# to unbundle, sh this file (in an empty directory)
echo docuptex.tex 1>&2
sed >docuptex.tex <<'-------cut here----- docuptex.tex' 's/^X//'
X%%
X%% SIAM Plain TeX macro documentation.
X%% Paul Duggan
X%% 9-15-94
X
X\input siamptex.sty
X
X% The definitions are to provide a verbatim text environment
X
X\def\uncatcodespecials{\def\do##1{\catcode`##1=12 }\dospecials}
X\def\setupverbatim{\tt%
X  \def\par{\leavevmode\endgraf}%
X  \obeylines\uncatcodespecials\obeyspaces}
X{\obeyspaces\global\let =\ }
X\def\doverbatim#1{\def\next##1#1{##1\endgroup}\next}
X\def\verbatim{\begingroup\setupverbatim\doverbatim}
X
X
X\overfullrule=0pt
X
X\topmatter
X\vol{1}
X\no{1, pp.~000--000}
X\SIMAC
X\date{October 1994}
X\copyyear{1994}
X\code{000}
X
X\title Using SIAM's \TeX\ Macros\endtitle
X
X\shorttitle{USING SIAM'S \TeX\ MACROS}
X
X\recdate{*}{August 9, 1994. 
XThis work was supported by the Society for Industrial
Xand Applied Mathematics}
X
X\author Paul Duggan\fnmark{$^{\dag}$}\endauthor
X
X\address{$^{\dag}$}{Society for Industrial and Applied Mathematics,
XPhiladelphia, Pennsylvania ({\tt duggan@siam.org}). Questions, comments,
Xor corrections to this document may be directed to that e-mail address} 
X
X\abstract{Documentation is given for use of the SIAM \TeX\ macros.
XInstructions and suggestions for compliance with SIAM style
Xstandards are also included.}
X\subjclass\endsubjclass
X\keywords\endkeywords
X
X
X\endtopmatter
X
X
X\heading{1}{Introduction}
XThis file is documentation for the SIAM \TeX\ macros and provides
Xinstruction for submission of files formatted in \TeX.
X
XTo accommodate authors who electronically typeset their manuscripts,
XSIAM supports the use of \TeX. To ensure quality typesetting according
Xto SIAM style standards, SIAM provides a \TeX\ macro style file.
XUsing \TeX\ to format a manuscript should simplify the editorial process
Xand lessen the author's proofreading burden. However,
Xit is still necessary to proofread the galley proofs with care.
X
XElectronic files should not be submitted until the paper has been
Xaccepted, and then not until requested to do so by someone in the SIAM
Xoffice. Once an article is slated for an issue,
Xsomeone from the SIAM office will contact the author about any or all
Xof the following: editorial and stylistic queries, 
Xsupplying the source files (and any supplementary macros)
Xfor the properly formatted article, and handling figures.
X
XWhen submitting electronic files  (electronic submissions) 
X(to {\tt tex@siam.org}) write  the journal, issue, and author's
Xname in the subject line of the message. 
XAuthors are responsible for ensuring  that the paper generated 
Xfrom the source files exactly matches the paper that
Xwas accepted for publication by the review editor. If it does not,
Xinformation on how it differs should be indicated in the transmission 
Xof the file. When submitting a file, please be sure to include any 
Xadditional macros (other than those provided by SIAM) that will be 
Xneeded to run the paper. 
X
XSIAM uses MS-DOS-based computers for \TeX\ processing. Therefore
Xall filenames should be restricted to eight characters or less,
Xplus a three character extension.
X
XOnce the files are corrected here at SIAM, we will mail the revised 
Xproofs to be read against the original edited hardcopy 
Xmanuscript. We are not
Xset up to shuttle back and forth varying electronic versions of each
Xpaper, so we must rely on hard copy of the galleys. The author's proofreading
Xis an important but easily overlooked step. Even if SIAM were not
Xto introduce a single editorial change into your manuscript, there
Xwould still be a need to check, because electronic transmission
Xcan introduce errors.
X
X
XThis distribution contains the following items: {\tt siamptex.sty}, the
Xmain macro package; this documentation file; and a sample file {\tt
Xpexample.tex}. The sample file is representative of the standard way 
Xto apply the macros. The rest of this paper emphasizes some aspects 
Xof applying the macros, points out options and special cases, and 
Xdescribes the SIAM style standards.
XThe SIAM macros make use of the small caps font, {\tt cmcsc}, which
Xis not installed in some systems. This font along with other 
XAMS-\TeX\ fonts can be retrieved from the American Mathematical
XSociety via anonymous FTP to {\tt e-math.ams.com}.
X
X\heading{2}{Headings}
XThe top matter of a journal is in a standard format. The macro
Xand initial definitions should be included as follows.
X
X\verbatim:
X\input siamptex.sty
X\overfullrule=0pt
X
X:
X
XThe \verbatim:\overfullrule: should be set wider than zero points
Xwhile still being edited by the author to assist in locating 
Xlines that extend beyond the margins.
X
X
XThe ``slugline,'' which contains the journal and copyright information, is
Xcreated by special commands following the \verbatim:\topmatter:.
X
X\verbatim:
X\SIMAX
X\vol{0}
X\no{0, pp.~000--000}
X\date{April 1995}
X\copyyear{1995}
X\code{000}
X
X:
XBecause authors will probably not know exact 
Xvolume, number, date, or code, insert zeros in their place as
Xabove. SIAM will place the appropriate information in
Xthe commands in production; however, the commands must be used.
X\verbatim:\SIMAX:, \verbatim:\SIAP:, \verbatim:\SICOMP:,
X\verbatim:\SICON:, \verbatim:\SIDMA:, \verbatim:\SIMA:,
X\verbatim:\SIMAX:, \verbatim:\SINUM:, \verbatim:\SIOPT:,
X\verbatim:\SISC:, and \verbatim:\SIREV: are the commands provided
Xto include the journal title in the heading. 
X
XThe title and author(s) of the paper are indicated by the
X\verbatim:\title\endtitle: and \verbatim:\author\endauthor: commands.
XAuthor support and address information is indicated as follows.
X
X\verbatim:
X\author A.~U. Thorone\fnmark{$^{\dag}$}  
X        \and A.~U. Thortwo\fnmark{$^{\ddag}$}\endauthor
X
X\address{$^{\dag}$}{Address of the first author and support}
X\address{$^{\ddag}$}{Address of the second author and support}
X
X:
X
XIf more than two authors are included, each should be separated
Xby a comma after the \verbatim:\fnmark{}: command. If more than one
Xauthor shares common footnote information, then a common footnote
Xand footnote mark should be used.
X 
XThe commands \verbatim:\shorttitle{}: and \verbatim:\shortauthor{}:
Xare used to designate the running heads for the paper.
X
XThe \verbatim:\abstract{}:, \verbatim:\keywords\endkeywords:, and
X\verbatim:\subjclass\endsubjclass:\break commands are used to include the
Xabstract, key words, and AMS subject classification numbers, respectively.
