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Abstract. The harmonic Lanczos bidiagonalization method can be used to compute the smallest singular triplets of a large matrix A. We prove that for good enough projection subspaces harmonic Ritz values converge if the columns of A are strongly linearly independent. On the other hand, harmonic Ritz values may miss some desired singular values when the columns of A almost linearly dependent. Furthermore, harmonic Ritz vectors may converge irregularly and even may fail to converge. Based on the refined projection principle for large matrix eigenproblems due to the first author, we propose a refined harmonic Lanczos bidiagonalization method that takes the Rayleigh quotients of the harmonic Ritz vectors as approximate singular values and extracts the best approximate singular vectors, called the refined harmonic Ritz approximations, from the given subspaces in the sense of residual minimizations. The refined approximations are shown to converge to the desired singular vectors once the subspaces are sufficiently good and the Rayleigh quotients converge. An implicitly restarted refined harmonic Lanczos bidiagonalization algorithm (IRRHLB) is developed. We study how to select the best possible shifts, and suggest refined harmonic shifts that are theoretically better than the harmonic shifts used within the implicitly restarted Lanczos bidiagonalization algorithm (IRHLB). We propose a novel procedure that can numerically compute the refined harmonic shifts efficiently and accurately. Numerical experiments are reported that compare IRRHLB with five other algorithms based on the Lanczos bidiagonalization process. It appears that IRRHLB is at least competitive with them and can be considerably more efficient when computing the smallest singular triplets.

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The phenomena of bifurcation and chaos are studied for a class of second order nonlinear nonautonomous ordinary differential equations, which may be formulated by the nonlinear radially symmetric motion of the dynamically loaded hyperelastic spherical membrane composed of the Rivlin–Saunders material model with a noninteger power-law exponent. Firstly, based on the variational principle, the governing equation describing the problem is obtained with the spherically symmetric deformation assumption. Then, the dynamic characteristics of the system are qualitatively analyzed in detail in terms of different values of material parameters. Particularly, for a given constant load, the parameter spaces describing the bifurcation behaviors of equilibrium curves are established and the characteristics of equilibrium points are presented; for a periodically perturbed load, the quasi-periodic and chaotic behaviors are discussed for the systems with two and three equilibrium points, respectively.