Accurate BABE Factorisation of Tridiagonal Matrices for Eigenproblems

Recently Fernando successfully resurrected a classical method for computing eigen vectors which goes back to the times of Cauchy This algorithm has been in the doldrums for nearly fty years because of a fundamental di culty highlighted by Wilkinson The algorithm is based on the solution of a nearly homogeneous system of equations J I z k ek zk for the approximate eigenvector z where is an eigenvalue shift k is a scalar and ek is a unit vector The best minimal residual approximation for z is obtained by choosing the k k n for which j k j is minimal and tiny If the LDU factorisation is computed from the top of the matrix J I and the UDL factorisation from the bottom then the residual k ap pear as the pivot where these two factorisations meet We study the properties of this BABE burn at both ends factorisation which are closely related to the properties of LDU and UDL factorisations We show that LDU UDL and BABE factorisations possess mixed stability with tiny relative perturbations However it is demonstrated that the computed eigenvectors are mixed stable if the matrix is real symmetric and backward stable if the matrix is not real symmetric If the matrix is real symmetric then the inertia counts given by the pivots of the LDL UDU t and BABE factorisa tions are also backward stable We also prove the monotonicity of the inertia count in oating point arithmetic with respect to the shift as given by the pivots of the BABE factorisation The monotonicity property was rst studied by Kahan in the context of determining accurate eigenvalues via the LDL factorisation but it is also imperative for computing accurate eigenvectors