XIf there is to be only one subject classification number, the
X\verbatim:\subjclass: command should be preceded by
X\verbatim:\oneclass:. (If the AMS numbers are unknown, leave that
Xinformation blank.) Authors are responsible for providing AMS numbers.
XThey can be found in the Annual Index of Math Reviews or
Xthrough {\tt e-Math} ({\tt telnet e-math.ams.com}; login
Xand password are both {\tt e-math}).
XComplete the topmatter section by including \verbatim:\endtopmatter:.
X
X 
X
X\heading{3}{Equations and mathematics}
XEquations and mathematics
Xare handled by standard \TeX\ commands. SIAM style is
Xfor numbered equations to appear flush with the left margin. The
X\verbatim:\leqno: and \verbatim:\leqalignno{}: commands are used for
Xthis purpose. If any letters indicating subequations are to be used
Xwith the numbers, they should be set in roman type.
X
XClear equation formatting using \TeX\ can be challenging. Aside from
Xthe regular \TeX\ documentation, authors will find Nicholas 
XJ. Higham's book {\it Handbook of Writing for the Mathematical
XSciences\/} [1] useful for guidelines and tips on formatting with
X\TeX. The book covers many other topics related to article
Xwriting as well.                
X
XAuthors commonly make mistakes by using
X \verbatim:<:, \verbatim:>:, \verbatim:\mid:, and 
X\verbatim:\parallel: as delimiters, instead of 
X\verbatim:\langle:, \verbatim:\rangle:, \verbatim:|:,
Xand \verbatim:\|:. The incorrect symbols have particular 
Xmeanings distinct from the correct ones and should not be confused.
X
X\bigskip
X\eightpoint
X\halign{#\hfil\quad&&#\hfil\quad\cr
X&{\bf Wrong} &&&& {\bf Right}&\cr
X&\verbatim:<x, y>: & $<x, y>$   &&&
X   \verbatim:\langle x, y\rangle: & $\langle x, y\rangle$ \cr
X&\verbatim:5 < \mid A \mid: &  $5 < \mid A \mid$ &&&
X   \verbatim:5 < |A|: &  $5 < |A|$ \cr
X&  \verbatim:6x = \parallel x: \cr
X&\verbatim:    - 1\parallel_{i}: & $6x = \parallel x - 1\parallel_{i}$ &&&
X   \verbatim:6x = \|x - 1\|_{i}: & $6x = \| x - 1\|_{i}$\cr}
X   \tenpoint\rm 
X\bigskip
X
XAnother common author error is to put large (and even medium sized)
Xmatrices in-line with the text, rather than displaying them. This
Xcreates unattractive line spacing problems, and should be assiduously
Xavoided. Text-sized matrices (like $({a \atop b} {b \atop c})$) might
Xbe used but anything much more complex than the example cited will
Xnot be easy to read and should be displayed.
X
XMore information on the formatting of equations and aligned
Xequations is found in Knuth [2]. Authors bear primary responsibility
Xfor formatting their equations within margins and in an aesthetically
Xpleasing and informative manner.
X
XThe SIAM macros include additional roman math words, or ``log-like"
Xfunctions, to those provided in standard \TeX.  The following
Xcommands are added: \verbatim:\const:, \verbatim:\diag:, \verbatim:\grad:,
X\verbatim:\Range:, \verbatim:\rank:, and \verbatim:\supp:.
XThese commands produce the same word as the command name
Xin math mode, in roman type.
X
X                     
XGroups of equations that are not directly related to each other
Xshould normally be centered independently. This may be done through
Xthe \TeX\ math command \verbatim:\displaylines{}:. Numbering 
Xindependently centered equations can be difficult, so Seroul and 
XLevy's [3] macro \verbatim:\ldisplaylinesno{}: has been included 
Xin {\tt siamptex.sty}. \verbatim:\ldisplaylinesno{}: works just 
Xlike \verbatim:\leqalignno:, except no ampersand is used to align 
Xthe equations, since they are to be centered.
X
X\heading{4}{Text formatting}
XSection and subsection headings are both included using the
X\verbatim:\heading{}{}: command, which requires two arguments. The 
Xfirst argument is for the number, and the second 
Xis the title of the section or
Xsubsection. 
X
XNo extra spacing should be placed between paragraphs. The
X\verbatim:\heading: command inserts the required spacing between 
Xsections.
X
XSIAM style does not normally make use of plain \TeX's 
X\verbatim:\item: command. The \verbatim:\meti: command is preferred for
Xlists of items beginning with, for instance, bullets ($\bullet$) or
Xroman numerals (iv). The \verbatim:\meti: command retains normal
Xparagraph shape but places all labels aligned flush right. For
Xexample:
X
X\verbatim:
X\meti{(i)} This is the first item.
X\meti{(ii)} This is the second item of the series.
X
X:
X
Xproduces
X
X\meti{(i)} This is the first item.
X\meti{(ii)} This is the second item of the series. 
X\medskip
X
XThe \verbatim:\meti: macro was adapted from Seroul and Levy [3].
X
X\heading{4.1}{Punctuation}
XAll standard punctuation and all numerals should be set in roman type
X(upright) even within italic text. 
XThe only exceptions are periods and commas. They may
Xbe set to match the surrounding text.
X
XReferences to sections should use the symbol \S, generated by
X\verbatim:\S:. (If the reference begins a sentence, the term ``Section''
Xshould be spelled out in full.) Authors should not redefine
X\verbatim:\S:, say, to be a calligraphic S, because \verbatim:\S:
Xmust be reserved for use as the section symbol.  
X
XAuthors sometimes confuse the use of various types of dashes.
XHyphens (\verbatim:-:, -) are used for some compound words (many
Xsuch words should have no hyphen but must be run together, 
Xlike ``nonzero,'' or split apart, like ``well defined.'' 
XMinus signs (\verbatim:$-$:, $-$)
Xshould be used in math to represent subtraction or negative numbers.
XEn dashes (\verbatim:--:, --) are used for ranges (like 3--5, 
XJune--August), or for joined names (like Runge--Kutta). Em dashes
X(\verbatim:---:, ---) are used to set off a clause---such as this 
Xone---from the rest of the sentence. 
X
X\heading{4.2}{Theorems, lemmas, and proofs}
XTheorems, lemmas, propositions, and so forth, have macros included
Xfor correct formatting. Below is an example.
X
X\verbatim:
X\thm{Theorem 4.1}
XSample theorem included for illustration.  
XNumbers and parentheses, like equation $(3.2)$, should be set 
Xin roman type.  Note that words in displayed equations, such as
X$$ x^2 = Y^2 \sin z^2 \hbox{ for all } x $$
Xwill appear in italic type in a theorem, though normally
Xthey should appear in roman.\endthm
X
X:
X
XThis sample produces Theorem 4.1 below.
X
X\thm{Theorem 4.1}
XSample theorem included for illustration.  
XNumbers and parentheses, like equation $(3.2)$, should be set 
Xin roman type.  Note that words in displayed equations, such as
X$$ x^2 = Y^2 \sin z^2 \hbox{ for all } x $$
Xwill appear in italic type in a theorem, though normally
Xthey should appear in roman.\endthm
X
X
XThe \verbatim:\cor:, \verbatim:\dfn:, \verbatim:\lem:, and 
X\verbatim:\prop:  commands
Xall work similarly. Named theorems should be designated with the
Xtitle in roman type, enclosed in parentheses.
