_n(\tilde r^{(0)},A^*)^*A\K_n(r^{(0)},A)$. Here we define the matrix $\K_n(v,A)=\bmatrix{v&Av&\cdots&A^{n-1}v\cr}$, a matrix whose columns span the Krylov space $\Kr_n(v,A)$. The following three theorems give exact conditions for breakdown of the above algorithms. Detailed proofs may be found in [3]. A result similar to Theorem 2 is found in [1]; see also [5]. \thm{Theorem 1 {\rm (Lanczos--Orthodir breakdown)}} Suppose Lanczos/Orthodir has successfully generated $u^{(n-1)}\not=u$. Then the following are equivalent: \meti{$\bullet$} The algorithm does not break down at step $n$. \meti{$\bullet$} The matrix $\K_n(\tilde r^{(0)},A^*)^*A\K_n(r^{(0)},A)$ is nonsingular. \meti{$\bullet$} There exists a unique iterate $u^{(n)}$ satisfying $(2)$. \endthm \thm{Theorem 2 {\rm (Lanczos--Orthomin breakdown)}} Suppose Lanczos/Orthomin has successfully generated $u^{(n-1)}\not=u$. Then the following are equivalent: \meti{$\bullet$} The algorithm breaks down at step $n$. \meti{$\bullet$} Either $\K_{n-1}(\tilde r^{(0)},A^*)^*\K_{n-1}(r^{(0)},A)$ or $\K_n(\tilde r^{(0)},A^*)^*A\K_n(r^{(0)},A)$ is singular. \endthm \prop{Proposition 3 {\rm (zero sets of polynomials)}} Let $\fK=\fR$ or $\fC$. If $P$ is a complex nonzero polynomial in the variables $x_1,x_2,\ldots ,x_N\in\fK$, then $P(x)\not=0$ for almost every $x=(x_1,x_2,\ldots,x_N)\in \fK^N$. \endprop \prf{Proof} If $\fK=\fR$ and $P$ is nonzero, then either $\Re P(z)$ or $\Im P(z)$ is a nonzero (real) polynomial; if $\fK=\fC$, we may decompose each $x_i$ into real and imaginary parts, giving $2N$ variables, and consider the real polynomial $P(x)^*P(x)$. In any case, we may assume without loss of generality that $P$ is a nonzero real polynomial of real variables. We know that for any point $x$, the polynomial $P$ is the zero polynomial if and only if the polynomial and all its derivatives are zero at $x$. Let $V_0$ denote the set of zeros of $P$ in $\fR^N$. Suppose the set $V_0$ has nonzero measure. We know from integration theory (see, for example, [6, pp.\ 128f]) that almost every point of $V_0$ is a point of density in each of the $N$ coordinate directions. We recall that $x\in\fR$ is a point of density of a measurable subset $S\subseteq\fR$ if for any sequence of intervals $I_n$ such that $x\in I_n$ with measure $m(I_n)\rightarrow 0$ we have $m(S\cap I_n)/m(I_n)\rightarrow 1$. It is easily seen that at such points in $V_0$, the first partial derivatives of $P$ must necessarily be zero. Let $V_1$ be the points of $V_0$ where all first derivatives are also zero. We have just shown that $V_0$ and $V_1$ both have the same nonzero measure. The argument may be repeated for $V_1$ to show all second partial derivatives of $f$ are zero at almost every point of $V_0$, and so forth, resulting in the fact that $P$ and all its derivatives are zero on a set which has nonzero measure. The proof is completed by selecting any one of these points. \qquad\endproof \thm{Theorem 4 {\rm (Lanczos breakdown, iterate $n$)}} Let $\fK=\fR$ or $\fC$, $A, \tilde Z\in\fK^{N\times N}$, and $n\leq d(A)$. Then exactly one of the following three conditions holds for the Lanczos method with $\tilde r^{(0)}=\tilde Z^* r^{(0)}$. \meti{\rm (i)} Hard breakdown at step $n$ occurs for every vector $r^{(0)}\in\sT_n(A)\cap\fK^N$ $($and thus