An Improved Structure- Preserving Doubling Algorithm For a Structured Palindromic Quadratic Eigenvalue Problem

In this paper, we present a numerical method to solve the palindromic quadratic eigenvalue problem (PQEP) (λA+λQ+A)z = 0 arising from the vibration analysis of high speed trains, where A, Q ∈ Cn×n have special structures: both Q and A are m × m block matrices with each block being k × k, and moreover they are complex symmetric, block tridiagonal and block Toeplitz, and also A has only one nonzero block in the (1,m)th block position. The method is an improved version of Guo’s and Lin’s efficient solvent approach [SIAM J. Matrix Anal. Appl., 31 (2010), 2784-2801] which solves the PQEP by computing the so-called stabilizing solution to the mk×mk nonlinear matrix equation X + ATX−1A = Q via the doubling algorithm. Here, we exploit the fact that the stabilizing solutionX differs from Q only in its (m,m)th block position and which had also been noted and exploited by Guo and Lin there, too. What distinguishes our method from theirs is that we devise a new nonlinear matrix equation X̃ + Ã T X̃−1Ã = Q̃ of only k × k in size just for computing the differing block. The new and much smaller matrix equation is also solved by the doubling algorithm at the same speed in terms of the number of doubling iterations as but about 4.8 times faster in flops than the doubling algorithm on the larger matrix equation, and its stabilizing solution X̃ is used to recover the bigger stabilizing solution X. Numerical examples are presented to show the effectiveness of the improved method.