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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