On sufficient and necessary conditions for the Jacobi matrix inverse eigenvalue problem
Copyright policy )On sufficient and necessary conditions for the Jacobi matrix inverse eigenvalue problem
The authors establish a sufficient condition, and some necessary conditions which complement those previously given by the first author and \textit{W. W. Sun} [Inverse Probl. 15, 977--987 (1999; Zbl 0942.65042)], for the existence of real numbers \(\beta _1 ,\dots ,\beta_{n-1}\), such that the \(n\times n\) Jacobi matrix with \(i\)th diagonal element \(1+\beta _{i-1}^2\), and \(i\)th superdiagonal element \(\beta _i\), where \(\beta _0 =0\), has a given set of eigenvalues. They also give an algorithm for computing some of the (generally non-unique) solutions. This matrix arises from the discretization of a certain Sturm--Liouville problem. When such discretizations are used to solve inverse Sturm-Liouville problems numerically, allowance must be made for the different asymptotic behaviour of the eigenvalues of the discrete and continuous problems. For some recent references on how this may be done, see the reviewer [Inverse Probl. 21, 223-238 (2005)].
- Xiamen University China (People's Republic of)
- University of Hong Kong China (People's Republic of)
Eigenvalues, singular values, and eigenvectors, algorithm, Inverse problems involving ordinary differential equations, inverse Sturm-Liouville problems, Numerical solutions to inverse eigenvalue problems, Inverse problems in linear algebra, Sturm-Liouville theory, inverse eigenvalue problems, existence of solutions, Numerical approximation of eigenvalues and of other parts of the spectrum of ordinary differential operators, Numerical solution of eigenvalue problems involving ordinary differential equations, Numerical solution of inverse problems involving ordinary differential equations, Jacobi matrices
Eigenvalues, singular values, and eigenvectors, algorithm, Inverse problems involving ordinary differential equations, inverse Sturm-Liouville problems, Numerical solutions to inverse eigenvalue problems, Inverse problems in linear algebra, Sturm-Liouville theory, inverse eigenvalue problems, existence of solutions, Numerical approximation of eigenvalues and of other parts of the spectrum of ordinary differential operators, Numerical solution of eigenvalue problems involving ordinary differential equations, Numerical solution of inverse problems involving ordinary differential equations, Jacobi matrices
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