Numerical solutions for large sparse quadratic eigenvalue problems Original Research Article

We study the quadratic eigenvalue problem (A + λB + λ2C) X = 0, where A, B, and C are symmetric real n × n matrices, and A, C are positive definite. We propose an efficient numerical algorithm to compute a few of the smallest positive eigenvalues of the problem and their associated eigenvectors. The new algorithm includes two parts. The first part gives iterative methods which can be used to compute the smallest positive eigenvalue. We develop a globally linearly convergent basic iteration and two locally quadratically convergent iterations. The second part uses the nonequivalence deflation technique. This technique allows us to transform the original problem to a new problem with different A, B, and C. The new problem has the same eigenvalues as the old problem except that the smallest positive eigenvalue of the old problem is replaced by zero. Therefore, the second smallest positive eigenvalue of the old problem becomes the smallest positive one for the new problem. Then the above proposed iterative methods can be applied again to find the second smallest positive eigenvalue. Proceeding in this way, we can find out the 3rd, 4th, … smallest positive eigenvalues. Our algorithm utilizes the symmetry and positivity of the given matrices, and avoids computing the undesired complex conjugate eigenvalues. Under some mild conditions, is efficient and reliable. The above process can also be used to find a few of the largest negative eigenvalues.