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-% 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