Abstract
We propose a new idea to construct an effective algorithm to compute the minimal positive solution of the nonsymmetric algebraic Riccati equations arising from transport theory. For a class of these equations, an important feature is that the minimal positive solution can be obtained by computing the minimal positive solution of a couple of fixed‐point equations with vector form. Based on the fixed‐point vector equations, we introduce a new algorithm, namely, two-step relaxation Newton, derived by combining two different relaxation Newton methods to compute the minimal positive solution. The monotone convergence of the solution sequence generated by this new algorithm is established. Numerical results are given to show the advantages of the new algorithm for the nonsymmetric algebraic Riccati equations in vector form.
Highlights
We are interested in iteratively solving the algebraic Riccati equation arising from transport theory see, e.g., 1–5 and references therein : XCX − XE − AX B 0, 1.1 where A, B, C, E ∈ Rn×n are given by
For the discussion on more general nonsymmetric algebraic Riccati equations arising in real world, we refer the interested reader to 13–17
In this paper, based on the fixed-point equations 1.7, we present an accelerated version of the NBJ scheme, namely, two-step relaxation Newton method, to compute the minimal positive solution
Summary
The parameters c and α satisfy 0 < c ≤ 1, 0 ≤ α < 1, and {ci}ni 1 and {ω}ni 1 are sets of the composite Gauss-Legendre weights and nodes, respectively, on the interval 0, 1 satisfying n. Based on the vector equations 1.7 , Lu 9 investigated the simple iteration SI method to compute the minimal positive solution as follows: uk 1 uk ◦ P vk e, vk 1 vk ◦ Quk e, v0 0, u0 0. In this paper, based on the fixed-point equations 1.7 , we present an accelerated version of the NBJ scheme, namely, two-step relaxation Newton method, to compute the minimal positive solution. In the end of the work of this paper, we have constructed another two-step relaxation Newton algorithm which performs much better than the SI, NBJ methods; at the moment we cannot theoretically prove the convergence of this method to the minimal positive solution of the vector equations 1.7.
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