We consider the solution of the large-scale nonsymmetric algebraic Riccati equation XCX - XD - AX + B = 0 from transport theory (Juang 1995), with M equivalent to [D, -C; -B, A] is an element of R(2nx2n) being a nonsingular M-matrix. In addition, A, D are rank-1 updates of diagonal matrices, with the products A(-1)u, A(-T)u, D(-1) v and D(-T) v computable in O(n) complexity, for some vectors u and v, and B, C are rank 1. The structure-preserving doubling algorithm by Guo et al. (2006) is adapted, with the appropriate applications of the Sherman-Morrison-Woodbury formula and the sparse-plus-low-rank representations of various iterates. The resulting large-scale doubling algorithm has an O(n) computational complexity and memory requirement per iteration and converges essentially quadratically, as illustrated by the numerical examples.