at least for almost every $r^{(0)}\in\fK^N)$. \meti{\rm (ii)} Hard breakdown at step $n$ occurs for a nonempty measure-zero set of vectors $r^{(0)}\in\sT_n(A)\cap\fK^N$ $($and thus a nonempty measure-zero set of vectors in $\fK^N)$. \meti{\rm (iii)} Hard breakdown at step $n$ occurs for no vectors $r^{(0)}\in\sT_n(A)\cap\fK^N$ $($and thus for at most a measure-zero set of vectors in $\fK^N)$. Furthermore, the same result holds if ``hard breakdown'' is replaced by ``soft breakdown'' in the statement of this theorem. \endthm \prf{Proof} For vectors $r^{(0)}\in\sT_n(A)\cap\fK^N$, breakdown is equivalent to singularity of an appropriate moment matrix. The set $\sT_n(A)\cap\fK^N$ amounts to almost every vector in $\fK^N$. Now, by Corollary 5, the set $S_n$ of vectors in $\fK^N$ for which the moment matrix of dimension $n$ is singular is either the set of all vectors or a subset of measure zero. If the moment matrix is singular for every vector (i.e., $S_n=\fK^N$), then it is singular for every vector in $\sT_n(A)\cap\fK^N$, giving case (i) above. Otherwise the set $S_n$ is measure zero in $\fK^N$. Thus $\sB_n\equiv S_n\cap(\sT_n(A)\cap\fK^N)$ is of measure zero and is either empty or nonempty. \qquad\endproof \heading{2}{Tables and figures} In Tables 1 and 2 we consider the unpreconditioned problem and also the (left) ILU- and MILU-preconditioned problem (see [2] and [4]). Runs for which convergence was not possible in ITMAX iterations are labeled by (--). \topinsert \hbox{\vbox{ \eightpoint {\parindent 0pt \centerline{\smc Table 1} \centerline{\it Model problem, $h^{-1}=128$, {\rm ITMAX=3000}. Number of iterations.}\vskip 6pt \hfil\vbox{\offinterlineskip \hrule \halign{&\vrule#&\strut\ \hfil#\ \cr height2pt&\omit&&\omit&&\omit&&\omit&&\omit &&\omit&&\omit&&\omit&&\omit&&\omit&&\omit&\cr &{\rm method $\backslash$ Dh: } & &0&&2${}^{-3}$&&2${}^{-2}$&&2${}^{-1}$&&2${}^{0}$& &2${}^{1}$&&2${}^{ 2}$&&2${}^{ 3}$&&2${}^{ 4}$&&2${}^{5}$&\cr height2pt&\omit&&\omit&&\omit&&\omit&&\omit &&\omit&&\omit&&\omit&&\omit&&\omit&&\omit&\cr \noalign{\hrule} height2pt&\omit&&\omit&&\omit&&\omit&&\omit &&\omit&&\omit&&\omit&&\omit&&\omit&&\omit&\cr &{GMRES}($\infty$) \hfill & & 290&& 269&& 245&& 220&& 200&& 189&& 186&& 189&& 207&& 249&\cr &{BCG} \hfill & & 308&& 341&& 299&&1518&& -- && -- && -- && -- && 533&& -- &\cr &{BCG}{\rm, random $u^{(0)}$} \hfill & & 309&& 354&& 300&& 310&& 313&& 301&& 299&& 302&& 290&& 293&\cr &{BCGNB} \hfill & & 308&& 353&& 284&& 338&& 253&& 240&& 243&& 240&& 302&& 962&\cr &{CGS} \hfill & & 272&& 254&& 222&& -- && -- && -- && -- && -- && -- && -- &\cr &{CGS}{\rm, random $u^{(0)}$} \hfill & & 193&& 189&& 200&& 192&& 193&& 175&& 225&& 212&& 216&& 197&\cr &{CGSNB} \hfill & & 272&& 284&& 212&& 196&& 151&& 162&& 158&& 173&& 156&& 256&\cr height1pt&\omit&&\omit&&\omit&&\omit&&\omit &&\omit&&\omit&&\omit&&\omit&&\omit&&\omit&\cr } \hrule}\hfil}}} \endinsert \topinsert \hbox{\vbox{ \eightpoint {\parindent 0pt \centerline{\smc Table 2} \centerline{\it Model Problem, $h^{-1}=128$,} \centerline{\it {\rm MILU}-preconditioning, {\rm ITMAX=500.} Number of iterations.