X
X\verbatim:
X\thm{Theorem 3.2 {\rm (sample theorem with title)}}
X
X:
X
XProofs are illustrated in the following example:
X
X\verbatim:
X\prf{Proof}
XThe body of the proof.
X\qquad\endproof
X
X:
X
XIf the proof ends with a displayed equation, the \verbatim:\endproof:
Xbox \endproof\  should appear two ems (\verbatim:\qquad:)
Xfrom the end of the equation on line with it horizontally. 
X
X\heading{5}{Figures and tables}
XFigures and tables are best handled in \TeX\ by putting them within a
X\verbatim:\topinsert \endinsert: or \verbatim:\midinsert \endinsert: 
Xenvironment. The appropriate amount
Xof space should be left for the figure, and the caption should be
Xformatted to be centered or as a paragraph if more than one line. Text
Xshould be italic, eight-point type, with the words ``Fig.~\#'' in small
Xcaps. A sample follows.
X
X\verbatim:
X\midinsert
X\vskip 22pc
X\centerline{\eightpoint{\smc Fig.~5.1}. \it Italic caption text.}
X\endinsert
X
X:
X
XSIAM tables should be formatted in eight-point type, with enough
Xspace left between entries and surrounding lines so that they do not
Xtouch. Take particular care with super- and subscript characters.
XSee Knuth [2] or Seroul and Levy [3] for more information on
Xthe formatting of tables in \TeX. Table captions are similar to
Xfigure captions, but the word ``Table'' and the number appear on 
Xa separate line from the caption text.
X
XSIAM supports the use of {\tt psfig} for including {\smc PostScript}
Xfigures. All {\smc Post\-Script} figures should be sent in separate
Xfiles. See the {\tt psfig} documentation (from wherever you acquired 
X{\tt psfig}) for more details on the use
Xof this style option. It is a good idea to submit high-quality
Xhardcopy of all {\smc Post\-Script} figures just in case there 
Xis difficulty in the reproduction of the figure. Figures produced 
Xby other non-\TeX\ methods should be included as high-quality 
Xhardcopy when  the manuscript is submitted.
X
X\heading{6}{Bibliographies}
XReferences are handled using the \verbatim:\Refs: command. 
XAll names are to be keyed initial upper case cap and small 
Xcaps. Only the first and middle initials, followed by the 
Xlast name, are to be used. Last names should never be listed first. 
XSome representative sample entries are illustrated below:
X
X\verbatim|
X\Refs
X
X\ref 1\\ {\smc A.~U Thorone}, {\it Title of paper with lower case
Xletters}, SIAM J. Abbrev. Correctly, 2 (1992), pp.~000--000.\endref
X
X\ref 2\\ \sameauthor, % generates a 3-em rule
X{\it Title of paper appearing in book}, in Book Title: 
XWith All Initial Caps, Publisher, Location, 1992.\endref
X
X\ref 3\\ {\smc W. Riter},
X{\it Title of another paper appearing in a book}, in The Book 
XTitle, E.~D. One, E.~D. Two, and A.~N. Othereditor, eds., 
XPublisher, Location, 1992, pp.~000--000.\endref
X
X\ref 4\\ {\smc A.~U. Thorone, A.~U. Thortwo, and A.~U. Thorthree},
X{\it Title of Book{\rm III:} Note Initial Caps}, 
XPublisher, Location, pp.~000--000, 1994.\endref
X
X\ref 5\\ {\smc A. Notherauth}, {\it Title of paper that's not yet
Xpublished}, SIAM J. Abbrev. Correctly, to appear.\endref
X|
X
XOther types of references fall into the same general pattern. See the
Xsample file or any SIAM journal for other examples. Authors must
Xcorrectly format their bibliography to be considered as having used
Xthe macros correctly. An incorrectly formatted bibliography is not 
Xonly time-consuming for SIAM to process 
Xbut it is possible that errors may be introduced by 
Xkeyboarders/copy editors.
X
XAs an alternative to the above style of reference, an alphanumeric
Xcode may be used in place of the number (e.g., [AUTh90]). The same
X\verbatim:\Refs: and \verbatim:\ref: commands are used, but the
Xcommand \verbatim:\resetrefindent{}: must be used before the 
X\verbatim:\Refs: command, with the widest expected alphanumeric code
Xas an argument.
X
XAnother alternative is no number, simply the authors' names and
Xthe year of publication following in parentheses. The rest of the
Xformat is identical. To get an entry with no number in brackets 
Xbefore it, use the \verbatim:\xref: command. This method is acceptable
Xbut not encouraged.
X
X\heading{7}{Conclusion} Many other style suggestions and tips
Xcould be given to help authors but are beyond the scope of this
Xdocument. Simple mistakes can be avoided by increasing your familiarity
Xwith how \TeX\ functions. The books referred to throughout this document
Xare also useful to the author who wants clear, beautiful typography
Xwith minimal mistakes. 
X
X\Refs
X
X\ref 1\\ {\smc N.~J. Higham}, {\it Handbook of Writing for
Xthe Mathematical Sciences}, Society for Industrial and Applied
XMathematics, Philadelphia, PA, 1993.\endref
X
X\ref 2\\ {\smc D.~E. Knuth}, {\it The \TeX book}, Addison Wesley,
XReading, MA, 1986.\endref
X
X\ref 3\\ {\smc R. Seroul and S. Levy}, {\it A Beginner's Book of
X{\rm \TeX}}, Springer-Verlag, Berlin, New York, 1991.\endref
X
X
X\bye 
-------cut here----- docuptex.tex
echo pexample.tex 1>&2
sed >pexample.tex <<'-------cut here----- pexample.tex' 's/^X//'
X% Sample file for SIAM's plain TeX macro package.
X% 9-14-94 Paul Duggan
X\input siamptex.sty
X
X% author defined macros included for illustrative purposes only.
X% symbols for real numbers, complex, ... (\Bbb font from AMS-TeX
X% fonts v2.x also usable)
X
X\def\fR{{\bf R}}
X\def\fC{{\bf C}}
X\def\fK{{\bf K}}
X
X% misc. operators
X\def\Span {\mathop{\hbox{\rm span}}\nolimits}
X\def\Range{\mathop{\hbox{\rm Range}}\nolimits}
X\def\Det  {\mathop{\hbox{\rm det}}}
X\def\Re   {\mathop{\hbox{\rm Re}}}
X\def\Im   {\mathop{\hbox{\rm Im}}}
X\def\Deg  {\mathop{\hbox{\rm deg}}}
X
X% misc.
X
X\def\Kr{\hbox{\bf  K}}
X\def\K {     {      K}}
X\def\sT{\hbox{$\cal T$}}
X\def\sB{\hbox{$\cal B$}}
X
X\def\bmatrix#1{\left[ \matrix{#1} \right]}
X
X% Each of the following commands have to be filled in with
X% something. If the data is unknown, the arguments can be
X% left blank.
