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.
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