} \medskip \hfil\vbox{\offinterlineskip \hrule \halign{&\vrule#&\strut\ \hfil#\ \cr height2pt&\omit&&\omit&&\omit&&\omit&&\omit &&\omit&&\omit&&\omit&&\omit&&\omit&&\omit&\cr &{\rm Method $\backslash$ Dh: } & &0&&2${}^{-3}$&&2${}^{-2}$&&2${}^{-1}$&&2${}^{0}$& &2${}^{1}$&&2${}^{ 2}$&&2${}^{ 3}$&&2${}^{ 4}$&&2${}^{5}$&\cr height2pt&\omit&&\omit&&\omit&&\omit&&\omit &&\omit&&\omit&&\omit&&\omit&&\omit&&\omit&\cr \noalign{\hrule} height2pt&\omit&&\omit&&\omit&&\omit&&\omit &&\omit&&\omit&&\omit&&\omit&&\omit&&\omit&\cr &{\rm {GMRES}($\infty$)} \hfill & & 27&& 25&& 24&& 26&& 28&& 28&& 25&& 19&& 14&& 10&\cr &{\rm {GMRES}($\infty$), random $u^{(0)}$} \hfill & & 33&& 29&& 28&& 29&& 31&& 31&& 29&& 24&& 19&& 14&\cr &{\rm {BCG}} \hfill & & 31&& 27&& 29&& 33&& 30&& 37&& 30&& 23&& 15&& 10&\cr % &{BCG}, random $u^{(0)}$, $\gamma=.1$ \hfill & % & 35&& 30&& 31&& 35&& 40&& 37&& 34&& 27&& 20&& 15&\cr &{\rm {BCG}, random $u^{(0)}$} \hfill & & 38&& 34&& 33&& 37&& 44&& 40&& 38&& 29&& 23&& 18&\cr &{\rm {BCGNB}} \hfill & & 28&& 27&& 29&& 30&& 34&& 35&& 30&& 23&& 15&& 10&\cr &{\rm {CGS}} \hfill & & 21&& 18&& 17&& 20&& 22&& 22&& 19&& 15&& 9&& 6&\cr &{\rm {CGS}, random $u^{(0)}$} \hfill & & 24&& 18&& 20&& 22&& 22&& 23&& 21&& 16&& 12&& 9&\cr &{\rm {CGSNB}} \hfill & & 21&& 18&& 17&& 20&& 22&& 27&& 20&& 15&& 9&& 6&\cr height1pt&\omit&&\omit&&\omit&&\omit&&\omit &&\omit&&\omit&&\omit&&\omit&&\omit&&\omit&\cr } \hrule}\hfil}}} \endinsert We make the following observations about these runs. \meti{$\bullet$} For the unpreconditioned problem, the standard {BCG} and {CGS} algorithms break down in a number of cases, but the use of random $u^{(0)}$ or the use of {BCGNB} or {CGSNB} resulted in convergence. Furthermore, the iteration counts for the algorithms {BCG} and {BCGNB} are in general comparatively close to those of the ``best'' method, {GMRES}($\infty$), while these algorithms have short economical recurrences, unlike {GMRES}($\infty$). This underscores the importance of the Lanczos algorithms as economical solution techniques. \meti{$\bullet$} For the ILU-preconditioned problems, in most cases all methods worked well. For the case of $Dh=1$, {BCG} gave an excessive number of iterations, but this was remedied significantly by {BCGNB} and much more so by the use of random $u^{(0)}$. Similarly, {CGS} could not converge, but {CGSNB} and {CGS} with random $u^{(0)}$ both converged. \meti{$\bullet$} For all of the MILU-preconditioned problems, all of the Lanczos-type algorithms performed quite well. In particular, the {BCG} algorithm gave approximately the same number of iterations as {GMRES}($\infty$). Figures 1 and 2 give representative plots of the convergence behavior of the algorithms for the case of $h^{-1}=128$, $Dh=4$, and no preconditioning. These results show that the new algorithms keep the residual size better behaved than the standard {BCG} and {CGS} algorithms over the course of the run. \topinsert \vskip 3.2in \centerline{\eightpoint\smc Fig.~1. \it Residual behavior: $h^{-1}=128$, $Dh=4$.} \endinsert \topinsert \vskip 3.2in \centerline{\eightpoint\smc Fig.~2. \it Residual behavior: $h^{-1}=128$, $Dh=4$.