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diff --git a/Master/texmf-dist/tex/plain/siam/pexample.tex b/Master/texmf-dist/tex/plain/siam/pexample.tex new file mode 100644 index 00000000000..eb2c2f50101 --- /dev/null +++ b/Master/texmf-dist/tex/plain/siam/pexample.tex @@ -0,0 +1,523 @@ +% 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 |