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\section*{Numerical methods for two-parameter eigenvalue problems} By {\sl Bor Plestenjak}.  
\medskip \noindent We consider the    \emph{two-parameter eigenvalue problem} \cite{Atkinson}
 \begin{eqnarray}   A_1x_1&=&\lambda B_1x_1+\mu C_1x_1,\nonumber\\[-2ex]   \label{problem} \\[-2ex]   A_2x_2&=&\lambda B_2x_2+\mu C_2x_2,\nonumber \end{eqnarray}
 where $A_i,B_i$, and $C_i$ are given $n_i\times n_i$ matrices over 
${\mathbb C}$, $\lambda,\mu\in{\mathbb C}$, and $x_i\in {\mathbb C}^{n_i}$ for $i=1,2$. A pair $(\lambda,\mu)$ is  an \emph{eigenvalue} if it satisfies (\ref{problem})  for nonzero vectors $x_1,x_2$. The tensor product $x_1\otimes x_2$ is then the corresponding \emph{eigenvector}. 
  On the tensor product space
 $S:= {\mathbb C}^{n_1}\otimes {\mathbb C}^{n_2}$ of the dimension $N:=n_1n_2$ we can define \emph{operator determinants} \begin{eqnarray*}   \Delta_0&=&B_1\otimes C_2-C_1\otimes B_2,\cr   \Delta_1&=&A_1\otimes C_2-C_1\otimes A_2,\cr   \Delta_2&=&B_1\otimes A_2-A_1\otimes B_2. \end{eqnarray*} The two-parameter problem $(\ref{problem})$ is  \emph{nonsingular} if its operator determinant $\Delta_0$ is invertible. In this case  $\Delta_0^{-1}\Delta_1$ and  $\Delta_0^{-1}\Delta_2$ commute and problem (\ref{problem})  is equivalent to the associated problem \begin{eqnarray} \Delta_1 z&=&\lambda \Delta_0 z,\nonumber\\[-2ex] \label{drugi}\\[-2ex] \Delta_2 z&=&\mu \Delta_0 z\nonumber \end{eqnarray} for decomposable tensors $z\in S$, $z=x_1\otimes x_2$.   Some numerical methods and a basic theory of the two-parameter eigenvalue problems will be presented. A possible approach is to solve the associated couple  of generalized eigenproblems (\ref{drugi}), but this is  only feasible for problems of low dimension because the size of the matrices  of (\ref{drugi}) is $N\times N$. For larger problems, if we are  interested in a part of the eigenvalues close to a given target,  the Jacobi--Davidson method \cite{HP,HP2,HP3} gives very good results.  
Several applications lead to singular two-parameter eigenvalue problems  where $\Delta_0$ is singular. Two such examples are model updating \cite{Cottin} and the quadratic two-parameter eigenvalue problem 

\begin{eqnarray}   (S_{00}+\lambda S_{10} +\mu S_{01} + \lambda^2 S_{20} +\lambda \mu S_{11} + \mu^2 S_{02})x&=&0\nonumber\\[-1.7ex]   \label{qepproblem} \\[-1.7ex]   (T_{00}+\lambda T_{10} +\mu T_{01} + \lambda^2 T_{20} +\lambda \mu T_{11} + \mu^2 T_{02})y&=&0.\nonumber \end{eqnarray} 

We can linearize (\ref{qepproblem}) as a singular two-parameter  eigenvalue problem, a  possible linearization is 
 $$\left(\left[\begin{matrix}S_{00} & S_{10} &S_{01} \cr 0 & -I & 0 \cr 0 & 0 & -I  \end{matrix}\right] 
+\lambda \left[\begin{matrix}0 & S_{20} &{1\over 2}S_{11} \cr I & 0& 0\cr 0& 0& 0\end{matrix}\right] +\mu \left[\begin{matrix} 0&{1\over 2}S_{11} &S_{02} \cr 0&0&0\cr I&0&0\end{matrix}\right]\right)\widetilde x=0 $$
 $$\left(\left[\begin{matrix}T_{00} & T_{10} &T_{01} \cr 0& -I & 0\cr 0& 0& -I\end{matrix}\right] +\lambda \left[\begin{matrix} 0& T_{20} &{1\over 2}T_{11} \cr I &0&0\cr 0&0&0\end{matrix}\right] +\mu \left[\begin{matrix} 0&{1\over 2}T_{11} &T_{02} \cr 0&0&0\cr I&0&0\end{matrix}\right]\right)\widetilde y=0, $$ 

where $\widetilde x=\left[\begin{matrix}x \cr \lambda x \cr  \mu x\end{matrix}\right]$ and   $\widetilde y=\left[\begin{matrix}y \cr \lambda y \cr  \mu y\end{matrix}\right]$.   
Some theoretical results and numerical methods for  singular two-parameter eigenvalue problems will be presented.  

\begin{thebibliography}{99} 
\bibitem{Atkinson} {\sc F.~V.~Atkinson},  	{\sl Multiparameter eigenvalue problems},    Academic Press, New York, 1972. 
\bibitem{Cottin} {\sc N.~Cottin},  	{\sl Dynamic model updating --- a multiparameter eigenvalue problem},  	{Mech. Syst. Signal Pr.}, {15}~(2001), pp.~649--665. 
\bibitem{HP} {\sc M.~E.~Hochstenbach and B.~Plestenjak},  	{\sl A Jacobi--Davidson type method for a right definite two-parameter  	eigenvalue problem},     SIAM J. Matrix Anal. Appl., 24~(2002), pp.~392--410. 
\bibitem{HP2} {\sc M.~E. Hochstenbach, T.~Ko{\v{s}}ir, and B.~Plestenjak},  	{\sl A   {J}acobi--{D}avidson type method for the nonsingular two-parameter eigenvalue   problem}, SIAM J. Matrix Anal. Appl., 26 (2005), pp.~477--497. 
\bibitem{HP3} {\sc M.~E.~Hochstenbach and B.~Plestenjak},  	{\sl Harmonic Rayleigh--Ritz extraction for the multiparameter     eigenvalue problem}, to appear in ETNA. \end{thebibliography}

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%two-parameter eigenvalue problem, Jacobi-Davidson method, model updating","65F15","15A18","","02:19:23","Sat Mar 08 2008","213.143.69.43"
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