  In this paper, we first propose a new parameterized definition of comparison
matrix of a given complex matrix, which generalizes the definition proposed by
\cite {Axe1}. Based on this, we propose a new class of complex nonsymmetric
algebraic Riccati equations (NAREs) which extends the class of nonsymmetric
algebraic Riccati equations proposed by \cite {Axe1}. We also generalize the
definition of the extremal solution of an NARE and show that the extremal
solution of an NARE exists and is unique. Some classical algorithms can be
applied to search for the extremal solution of an NARE, including Newton's
method, some fixed-point iterative methods and doubling algorithms. Besides, we
show that Newton's method is quadratically convergent and the fixed-point
iterative method is linearly convergent. We also give some concrete strategies
for choosing suitable parameters such that the doubling algorithms can be used
to deliver the extremal solutions, and show that the two doubling algorithms
with suitable parameters are quadratically convergent. Numerical experiments
show that our strategies for parameters are effective.