} \endinsert We now consider a more difficult class of finite difference problems, namely, central finite differencing applied to the Dirichlet problem $$ -u_{xx}(x,y) - u_{yy}(x,y) + D[(y-\textstyle{1\over 2}\displaystyle) u_x(x,y) + (x-\textstyle{1\over 3}\displaystyle) (x-\textstyle{2\over 3}\displaystyle) u_y(x,y)], $$ $$ - 43\pi^2u(x,y) = G(x,y) \quad {\rm on}\ \Omega=[0,1]^2,$$ $$u(x,y) = 1 + xy \quad \hbox{\rm on}\ \partial\Omega,$$ with $G(x,y)$ chosen as before so that the true solution is $u(x,y)=1+xy$. Again, we let $h$ denote the mesh size in each direction. For $D=0$ and $h$ small, the matrix generated by this problem is a symmetric indefinite matrix with 16 distinct negative eigenvalues and the rest of the spectrum positive. The standard conjugate residual algorithm applied to this problem with $h^{-1}=128$ and $D=0$ requires 766 iterations to converge to $||r^{(n)}||/||b||<\zeta=10^{-6}$. In any case, this is a difficult problem to solve. \def\qed{\vrule height8pt width4pt depth0pt\par\medskip} \def\Zero{{\bf 0}} \def\dis{\displaystyle} \def\b{\beta} \def\r{\rho} \def\X{{\bf X}} \def\Y{{\bf Y}} \def\bb{{\bar \beta}} \def\tbcr{\theta\bb c_h \rho_h} \def\ep{\varepsilon} Figures 3(a) and 3(b) show the compartmental diagrams for SI models without and with deaths due to the disease, for the situation in which the infectious period has only one stage. Figures 4(a) and 4(b) give the corresponding models with $m$ stages of infection. Venereal warts, caused by the human papilloma virus, and ordinary herpes are examples of sexually transmitted diseases without deaths due to the disease, although both are not quite SI diseases because of partial immunity. AIDS is the example of an SI disease with death due to the disease. Although our main focus is on the latter, we present results on SI models without deaths due to the disease because the simplification in the dynamics of such models throws light on the case with disease-related deaths. \topinsert \vskip 2in \centerline{\eightpoint {\smc Fig.} 3(a). SI {\it model for subgroup $i$, without death due to the disease.}} \vskip 2in \centerline{\eightpoint {\smc Fig.} 3(b). SI {\it model with death due to the disease.}} \endinsert \topinsert \vskip 2in \centerline{\eightpoint {\smc Fig.} 4(a). SI {\it model without deaths due to the disease with $m$ stages of infection.}} \vskip 2in \centerline{\eightpoint {\smc Fig.} 4(b). SI {\it model with deaths due to the disease, with $m$ stages of infection.}} \endinsert \heading{3}{Equations and alignments} The equations for the system follow directly from the definitions and the compartmental diagrams. For one infected stage with no disease-related deaths, the equations are $$ \dot X_i=-X_ig_i-\mu X_i+U_i, \leqno(1)$$ $$ \dot Y_i=X_ig_i-\mu Y_i. \leqno(2)$$ If there are multiple stages to the infection, (2) is replaced by (3)--(5) as follows: $$\leqalignno{\dot Y_{i1}&=X_ig_i-(k+\mu)Y_{i1}, &(3)\cr \dot Y_{ir}&=kY_{i,r-1}-(k+\mu)Y_{ir},\qquad r=2,\ldots,m-1 &(4)\cr \dot Y_{im}&=kY_{i,m-1}-\mu Y_{im}. &(5)\cr }$$ \heading{3.1}{The SI model with structured mixing} In this subsection we write the equations for the SI model with structured mixing, with one infected stage and with deaths due to the disease. The equations for multiple infected stages follow easily, as do those for SI models without death due to the disease. Recall that $f_{is}$ gives the fraction of population subgroup $i$'s contacts that are made in activity group $s$. The total contact rate of susceptibles from population subgroup $i$ in activity group $s$ must be $c_iX_if_{is}$. Let $\rho_{ij}(s)$ be the fraction of the