Abstract
Finding arbitrary roots of polynomials is a fundamental problem in various areas of science and engineering. A myriad of methods was suggested to address this problem, such as the sequential Newton’s method and the Durand–Kerner (D–K) simultaneous iterative method. The sequential iterative methods, on the one hand, need to use a deflation procedure in order to compute approximations to all the roots of a given polynomial, which can produce inaccurate results due to the accumulation of rounding errors. On the other hand, the simultaneous iterative methods require good initial guesses to converge. However, Artificial Neural Networks (ANNs) are widely known by their capacity to find complex mappings between the dependent and independent variables. In view of this, this paper aims to determine, based on comparative results, whether ANNs can be used to compute approximations to the real and complex roots of a given polynomial, as an alternative to simultaneous iterative algorithms like the D–K method. Although the results are very encouraging and demonstrate the viability and potentiality of the suggested approach, the ANNs were not able to surpass the accuracy of the D–K method. The results indicated, however, that the use of the approximations computed by the ANNs as the initial guesses for the D–K method can be beneficial to the accuracy of this method.
Highlights
Finding arbitrary roots of a given polynomial is a fundamental problem in different areas of science and engineering
Since traditional Artificial Neural Networks (ANNs) are generally known for their capability to discover complex nonlinear input–output mappings, and find good approximations for complex problems, this paper suggests and tests a neural network-based approach for finding real and complex roots of polynomials, aiming to assess its potential and limitations regarding efficiency and accuracy. (It is important to note that this approach uses inductive inference in order to find the roots of a given polynomial simultaneously.) The ANN-based approach is compared with the D–K method, one of the most traditional simultaneous iterative methods for finding all the roots of a given polynomial
The results obtained with the adopted approach are presented, along with comparisons with the numerical approximations provided by the D–K method, in terms of accuracy and execution time, when the polynomials have real and complex roots
Summary
Finding arbitrary (real or complex) roots of a given polynomial is a fundamental problem in different areas of science and engineering. Many iterative methods for finding all the roots of a given polynomial already exist, e.g., the Newton’s method and the Durand–Kerner (D–K) method [1,2], they usually require: (a) repeated deflations, which may cause very inaccurate results because of the accumulation of floating point rounding errors, (b) good initial approximations to the roots for the algorithm converge, or (c) the computation of first or second order derivatives, which, besides being computationally intensive, it is not always possible Due to those drawbacks, and since traditional Artificial Neural Networks (ANNs) are generally known for their capability to discover complex nonlinear input–output mappings, and find good approximations for complex problems, this paper suggests and tests a neural network-based approach for finding real and complex roots of polynomials, aiming to assess its potential and limitations regarding efficiency and accuracy. In 2001, Huang and Chi [4,5] used that ANN architecture and added a priori knowledge about the relationships between the roots and coefficients in the training algorithm aiming to find the real and complex roots of a given polynomial. (This is considered by the authors to be the first work that addressed the root-finding problem using ANNs directly.)
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