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A constructive approach for finding arbitrary roots of polynomials by neural networks
1Institute of Intelligent Machines, Chinese Academy of Sciences, P.O. Box 1130, Hefei, Anhui 230031, China. dshuang@iim.ac.cn
IEEE Transactions on Neural Networks
|September 24, 2004
Summary
This study introduces a neural network method using a constrained learning algorithm (CLA) to efficiently find real and complex polynomial roots. The approach enhances computational efficiency for higher-order polynomials compared to traditional methods.
Area of Science:
- Numerical Analysis
- Computational Mathematics
- Artificial Intelligence
Background:
- Finding roots of polynomials is a fundamental problem in mathematics.
- Existing numerical methods can be computationally intensive, especially for higher-order polynomials.
- Neural networks offer alternative computational paradigms.
Purpose of the Study:
- To propose a novel constructive approach for finding arbitrary polynomial roots.
- To enhance the efficiency and feasibility of polynomial root finding using neural networks.
Main Methods:
- Utilizing a multilayer perceptron network (MLPN) with a constrained learning algorithm (CLA).
- Encoding a priori information of constraint relations between root moments and polynomial coefficients into the backpropagation algorithm (BPA).
- Simplifying the root moment method (RMM) into a recursive version to decrease computational complexity.
Main Results:
- The proposed neural connectionism approach demonstrates superior efficiency and feasibility.
- Successfully finds arbitrary real and complex roots of arbitrary polynomials.
- Reduced computational complexity for higher-order polynomial root finding.
Conclusions:
- The developed MLPN with CLA offers an effective and efficient method for polynomial root finding.
- This neural network-based approach outperforms traditional non-neural methods.
- The recursive RMM simplification further enhances computational tractability.