X
X\topmatter
X\journal{SIAM J. E{\smc XAMPLE} F{\smc ILES}}
X\vol{1}
X\no{1, pp.~000--000}
X\date{October 1994}
X\copyyear{1994}
X\code{000}
X
X
X\title SAMPLE FILE FOR SIAM PLAIN \TeX\ MACRO
XPACKAGE\endtitle 
X
X\shorttitle{SIAM MACRO EXAMPLE}
X
X\recdate{*}{October 1, 1994; accepted by the editors Month, x,
Xxxxx. This work was supported by the Society for Industrial
Xand Applied Mathematics, Philadelphia, Pennsylvania}
X
X\author Paul Duggan\fnmark{$^{\dag}$} \and Various A.~U.
XThors\fnmark{$^{\ddag}$}\endauthor
X
X\address{$^{\dag}$}{Composition Department, Society for
XIndustrial and Applied Mathematics, 3600 University City
XScience Center, Philadelphia, Pennsylvania, 19104-2688
X({\tt duggan@siam.org})}
X
X\address{$^{\ddag}$}{Various affiliations, supported by
Xvarious foundation grants}
X
X\abstract{An example of SIAM \TeX\ macros is presented.
XVarious aspects of composing manuscripts for SIAM's journals
Xare illustrated with actual examples from accepted
Xmanuscripts. SIAM's stylistic standards are adhered to
Xthroughout, and illustrated.}
X
X\keywords polynomials, SI model\endkeywords
X
X\subjclass 33H40, 35C01\endsubjclass
X
X% if there is only one AMS subject number, the
X% command \oneclass should precede the \subjclass command.
X
X\endtopmatter
X
X\heading{1}{Introduction and examples}
XThis paper presents a sample file for the use of SIAM's
X\TeX\ macro package. It illustrates the features of the
Xmacro package, using actual examples culled from various
Xpapers published in SIAM's journals. This sample will provide 
Xexamples of how to use the
Xmacros to generate standard elements of journal papers,
Xe.g., equations, theorems, or figures. This paper also
Xserves as an exmple of SIAM's stylistic preferences for the
Xformatting of such elements as bibliographic references,
Xdisplayed equations, and aligned equations, among others.
XSome special circumstances are not dealt with this the
Xsample file; for that information, please see the
Xassociated documentation file.
X
X{\it Note}. This paper is not to be read in any form for
Xcontent. The conglomeration of equations, lemmas, and other
Xtext elements were put together solely for typographic
Xillustrative purposes.
X
XFor theoretical reasons, it is desirable to find characterizations of the
Xconditions of breakdown of the algorithms that are based on the key {\it
Xspaces} $\Kr_n(r^{(0)},A)$ and $\Kr_n(\tilde r^{(0)},A^*)$ rather than
Xthe {\it formulas} for the algorithms.  In particular, we will
Xcharacterize breakdown of the three Lanczos algorithms in terms of the
X{\it moment matrices} $\K_n(\tilde r^{(0)},A^*)^*\K_n(r^{(0)},A)$ and
X$\K_n(\tilde r^{(0)},A^*)^*A\K_n(r^{(0)},A)$.  Here we define the matrix
X$\K_n(v,A)=\bmatrix{v&Av&\cdots&A^{n-1}v\cr}$, a matrix whose columns span
Xthe Krylov space $\Kr_n(v,A)$.
X
XThe following three theorems give exact conditions for breakdown of the
Xabove algorithms.  Detailed proofs may be found in [3].  A
Xresult similar to Theorem 2 is found in [1]; see also [5].
X
X
X\thm{Theorem 1 {\rm (Lanczos--Orthodir breakdown)}}
XSuppose Lanczos/Orthodir has successfully generated
X$u^{(n-1)}\not=u$.  Then the following are equivalent:
X
X\meti{$\bullet$} The algorithm does not break down at step $n$.
X
X\meti{$\bullet$} The matrix $\K_n(\tilde r^{(0)},A^*)^*A\K_n(r^{(0)},A)$
Xis nonsingular.
X
X\meti{$\bullet$} There exists a unique iterate $u^{(n)}$ satisfying $(2)$.
X\endthm
X
X
X\thm{Theorem 2 {\rm (Lanczos--Orthomin breakdown)}}
XSuppose Lanczos/Orthomin has successfully generated $u^{(n-1)}\not=u$.
XThen the following are equivalent:
X
X\meti{$\bullet$} The algorithm breaks down at step $n$.
X
X\meti{$\bullet$} Either
X$\K_{n-1}(\tilde r^{(0)},A^*)^*\K_{n-1}(r^{(0)},A)$ or
X$\K_n(\tilde r^{(0)},A^*)^*A\K_n(r^{(0)},A)$ is singular.
X\endthm
X
X
X\prop{Proposition 3 {\rm (zero sets of polynomials)}}
XLet $\fK=\fR$ or $\fC$.  If $P$ is a complex nonzero polynomial in the
Xvariables $x_1,x_2,\ldots ,x_N\in\fK$, then $P(x)\not=0$ for almost every
X$x=(x_1,x_2,\ldots,x_N)\in \fK^N$.
X\endprop
X
X\prf{Proof}
XIf $\fK=\fR$ and $P$ is nonzero, then either $\Re P(z)$ or $\Im P(z)$
Xis a nonzero (real) polynomial; if $\fK=\fC$, we may decompose each $x_i$
Xinto real and imaginary parts, giving $2N$ variables, and consider the
Xreal polynomial $P(x)^*P(x)$.  In any case, we may assume without loss of
Xgenerality that $P$ is a nonzero real polynomial of real variables.
X
XWe know that for any point $x$, the polynomial $P$ is the zero polynomial
Xif and only if the polynomial and all its derivatives are zero at $x$.
XLet $V_0$ denote the set of zeros of $P$ in $\fR^N$.  Suppose the set
X$V_0$ has nonzero measure.  We know from integration theory (see, for
Xexample, [6, pp.\ 128f]) that almost every point of $V_0$ is
Xa point of density in each of the $N$ coordinate directions.  We recall
Xthat $x\in\fR$ is a point of density of a measurable subset
X$S\subseteq\fR$ if for any sequence of intervals $I_n$ such that
X$x\in I_n$ with measure $m(I_n)\rightarrow 0$ we have
X$m(S\cap I_n)/m(I_n)\rightarrow 1$.
X
XIt is easily seen that at such points in $V_0$, the first
Xpartial derivatives of $P$ must necessarily be zero.  Let $V_1$ be the
Xpoints of $V_0$ where all first derivatives are also zero.  We have just
Xshown that $V_0$ and $V_1$ both have the same nonzero measure.  The
Xargument
Xmay be repeated for $V_1$ to show all second partial derivatives of $f$
Xare zero at almost every point of $V_0$, and so forth, resulting in the
Xfact that $P$ and all its derivatives are zero on a set which has nonzero
Xmeasure.  The proof is completed by selecting any one of these points.
X\qquad\endproof
X
X\thm{Theorem 4 {\rm (Lanczos breakdown, iterate $n$)}}
XLet $\fK=\fR$ or $\fC$, $A, \tilde Z\in\fK^{N\times N}$, and $n\leq d(A)$.