contacts of group $i$ that are with members of group $j$, within activity group $s$. Assuming random allocation of the susceptibles and infecteds from each population subgroup to the activity groups, the fraction infected in group $j$ in activity group $s$ must be $Y_j/N_j$, giving $$ c_iX_if_{is}\rho_{ij}(s)\beta_j{Y_j \over N_j}\leqno(6) $$ for the rate at which susceptibles in $i$ are infected by contacts with infecteds from $j$ in activity group $s$. Thus, in this case, $g_i$ is given by $$ g_i=c_i\sum_sf_{is}\sum_j\rho_{ij}(s)\beta_j{Y_j \over N_j}, \leqno(7) $$ and (1a) and (1b) become $$ \dot X_i=-c_iX_i\sum_sf_{is}\sum_j\rho_{ij}(s)\beta_j{Y_j \over N_j}-\mu X_i+U_i, \leqno(8) $$ $$ \dot Y_i=c_iX_i\sum_sf_{is}\sum_j\rho_{ij}(s)\beta_j{Y_j \over N_j}-(\mu+k)Y_i. \leqno(9) $$ \heading{3.2}{Structured mixing within activity groups} If the mixing within activity groups is proportional mixing, then $\rho_{ij}(s)$ is given by (10): $$\rho_{ij}(s)={f_{js}c_jN_j\over \sum_pf_{ps}c_pN_p}, \leqno(10)$$ and (8) and (9) become (11) and (12): $$\dot X_i=-c_iX_i\sum_sf_{is}{\sum_jf_{js}c_j\beta_jY_j \over \sum_jf_{js}c_jN_j}-\mu X_i+U_i \leqno(11)$$ $$\dot Y_i=c_iX_i\sum_sf_{is}{\sum_jf_{js}c_j\beta_jY_j \over \sum_jf_{js}c_jN_j}-(k+\mu)Y_i. \leqno(12)$$ Expressions (11) and (12) show an important consequence of death due to the disease. If there are no deaths due to the disease, $N_j$ is constant on the asymptotically stable invariant subspace $U_j=\mu N_j$ for all $j$, and the first term, the nonlinear term, in (11) and (12) is a sum of {\it quadratic} terms. If there are deaths due to the disease, $N_j$ is no longer constant and the first term is a sum of rational expressions, each homogeneous of degree one. This observation extends to SIS, SIR, and SIRS models. \Refs \ref 1\\ {\smc R. Fletcher}, {\it Conjugate gradient methods for indefinite systems}, in Numerical Analysis Dundee 1975, G.~A. Watson, ed., Springer-Verlag, New York, Lecture Notes in Math. 506, 1976, pp. 73--89. \endref \ref 2\\ {\smc I. Gustafsson}, {\it Stability and rate of convergence of modified incomplete Cholesky factorization methods}, Ph.D. thesis, Chalmers University of Technology and the University of Goteborg, Goteborg, Sweden, 1979. \endref \ref 3\\ {\smc W.~D. Joubert}, {\it Generalized conjugate gradient and Lanczos methods for the solution of nonsymmetric systems of linear equations}, Ph.D. thesis and Report CNA-238, Center for Numerical Analysis, University of Texas, Austin, TX, January 1990. \endref \ref 4\\ {\smc J.~A. Meijerink and H.~A. van der Vorst}, {\it An iterative solution method for linear systems of which the coefficient matrix is a symmetric $M$-matrix}, Math. Comp., 31 (1977), pp.~148--162. \endref \ref 5\\ {\smc Y.~Saad}, {\it The Lanczos biorthogonalization algorithm and other oblique projection methods for solving large unsymmetric systems}, SIAM J. Numer. Anal., 19 (1982), pp. 485--506. \endref \ref 6\\ {\smc S. Saks}, {\it The Theory of the Integral}, G.~E. Stechert, New York, 1937. \endref \ref 7\\ {\smc M. Tinkham}, {\it Introduction to Superconductivity}, McGraw-Hill, New York, 1975. \endref \bye ficient matrix is a symmetric $M$-matrix}, Math. Comp., 31 (1977), pp.~148--162. \endref \ref 5\\ {\smc Y.