  This paper analyzes a special instance of nonsymmetric algebraic matrix
Riccati equations arising from transport theory. Traditional approaches for
finding the minimal nonnegative solution of the matrix Riccati equations are
based on the fixed point iteration and the speed of the convergence is linear.
Relying on simultaneously matrix computation, a structure-preserving doubling
algorithm (SDA) with quadratic convergence is designed for improving the speed
of convergence. The difficulty is that the double algorithm with quadratic
convergence cannot guarantee to work all the time. Our main trust in this work
is to show that applied with a suitable shifted technique, the SDA is
guaranteed to converge quadratically with no breakdown. Also, we modify the
conventional simple iteration algorithm in the critical case to dramatically
improve the speed of convergence. Numerical experiments strongly suggest that
the total number of computational steps can be significantly reduced via the
shifting procedure.
