For the steady-state solution of a differential equation from a one-dimensional multistate model in transport theory, we shall derive and study a nonsymmetric algebraic Riccati equation <i>B</i> <i>XF</i> <i>F</i>⁺<i>X</i> + <i>XB</i>⁺<i>X</i> = 0, where <i>F</i> (<i>I</i> <i>F</i>)<i>D</i> and <i>B</i> <i>BD</i> with positive diagonal matrices <i>D</i> and possibly low-ranked matrices <i>F</i> and <i>B</i>. We prove the existence of the minimal positive solution <i>X</i>^* under a set of physically reasonable assumptions and study its numerical computation by fixed-point iteration, Newtons method and the doubling algorithm. We shall also study several special cases. For example when <i>B</i> and <i>F</i> are low ranked then X*=(i=14UiViT) with low-ranked <i>U_i</i> and <i>V_i</i> that can be computed using more efficient iterative processes. Numerical examples will be given to illustrate our theoretical results.

For the steady-state solution of an integraldifferential equation from a two-dimensional model in transport theory, we shall derive and study a nonsymmetric algebraic Riccati equation B--XF--F+ X+ XB+ X= 0, where F I-s PD, B-(b I+ s P) D-and B+ b I+ s PD+ with a nonnegative matrix P, positive diagonal matrices D, and nonnegative parameters f, b b/(1-f) and s s/(1-f). We prove the existence of the minimal nonnegative solution X under the physically reasonable assumption f+ b+ s P (D++ D-)&lt; 1, and study its numerical computation by fixed-point iteration, Newtons method and doubling. We shall also study several special cases; eg when b = 0 and P is low-ranked, then X= s 2 UV is low-ranked and can be computed using more efficient iterative processes in U and V. Numerical examples will be given to illustrate our theoretical results.

We consider the solution of large-scale discrete-time algebraic Riccati equations with numerically low-ranked solutions. The structure-preserving doubling algorithm will be adapted, with the iterates for A not explicitly computed but in the recursive form Ak= A2 k 1 D (1) k

We propose an efficient eigensolver for computing densely distributed spectra of the two-dimensional transmission eigenvalue problem (TEP), which is derived from Maxwell's equations with Tellegen media and the transverse magnetic mode. The governing equations, when discretized by the standard piecewise linear finite element method, give rise to a large-scale quadratic eigenvalue problem (QEP). Our numerical simulation shows that half of the positive eigenvalues of the QEP are densely distributed in some interval near the origin. The quadratic JacobiDavidson method with a so-called non-equivalence deflation technique is proposed to compute the dense spectrum of the QEP. Extensive numerical simulations show that our proposed method processes the convergence efficiently, even when it needs to compute more than 5000 desired eigenpairs. Numerical results also illustrate that the computed

In this paper, we present an efficient QR algorithm for solving the linear response eigenvalue problem H x= x, where H is -symmetric with respect to 0= diag (I n, I n). Based on newly introduced -orthogonal transformations, the QR algorithm preserves the -symmetric structure of H throughout the whole process, and thus guarantees the computed eigenvalues to appear pairwise (, ) as they should. With the help of a newly established implicit -orthogonality theorem, we incorporate the implicit multi-shift technique to accelerate the convergence of the QR algorithm. Numerical experiments are given to show the effectiveness of the algorithm.