XThen exactly one of the following three conditions holds for the Lanczos
Xmethod with $\tilde r^{(0)}=\tilde Z^* r^{(0)}$.
X
X\meti{\rm (i)} Hard breakdown at step $n$ occurs for every vector
X$r^{(0)}\in\sT_n(A)\cap\fK^N$ $($and thus at least for almost every
X$r^{(0)}\in\fK^N)$.
X
X\meti{\rm (ii)} Hard breakdown at step $n$ occurs for a nonempty measure-zero
Xset of vectors $r^{(0)}\in\sT_n(A)\cap\fK^N$
X$($and thus a nonempty measure-zero set of vectors in $\fK^N)$.
X
X\meti{\rm (iii)} Hard breakdown at step $n$ occurs for no vectors
X$r^{(0)}\in\sT_n(A)\cap\fK^N$ $($and thus for at most a measure-zero set of
Xvectors in $\fK^N)$.
X
XFurthermore, the same result holds if ``hard breakdown'' is replaced by
X``soft breakdown'' in the statement of this theorem.
X\endthm
X
X
X\prf{Proof}
XFor vectors $r^{(0)}\in\sT_n(A)\cap\fK^N$, breakdown is equivalent to
Xsingularity of an appropriate moment matrix.  The set $\sT_n(A)\cap\fK^N$
Xamounts to almost
Xevery vector in $\fK^N$.  Now, by Corollary 5, the set $S_n$ of vectors in
X$\fK^N$ for which the moment matrix of dimension $n$ is singular is either
Xthe set of all vectors or a subset of measure zero.  If the moment matrix
Xis singular for every vector (i.e., $S_n=\fK^N$), then it is singular for
Xevery vector in $\sT_n(A)\cap\fK^N$, giving case (i) above.  Otherwise the
Xset $S_n$ is measure zero in $\fK^N$.  Thus
X$\sB_n\equiv S_n\cap(\sT_n(A)\cap\fK^N)$ is of measure zero
Xand is either empty or nonempty.
X\qquad\endproof
X
X\heading{2}{Tables and figures}
XIn Tables 1 and 2 we consider the unpreconditioned problem and also the (left) 
XILU- and MILU-preconditioned problem (see [2] and [4]).  Runs for which 
Xconvergence was not possible in ITMAX iterations are labeled by (--).
X
X
X\topinsert
X\hbox{\vbox{ \eightpoint
X{\parindent 0pt
X\centerline{\smc Table 1}
X\centerline{\it Model problem, $h^{-1}=128$, {\rm ITMAX=3000}.
X            Number of iterations.}\vskip 6pt
X\hfil\vbox{\offinterlineskip
X\hrule
X\halign{&\vrule#&\strut\ \hfil#\ \cr
Xheight2pt&\omit&&\omit&&\omit&&\omit&&\omit
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X&{\rm method $\backslash$ Dh: } &
X                   &0&&2${}^{-3}$&&2${}^{-2}$&&2${}^{-1}$&&2${}^{0}$&
X           &2${}^{1}$&&2${}^{ 2}$&&2${}^{ 3}$&&2${}^{ 4}$&&2${}^{5}$&\cr
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X&{GMRES}($\infty$) \hfill &
X& 290&& 269&& 245&& 220&& 200&& 189&& 186&& 189&& 207&& 249&\cr
X&{BCG} \hfill &
X& 308&& 341&& 299&&1518&& -- && -- && -- && -- && 533&& -- &\cr
X&{BCG}{\rm, random $u^{(0)}$} \hfill &
X& 309&& 354&& 300&& 310&& 313&& 301&& 299&& 302&& 290&& 293&\cr
X&{BCGNB} \hfill &
X& 308&& 353&& 284&& 338&& 253&& 240&& 243&& 240&& 302&& 962&\cr
X&{CGS} \hfill &
X& 272&& 254&& 222&& -- && -- && -- && -- && -- && -- && -- &\cr
X&{CGS}{\rm, random $u^{(0)}$} \hfill &
X& 193&& 189&& 200&& 192&& 193&& 175&& 225&& 212&& 216&& 197&\cr
X&{CGSNB} \hfill &
X& 272&& 284&& 212&& 196&& 151&& 162&& 158&& 173&& 156&& 256&\cr
Xheight1pt&\omit&&\omit&&\omit&&\omit&&\omit
X &&\omit&&\omit&&\omit&&\omit&&\omit&&\omit&\cr
X} \hrule}\hfil}}}
X\endinsert
X
X
X\topinsert
X
X\hbox{\vbox{ \eightpoint
X{\parindent 0pt
X\centerline{\smc Table 2}
X
X\centerline{\it Model Problem, $h^{-1}=128$,} 
X\centerline{\it {\rm MILU}-preconditioning, {\rm ITMAX=500.} 
XNumber of iterations.}
X
X\medskip
X
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X                   &0&&2${}^{-3}$&&2${}^{-2}$&&2${}^{-1}$&&2${}^{0}$&
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Xheight2pt&\omit&&\omit&&\omit&&\omit&&\omit
X &&\omit&&\omit&&\omit&&\omit&&\omit&&\omit&\cr
X&{\rm {GMRES}($\infty$)} \hfill &
X&  27&&  25&&  24&&  26&&  28&&  28&&  25&&  19&&  14&&  10&\cr
X&{\rm {GMRES}($\infty$), random $u^{(0)}$} \hfill &
X&  33&&  29&&  28&&  29&&  31&&  31&&  29&&  24&&  19&&  14&\cr
X&{\rm {BCG}} \hfill &
X&  31&&  27&&  29&&  33&&  30&&  37&&  30&&  23&&  15&&  10&\cr
X% &{BCG}, random $u^{(0)}$, $\gamma=.1$ \hfill &
X% &  35&&  30&&  31&&  35&&  40&&  37&&  34&&  27&&  20&&  15&\cr
X&{\rm {BCG}, random $u^{(0)}$} \hfill &
X&  38&&  34&&  33&&  37&&  44&&  40&&  38&&  29&&  23&&  18&\cr
X&{\rm {BCGNB}} \hfill &
X&  28&&  27&&  29&&  30&&  34&&  35&&  30&&  23&&  15&&  10&\cr
X&{\rm {CGS}} \hfill &
X&  21&&  18&&  17&&  20&&  22&&  22&&  19&&  15&&   9&&   6&\cr
X&{\rm {CGS}, random $u^{(0)}$} \hfill &
X&  24&&  18&&  20&&  22&&  22&&  23&&  21&&  16&&  12&&   9&\cr
X&{\rm {CGSNB}} \hfill &
X&  21&&  18&&  17&&  20&&  22&&  27&&  20&&  15&&   9&&   6&\cr
Xheight1pt&\omit&&\omit&&\omit&&\omit&&\omit
X &&\omit&&\omit&&\omit&&\omit&&\omit&&\omit&\cr
X} \hrule}\hfil}}}
X\endinsert
X
XWe make the following observations about these runs.
