Full spectrum of a large sparse $\top$-palindromic quadratic eigenvalue problem ($\top$-PQEP) is considered arguably for the first time in this article. Such a problem is posed by calculation of surface Green's functions (SGFs) of mesoscopic transistors with a tremendous non-periodic cross-section. For this problem, general purpose eigensolvers are not efficient, nor is advisable to resort to the decimation method \etc to obtain the Wiener-Hopf factorization. After reviewing some rigorous understanding of SGF calculation from the perspective of $\top$-PQEP and nonlinear matrix equation, we present our new approach to this problem. In a nutshell, the unit disk where the spectrum of interest lies is broken down adaptively into pieces small enough that they each can be locally tackled by the generalized $\top$-skew-Hamiltonian implicitly restarted shift-and-invert Arnoldi (G$\top$SHIRA) algorithm with suitable shifts and other parameters, and the eigenvalues missed by this divide-and-conquer strategy can be recovered thanks to the accurate estimation provided by our newly developed scheme. Notably the novel non-equivalence deflation is proposed to avoid as much as possible duplication of nearby known eigenvalues when a new shift of G$\top$SHIRA is determined. We demonstrate our new approach by calculating the SGF of a realistic nanowire whose unit cell is described by a matrix of size $4000\times 4000$ at the density functional tight binding level, corresponding to a $8\times 8\; \mbox{nm}^{2}$ cross-section. We believe that quantum transport simulation of realistic nano-devices in the mesoscopic regime will greatly benefit from this work.

In this paper, we first study \ell_q minimization and its associated iterative reweighted algorithm for recovering sparse vectors. Unlike most existing work, we focus on <i>unconstrained</i> \ell_q minimization, for which we show a few advantages on noisy measurements and/or approximately sparse vectors. Inspired by the results in [Daubechies et al., <i>Comm. Pure Appl. Math.</i>, 63 (2010), pp. 1--38] for <i>constrained</i> \ell_q minimization, we start with a preliminary yet novel analysis for <i>unconstrained</i> \ell_q minimization, which includes convergence, error bound, and local convergence behavior. Then, the algorithm and analysis are extended to the recovery of low-rank matrices. The algorithms for both vector and matrix recovery have been compared to some state-of-the-art algorithms and show superior performance on recovering sparse vectors and low-rank matrices.

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