~Saad}, {\it The Lanczos biorthogonalization algorithm and other oblique projection methods for solving large unsymmetric systems}, SIAM J. Numer. Anal., 19 (1982), pp. 485--506. \endref \ref 6siamptex.sty 664 144 7 27231 6124605441 6511 % SIAMPTEX.STY; 12-11-92; Paul Duggan, Society for Industrial % and Applied Mathematics. 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\def\plainoutput{\shipout\vbox{\makeheadline\pagebody\makefootline}% \advancepageno \ifnum\pageno>1 \global\footline={\hfill}% \fi \ifodd\pageno \global\headline={\hss\rightrh\hss{\tenpoint\rm\folio}}% \else \global\headline={\hskip-8pt{\tenpoint\rm\folio}\hss\leftrh\hss}% \fi \ifnum\outputpenalty>-\@MM \else\dosupereject\fi} \def\pagebody{\vbox to\vsize{\boxmaxdepth\maxdepth \pagecontents}} \def\makeheadline{\vbox to\z@{\vskip-22.5\p@ \line{\vbox to8.5\p@{}\rheadfont\the\headline}\vss}% \nointerlineskip} \def\makefootline{\baselineskip24\p@\vskip-8\p@\line{\the\footline}} \def\dosupereject{\ifnum\insertpenalties>\z@ % something is being held over \line{}\kern-\topskip\nobreak\vfill\supereject\fi} \def\relaxnext@{\let\next\relax} \def\footmarkform@#1{\ifmmode {}^{#1}\else$^{#1}$\fi } \let\thefootnotemark\footmarkform@ \def\makefootnote@#1#2{\insert\footins {\interlinepenalty\interfootnotelinepenalty \eightpoint \splittopskip=\ht\strutbox \splitmaxdepth=\dp\strutbox \floatingpenalty=\@MM \leftskip=\z@ \rightskip=\z@ \spaceskip=\z@ \xspaceskip=\z@ \leavevmode{#1}\footstrut\ignorespaces#2\unskip \lower\dp\strutbox\vbox to\dp\strutbox{}}} \newcount\footmarkcount@ \footmarkcount@=\z@ % Initialization \def\footnotemark{\let\@sf=\empty \relaxnext@ \ifhmode \edef\@sf{\spacefactor=\the\spacefactor}\/\fi \def\next@{\ifx[\next \let\next=\nextii@ \else \ifx"\next \let\next=\nextiii@ \else \let\next=\nextiv@ \fi\fi\next}% \def\nextii@[##1]{\footmarkform@{##1}\@sf}% \def\nextiii@"##1"{{##1}\@sf}% \def\nextiv@{\global\advance\footmarkcount@\@ne \footmarkform@{\number\footmarkcount@}\@sf}% \futurelet\next\next@} \def\footnotetext{\relaxnext@ \def\next@{\ifx[\next \let\next=\nextii@ \else \ifx"\next \let\next=\nextiii@ \else \let\next=\nextiv@ \fi\fi\next}% \def\nextii@[##1]##2{\makefootnote@{\footmarkform@{##1}}{##2}}% \def\nextiii@"##1"##2{\makefootnote@{##1}{##2}}% \def\nextiv@##1{\makefootnote@{\footmarkform@{% \number\footmarkcount@}}{##1}}% \futurelet\next\next@} \def\footnote{\let\@sf=\empty \relaxnext@ \ifhmode \edef\@sf{\spacefactor\the\spacefactor}\/\fi \def\next@{\ifx[\next \let\next=\nextii@ \else \ifx"\next \let\next=\nextiii@ \else \let\next=\nextiv@ \fi\fi\next}% \def\nextii@[##1]##2{\footnotemark[##1]\footnotetext[##1]{##2}}% \def\nextiii@"##1"##2{\footnotemark"##1"\footnotetext"##1"{##2}}% \def\nextiv@##1{\footnotemark\footnotetext{##1}}% \futurelet\next\next@} \def\adjustfootnotemark#1{\advance\footmarkcount@#1\relax} \skip\footins=18\p@ plus6\p@ minus6\p@ \def\footnoterule{\kern -4\p@\hrule width 3pc \kern 3.6\p@ } % rule = .4 pt high % Turn off @ as being a letter. % \catcode`\@=12 \let\next=\nextiv@ \fi\fi\next}% \def\nextii@[##1]##2{\footnotemark[##1]\footnotetext[##1]{##2}}% \def\nextiii@"##1"##2{\footnotemark"##1"\footnotetext"##1"{##2}}% \def\nextiv@##1{\footnotemark\footnotetext{##1}}% \futurelet\next\next@} \def\adjustfootnotemark#1{\advance\footmarkcount@#1\relax} \skip\footins=18\p@ plus6\