X
X\meti{$\bullet$} For the unpreconditioned problem, the standard
X{BCG} and {CGS} algorithms break down in a number of cases, but the use 
Xof random $u^{(0)}$ or the use of {BCGNB} or {CGSNB}
Xresulted in convergence.  Furthermore, the iteration counts for the
Xalgorithms {BCG} and {BCGNB} are in
Xgeneral comparatively close to those of the ``best'' method,
X{GMRES}($\infty$), while these algorithms have short economical
Xrecurrences, unlike {GMRES}($\infty$).  This underscores the 
Ximportance of the Lanczos algorithms as economical solution techniques.
X
X\meti{$\bullet$} For the ILU-preconditioned problems, in most cases
Xall methods worked well.  For the case of $Dh=1$, {BCG} gave
Xan excessive number of iterations, but this was remedied significantly
Xby {BCGNB} and much more so
Xby the use of random $u^{(0)}$.  Similarly, {CGS} could
Xnot converge, but {CGSNB} and {CGS} with random
X$u^{(0)}$ both converged.
X
X\meti{$\bullet$} For all of the MILU-preconditioned problems, all of
Xthe Lanczos-type algorithms performed quite well.  In particular, the
X{BCG} algorithm gave approximately the same number of
Xiterations as {GMRES}($\infty$).
X
XFigures 1 and 2 give representative plots of the convergence behavior of the
Xalgorithms for the case of $h^{-1}=128$, $Dh=4$, and no preconditioning.
XThese results show that the new algorithms keep the residual size 
Xbetter behaved than the standard {BCG} and {CGS}
Xalgorithms over the course of the run.
X
X\topinsert
X\vskip 3.2in
X\centerline{\eightpoint\smc Fig.~1. \it Residual 
Xbehavior: $h^{-1}=128$, $Dh=4$.}
X\endinsert
X
X
X\topinsert
X \vskip 3.2in
X\centerline{\eightpoint\smc Fig.~2. \it  Residual 
Xbehavior: $h^{-1}=128$, $Dh=4$.}
X\endinsert
X
XWe now consider a more difficult class of finite difference problems,
Xnamely, central finite differencing applied to the Dirichlet problem
X$$ -u_{xx}(x,y) - u_{yy}(x,y) +
X   D[(y-\textstyle{1\over 2}\displaystyle) u_x(x,y) +
X     (x-\textstyle{1\over 3}\displaystyle)
X     (x-\textstyle{2\over 3}\displaystyle) u_y(x,y)], $$ 
X$$ - 43\pi^2u(x,y) = G(x,y) \quad {\rm on}\ \Omega=[0,1]^2,$$
X$$u(x,y) = 1 + xy \quad \hbox{\rm on}\ \partial\Omega,$$
Xwith $G(x,y)$ chosen as before so that the true solution is $u(x,y)=1+xy$.
XAgain, we let $h$ denote the mesh size in each direction.  For $D=0$
Xand $h$ small, the matrix generated by this problem is a symmetric
Xindefinite matrix with 16 distinct negative eigenvalues and the rest
Xof the spectrum positive.
X
XThe standard conjugate residual algorithm applied to this problem with
X$h^{-1}=128$ and $D=0$ requires 766 iterations to converge to
X$||r^{(n)}||/||b||<\zeta=10^{-6}$.  In any case, this is a difficult
Xproblem to solve.
X
X  \def\qed{\vrule height8pt width4pt depth0pt\par\medskip} 
X  \def\Zero{{\bf 0}} 
X  \def\dis{\displaystyle} 
X  \def\b{\beta} 
X  \def\r{\rho} 
X  \def\X{{\bf X}} 
X  \def\Y{{\bf Y}} 
X  \def\bb{{\bar \beta}} 
X   \def\tbcr{\theta\bb c_h \rho_h} 
X  \def\ep{\varepsilon} 
X
X
X 
XFigures 3(a) and 3(b) show the compartmental diagrams for SI models without 
Xand with deaths due to the disease, for  the situation in which the infectious
Xperiod has only one stage. Figures 4(a) and 4(b) give the corresponding models 
Xwith $m$ stages of infection. Venereal 
Xwarts, caused by the human papilloma virus, and ordinary herpes are examples 
Xof sexually transmitted diseases without deaths 
Xdue to the disease, although both are not quite SI diseases because of 
Xpartial immunity. AIDS is the example of an SI disease with death due to 
Xthe disease. Although our main focus is on the latter, we present results 
Xon SI models without deaths due to the 
Xdisease because the simplification in the dynamics of such models 
X throws light on the case with disease-related deaths. 
X 
X\topinsert
X\vskip 2in
X\centerline{\eightpoint {\smc Fig.} 3(a). SI {\it model for subgroup $i$, without death
Xdue to the disease.}}
X\vskip 2in
X\centerline{\eightpoint {\smc Fig.} 3(b). SI {\it model with death due to the disease.}} 
X\endinsert
X
X\topinsert
X\vskip 2in
X\centerline{\eightpoint {\smc Fig.} 4(a). SI {\it model without deaths due to the
Xdisease with $m$ stages of infection.}}
X\vskip 2in
X\centerline{\eightpoint {\smc Fig.} 4(b). SI {\it model with deaths due to the disease,
Xwith $m$ stages of infection.}} 
X\endinsert
X
X\heading{3}{Equations and alignments} 
XThe equations for the system follow directly from the definitions and the 
Xcompartmental diagrams. For one infected stage with no disease-related 
Xdeaths, the equations are
X$$ \dot X_i=-X_ig_i-\mu X_i+U_i, \leqno(1)$$ 
X$$ \dot Y_i=X_ig_i-\mu Y_i. \leqno(2)$$ 
XIf there are multiple stages to the infection, (2) is replaced by 
X(3)--(5) as follows:
X$$\leqalignno{\dot Y_{i1}&=X_ig_i-(k+\mu)Y_{i1}, &(3)\cr 
X\dot Y_{ir}&=kY_{i,r-1}-(k+\mu)Y_{ir},\qquad  r=2,\ldots,m-1 &(4)\cr 
X\dot Y_{im}&=kY_{i,m-1}-\mu Y_{im}. &(5)\cr }$$ 
X 
X 
X
X\heading{3.1}{The SI model with structured mixing}
XIn this subsection we write the equations for the SI model with 
Xstructured mixing, with one infected stage and with deaths due to the 
Xdisease. The equations for multiple infected stages follow easily, as do 
Xthose for SI models without death due to the disease. Recall that $f_{is}$ 
Xgives  the fraction of population subgroup $i$'s 
Xcontacts that are made in activity group $s$. The total contact rate of 
Xsusceptibles from population subgroup $i$ in activity group $s$ must be 
X$c_iX_if_{is}$. Let $\rho_{ij}(s)$ be the fraction of the contacts of group 
X$i$ that are with members of group $j$, within activity group $s$. 
