Computation of Partial Derivative of Matrix Determinant Arises in Multiparameter Eigenvalue Problems

This chapter considers an iterative scheme based on Newton’s method to find the solution of eigenvalues of Linear Multiparameter Matrix Eigenvalue Problems(LMEP). This chapter is also intended to review some iterative algorithms for computation of partial derivatives of matrix determinant involved in Newton’s Method. First algorithm is based on standard Jacobi formula and second one is based on LU-decomposition Method together with an algorithm to compute directly the entries of the matrices involved in decomposition. Finally, an implicit determinant method is used for the computation of the partial derivatives of matrix determinant. Although the algorithms can be used to find the approximate eigenvalues of LMEPs, but the numerical works are performed by considering three-parameter case for better convenience and to relax computational cost and time. Numerical example is presented to test the efficiency of each iterative algorithms. Errors in computed eigenvalues are also compared with exact eigenvalues evaluated by Δ-Method, adopted by Atkinson.

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