# An Enhancement Algorithm of Cyclic Adaptive Fourier Decomposition

## TL;DR

This paper introduces an enhanced cyclic adaptive Fourier decomposition algorithm that combines a fast search for initial pole estimation with gradient descent optimization, achieving higher precision in rational approximation with reduced computational effort.

## Contribution

It proposes a novel enhancement algorithm that improves the accuracy and efficiency of cyclic adaptive Fourier decomposition by integrating fast search and gradient descent methods.

## Key findings

- Achieves higher approximation precision with less computation.
- Validates effectiveness through multiple examples.
- Outperforms traditional grid search-based methods.

## Abstract

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.