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A Jacobi–Davidson type method for a right definite two-parameter eigenvalue problem. (English) Zbl 1025.65023

The authors present a new numerical iterative method for computing the eigenvalues of a right definite two-parameter eigenvalue problem defined by \[ \begin{cases} A_{1}x=\lambda B_{1}x+\mu C_{1}y \\ A_{2}y=\lambda B_{2}x+\mu C_{2}y \end{cases} \] where \(A_{i},\) \(B_{i}\) and \(C_{i}\) are \(n_{i}\times n_{i}\) for \(i=1,2.\) For the system to be right definite, the determinant \(\det \left|\begin{matrix} x^{t}B_{1}x x^{t}C_{1}x \\ y^{t}B_{2}y y^{t}C_{2}y \end{matrix} \right|\)should be positive. It is also known that if the matrices are symmetric then we have the existence of \(n_{1}n_{2}\) linearly independent eigenvectors \(x\otimes y\). The algorithm is based on a variant of the subspace method where correction equations are used to expand the search spaces. Numerical examples with large matrices \( n_{i}=100\) are presented at the end.

MSC:

65F15 Numerical computation of eigenvalues and eigenvectors of matrices

Software:

JDQZ; JDQR
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