XAssuming random allocation of the susceptibles and infecteds from each 
Xpopulation subgroup to the activity groups, the fraction infected in group 
X$j$ in activity group $s$ must be $Y_j/N_j$, giving 
X$$ c_iX_if_{is}\rho_{ij}(s)\beta_j{Y_j \over N_j}\leqno(6)  $$ 
Xfor the rate at which susceptibles in $i$ are infected by contacts 
Xwith infecteds from $j$ in activity group $s$. Thus, in this case, $g_i$ is 
Xgiven by 
X$$ 
X g_i=c_i\sum_sf_{is}\sum_j\rho_{ij}(s)\beta_j{Y_j \over N_j}, 
X\leqno(7) 
X $$ 
Xand (1a) and (1b) become
X$$  \dot X_i=-c_iX_i\sum_sf_{is}\sum_j\rho_{ij}(s)\beta_j{Y_j \over 
XN_j}-\mu X_i+U_i, \leqno(8)  $$ 
X$$  \dot Y_i=c_iX_i\sum_sf_{is}\sum_j\rho_{ij}(s)\beta_j{Y_j \over 
XN_j}-(\mu+k)Y_i. \leqno(9)  $$ 
X 
X\heading{3.2}{Structured mixing within  activity groups} 
XIf the mixing within activity groups is proportional mixing, then 
X$\rho_{ij}(s)$ is given by (10): 
X$$\rho_{ij}(s)={f_{js}c_jN_j\over \sum_pf_{ps}c_pN_p}, \leqno(10)$$ 
Xand (8) and (9) become (11) and (12): 
X$$\dot X_i=-c_iX_i\sum_sf_{is}{\sum_jf_{js}c_j\beta_jY_j \over 
X\sum_jf_{js}c_jN_j}-\mu X_i+U_i \leqno(11)$$ 
X$$\dot Y_i=c_iX_i\sum_sf_{is}{\sum_jf_{js}c_j\beta_jY_j \over 
X\sum_jf_{js}c_jN_j}-(k+\mu)Y_i. \leqno(12)$$ 
X 
XExpressions (11) and (12) show an important consequence of death due 
Xto the disease. If there are no deaths due to the disease, $N_j$ is 
Xconstant on the asymptotically stable invariant subspace $U_j=\mu 
XN_j$ for all $j$, and the first term, the nonlinear term, in 
X(11) and (12) is a sum of {\it quadratic} terms. If there are deaths 
Xdue to the disease, $N_j$ is no longer constant and the first term is 
Xa sum of rational expressions, each homogeneous of degree one. This 
Xobservation extends to SIS, SIR, and SIRS models. 
X 
X 
X
X\Refs
X
X
X\ref 1\\
X{\smc R. Fletcher}, {\it Conjugate gradient methods for indefinite
Xsystems}, in Numerical Analysis Dundee 1975, G.~A. Watson, ed.,
XSpringer-Verlag, New York, Lecture Notes in Math. 506, 
X1976, pp. 73--89.
X\endref
X
X
X\ref 2\\
X{\smc I. Gustafsson}, {\it Stability and rate of convergence of
Xmodified incomplete Cholesky factorization methods}, Ph.D. thesis,
XChalmers University of Technology and the University of Goteborg,
XGoteborg, Sweden, 1979.
X\endref
X
X
X\ref 3\\
X{\smc W.~D. Joubert}, {\it Generalized conjugate gradient and
XLanczos methods for the solution of nonsymmetric systems of linear
Xequations},  Ph.D. thesis and Report
XCNA-238, Center for Numerical Analysis, University of Texas, 
XAustin, TX, January 1990.
X\endref
X
X
X\ref 4\\
X{\smc J.~A. Meijerink and H.~A. van der Vorst}, {\it An iterative
Xsolution method for linear systems of which the coefficient matrix is
Xa symmetric $M$-matrix}, Math. Comp., 31 (1977), pp.~148--162.
X\endref
X
X
X
X\ref 5\\
X{\smc Y.~Saad}, {\it The Lanczos biorthogonalization algorithm and
Xother oblique projection methods for solving large unsymmetric systems},
XSIAM J. Numer. Anal., 19 (1982), pp. 485--506.
X\endref
X
X
X\ref 6\\
X{\smc S. Saks}, {\it The Theory of the Integral}, G.~E. Stechert, 
XNew York, 1937.
X\endref
X
X\ref 7\\ 
X{\smc M. Tinkham}, {\it Introduction to
XSuperconductivity}, McGraw-Hill, New York, 1975.
X\endref
X
X\bye
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X\def\copyright#1{{\ooalign{\hfil\raise.07ex\hbox{c}\hfil\crcr#1\char"0D}}}
X
X\def\recdate#1#2{\global\recdate@true
X        \gdef\recdate@{#1Received by the editors \ignorespaces#2\unskip.}%
X        \makefootnote@{}{\recdate@}}
X                    
X\let\topmatter=\relax
X\def\endtopmatter{%
X        \global\headline={\hss\vtop to \z@ {%
X                \sixpoint\rm\noindent\thejournal \hfill%
X                \rm\copyright{\sevensy}\rm%
X                \ \thecopyyear\ Society for Industrial and Applied Mathematics\par
X                \noindent Vol. \thevol , No. \theno , \thedate\hfill\thecode\par
X        \vss}\hss}%                                   
X        \leavevmode\vskip8pt
X        \vbox{\noindent\raggedcenter\let\\=\break\titlefont\thetitle
X            \ifrecdate@*\fi}
X        \vskip6pt
X        \vbox{\noindent\raggedcenter\let\\=\break\authorfont\theauthor}
X        \ifdedicate@
X                \vskip6pt
X                \vbox{\noindent\raggedcenter\let\\=\break\dedicatefont
X                \thededicate}
X        \fi
X        \vskip14pt
X        {\eightpoint{\bf Abstract.\ }\theabstract\par
X        \ifkeywords@
X                \vskip10pt
X                {\bf Key words.} \thekeywords\par
X        \fi
X        \ifsubjclass@
X                \vskip10pt
X                {\bf AMS subject classification\ifoneclass@\else s\fi. }%
X                \thesubjclass\par
X        \fi}
X        \vskip8pt
X        \tenpoint}
X
X
X%%% THEOREM AND OTHER MACROS
X%%%
X\def\heading#1#2{%
X        \vskip6pt{\bf #1.\enspace\ignorespaces#2.\enspace}\ignorespaces}
X
X\def\thm#1{{\smc #1.}\begingroup\it\enspace\ignorespaces}
X\let\lem=\thm
X\let\cor=\thm
X\let\prop=\thm
X\let\dfn=\thm
X
X\def\endthm{\endgroup}
X\let\endlem=\endthm
X\let\endcor=\endthm
X\let\endprop=\endthm
X\let\enddfn=\endthm
X
X\def\prf#1{{\it #1}.\rm\enspace\ignorespaces}
X\let\rem=\prf
X\let\case=\prf
X
X\def\endproof{\vbox{\hrule height0.8pt\hbox{%
X   \vrule height1.3ex width0.8pt\hskip0.8ex
X   \vrule width0.8pt}\hrule height0.8pt
X  }}
X
X
