[Paper Review] GPGCD, an Iterative Method for Calculating Approximate GCD of Univariate Polynomials, with the Complex Coefficients

Akira Terui|arXiv (Cornell University)|Jul 12, 2010

Polynomial and algebraic computation15 references3 citations

TL;DR

This paper extends the GPGCD method—an iterative optimization approach for computing the approximate GCD of univariate polynomials with real coefficients—to handle polynomials with complex coefficients. By reformulating the approximate GCD problem as a constrained minimization and solving it via a modified Newton method, the method achieves perturbations comparable to STLN-based approaches but with up to 30 times faster performance on polynomials of moderate degree.

ABSTRACT

We present an extension of our GPGCD method, an iterative method for calculating approximate greatest common divisor (GCD) of univariate polynomials, to polynomials with the complex coefficients. For a given pair of polynomials and a degree, our algorithm finds a pair of polynomials which has a GCD of the given degree and whose coefficients are perturbed from those in the original inputs, making the perturbations as small as possible, along with the GCD. In our GPGCD method, the problem of approximate GCD is transfered to a constrained minimization problem, then solved with a so-called modified Newton method, which is a generalization of the gradient-projection method, by searching the solution iteratively. While our original method is designed for polynomials with the real coefficients, we extend it to accept polynomials with the complex coefficients in this paper.

Motivation & Objective

• To extend the GPGCD method, originally designed for real-coefficient polynomials, to handle polynomials with complex coefficients.
• To maintain minimal coefficient perturbations while computing an approximate GCD of a specified degree for complex-coefficient polynomials.
• To improve computational efficiency compared to existing methods such as STLN-based algorithms for the same problem.
• To ensure robust convergence and accuracy in the presence of noise and ill-conditioning in complex polynomial inputs.

Proposed method

• The approximate GCD problem is transformed into a constrained minimization problem minimizing the 2-norm of coefficient perturbations.
• The method uses a modified Newton method—a generalization of the gradient-projection method—to iteratively solve the constrained optimization problem.
• The solution relies on the subresultant condition: the (d−1)-th subresultant of the perturbed polynomials must be zero, ensuring a GCD of degree d.
• Complex coefficients are handled by separating real and imaginary parts, treating them as independent variables in the optimization framework.
• The algorithm computes the approximate GCD, the perturbations ΔF and ΔG, and the cofactors A and B that satisfy the linear relation A·F̃ + B·G̃ = 0.
• The method is implemented with a threshold of ε = 1.0×10⁻⁸ on the update vector norm to control convergence.

Experimental results

Research questions

• RQ1 Can the GPGCD method be effectively extended to polynomials with complex coefficients while preserving accuracy?
• RQ2 How does the performance of the complex GPGCD method compare to STLN-based methods in terms of perturbation size and computation time?
• RQ3 Does the modified Newton method ensure convergence and stability for complex polynomial approximate GCD problems?
• RQ4 What is the impact of polynomial degree and noise level on the accuracy and efficiency of the GPGCD method for complex coefficients?

Key findings

• The GPGCD method computes approximate GCDs with perturbations that are nearly identical to those produced by STLN-based methods, with errors ranging from 3.72×10⁻³ to 4.84×10⁻³ across all test cases.
• The GPGCD method achieves a 10- to 30-fold speedup over the STLN-based method, with computation times ranging from 0.15 to 6.00 seconds versus 1.79 to 199.48 seconds, respectively.
• Both methods converged in all 10 test cases, indicating robustness and reliability for polynomials of degrees 10 to 100 with approximate GCDs of degree 5 to 50.
• The number of iterations remained nearly identical between GPGCD and STLN, suggesting similar convergence behavior despite the performance gap in execution time.
• The method maintains high accuracy even as polynomial degree increases, with perturbation levels stabilizing around 4.5×10⁻³ for degrees 30 and above.