Least squares solutions of quadratic inverse eigenvalue problem with partially bisymmetric matrices under prescribed submatrix constraints

The inverse eigenvalue problems play an important role in broad application areas such as system identification, Hopfield neural networks, control design, mass-spring system and molecular spectroscopy. This paper proposes an algorithm that yields a new method to efficiently and accurately compute the partially bisymmetric solutions (M,C,K) under prescribed submatrix constraints of the quadratic inverse eigenvalue problem MXΛ2+CXΛ+KX=0. The algorithm is developed based on the conjugate gradient normal equations residual minimizing (CGNR) method. We discuss the convergence properties of the algorithm. Finally, the performance of the algorithm is tested on two numerical examples and compared to the previous algorithm.