X%%% FORMATTING AND MATH
X%%% 
X%%% NEW ROMAN MATH OPERATORS
X%%%
X\def\const{\mathop{\rm const}\nolimits}
X\def\diag{\mathop{\rm diag}\nolimits}
X\def\grad{\mathop{\rm grad}\nolimits}
X\def\Range{\mathop{\rm Range}\nolimits}
X\def\rank{\mathop{\rm rank}\nolimits}
X\def\supp{\mathop{\rm supp}\nolimits}
X
X%%% Centered displays, with numbering
X%%%
X
X\def\ldisplaylinesno#1{\displ@y\halign{
X  \hbox to\displaywidth{$\@lign\hfil\displaystyle##\hfil$}&
X    \kern-\displaywidth\rlap{$##$}\kern\displaywidth\crcr
X  #1\crcr}}
X
X%%% \meti to be used instead of \item
X%%%
X\def\meti#1{\parindent=2\parindent 
X\par\indent\llap{#1\enspace}\ignorespaces\parindent=.5\parindent}
X\def\metimeti#1{\parindent=2\parindent 
X\par\indent\llap{#1\enspace}\ignorespaces\parindent=.5\parindent}
X
X
X%%% Miscellaneous
X
X%%% REFERENCES
X
X\newdimen\refindent@
X\newdimen\refhangindent@
X\newbox\refbox@
X\setbox\refbox@=\hbox{\eightpoint\rm [00]}%   Default 2 digits
X\refindent@=\wd\refbox@
X
X\def\resetrefindent#1{%
X        \setbox\refbox@=\hbox{\eightpoint\rm [#1]}%
X        \refindent@=\wd\refbox@}
X
X\def\Refs{%
X        \unskip\vskip2pc
X        \centerline{\eightpoint\rm REFERENCES}%
X        \penalty10000
X        \vskip4pt
X        \penalty10000
X        \refhangindent@=\refindent@
X        \global\advance\refhangindent@ by 2em
X        \global\everypar{\hangindent\refhangindent@}%
X        \parindent=0pt\eightpoint\rm}
X
X\def\XRefs{%
X        \unskip\vskip2pc
X        \centerline{\eightpoint\rm REFERENCES}%
X        \penalty10000
X        \vskip4pt
X        \penalty10000
X        \refhangindent@=\refindent@
X        \global\everypar{\hangindent\refhangindent@}%
X        \parindent=0pt\eightpoint\rm}
X                                      
X\def\sameauthor{\leavevmode\vbox to 1ex{\vskip 0pt plus 100pt
X    \hbox to 3em{\leaders\hrule\hfil}\vskip 0pt plus 300pt}}
X    
X\def\ref#1\\#2\endref{\leavevmode\hbox to \refindent@{\hfil[#1]}\enspace #2\par}
X
X\def\xref\\#1\endref{\leavevmode #1\par}
X
X%%% OUTPUT
X
X\global\footline={\hss\eightpoint\rm\folio\hss}% first page
X\output{\plainoutput}
X\def\plainoutput{\shipout\vbox{\makeheadline\pagebody\makefootline}%
X  \advancepageno
X  \ifnum\pageno>1
X        \global\footline={\hfill}%
X  \fi     
X  \ifodd\pageno
X        \global\headline={\hss\rightrh\hss{\tenpoint\rm\folio}}%
X  \else
X        \global\headline={\hskip-8pt{\tenpoint\rm\folio}\hss\leftrh\hss}%
X  \fi
X  \ifnum\outputpenalty>-\@MM \else\dosupereject\fi}
X\def\pagebody{\vbox to\vsize{\boxmaxdepth\maxdepth \pagecontents}}
X\def\makeheadline{\vbox to\z@{\vskip-22.5\p@
X  \line{\vbox to8.5\p@{}\rheadfont\the\headline}\vss}%
X    \nointerlineskip}
X\def\makefootline{\baselineskip24\p@\vskip-8\p@\line{\the\footline}}
X\def\dosupereject{\ifnum\insertpenalties>\z@ % something is being held over
X  \line{}\kern-\topskip\nobreak\vfill\supereject\fi}
X
X\def\relaxnext@{\let\next\relax}
X\def\footmarkform@#1{\ifmmode {}^{#1}\else$^{#1}$\fi }
X\let\thefootnotemark\footmarkform@
X\def\makefootnote@#1#2{\insert\footins
X        {\interlinepenalty\interfootnotelinepenalty
X        \eightpoint \splittopskip=\ht\strutbox
X      \splitmaxdepth=\dp\strutbox
X        \floatingpenalty=\@MM 
X   \leftskip=\z@ \rightskip=\z@
X        \spaceskip=\z@ \xspaceskip=\z@
X        \leavevmode{#1}\footstrut\ignorespaces#2\unskip
X        \lower\dp\strutbox\vbox to\dp\strutbox{}}}
X\newcount\footmarkcount@
X \footmarkcount@=\z@                    % Initialization
X\def\footnotemark{\let\@sf=\empty \relaxnext@
X        \ifhmode \edef\@sf{\spacefactor=\the\spacefactor}\/\fi
X        \def\next@{\ifx[\next \let\next=\nextii@ \else
X                \ifx"\next \let\next=\nextiii@ \else
X                        \let\next=\nextiv@ \fi\fi\next}%
X        \def\nextii@[##1]{\footmarkform@{##1}\@sf}%
X        \def\nextiii@"##1"{{##1}\@sf}%
X        \def\nextiv@{\global\advance\footmarkcount@\@ne
X                \footmarkform@{\number\footmarkcount@}\@sf}%
X        \futurelet\next\next@}
X\def\footnotetext{\relaxnext@
X        \def\next@{\ifx[\next \let\next=\nextii@ \else
X                \ifx"\next \let\next=\nextiii@ \else
X                        \let\next=\nextiv@ \fi\fi\next}%
X        \def\nextii@[##1]##2{\makefootnote@{\footmarkform@{##1}}{##2}}%
X        \def\nextiii@"##1"##2{\makefootnote@{##1}{##2}}% 
X        \def\nextiv@##1{\makefootnote@{\footmarkform@{%
X                        \number\footmarkcount@}}{##1}}%
X        \futurelet\next\next@}
X\def\footnote{\let\@sf=\empty \relaxnext@
X        \ifhmode \edef\@sf{\spacefactor\the\spacefactor}\/\fi
X        \def\next@{\ifx[\next \let\next=\nextii@ \else
X                \ifx"\next \let\next=\nextiii@ \else
X                        \let\next=\nextiv@ \fi\fi\next}%
X        \def\nextii@[##1]##2{\footnotemark[##1]\footnotetext[##1]{##2}}%
X        \def\nextiii@"##1"##2{\footnotemark"##1"\footnotetext"##1"{##2}}%
X        \def\nextiv@##1{\footnotemark\footnotetext{##1}}%
X        \futurelet\next\next@}
X\def\adjustfootnotemark#1{\advance\footmarkcount@#1\relax}
X
X\skip\footins=18\p@ plus6\p@ minus6\p@
X
X\def\footnoterule{\kern -4\p@\hrule width 3pc \kern 3.6\p@ } % rule = .4 pt high
X
X
X
X%       Turn off @ as being a letter.
X%
X\catcode`\@=12
-------cut here----- siamptex.sty

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