  The paper investigates the complex gradient descent method (CGD) for the best
rational approximation of a given order to a function in the Hardy space on the
unit disk. It is equivalent to finding the best Blaschke form with free poles.
The adaptive Fourier decomposition (AFD) and the cyclic AFD methods in
literature are based on the grid search technique. The precision of these
methods is limited by the grid spacing. The proposed method employs a fast
search algorithm to find the initial for CGD, then finds the target poles by
gradient descent optimization. Hence, it can reach higher precision with less
computation cost. Its validity and effectiveness are confirmed by several
examples.
