# Numerical algorithm for the standard pairing problem based on the Heine-Stieltjes correspondence and the polynomial approach

#### Abstract

We present a detailed study of the computational complexity of a numerical algorithm based on the Heine-Stieltjes correspondence following the new approach we proposed recently for solving the Bethe ansatz (Gaudin-Richardson) equations of the standard pairing problem. For k pairs of valence nucleons in n non-degenerate single-particle energy levels, the approach utilizes that solutions of the Bethe ansatz equations can be obtained from two matrices of dimensions (k+1)×(k+1) and (n-1)×(k+1), which are associated with the extended Heine-Stieltjes and Van Vleck polynomials, respectively. Since the coefficients in these polynomials are free from divergence with variations in contrast to the original Bethe ansatz equations, the approach provides an efficient and systematic way to solve the problem. The method reduces to solving a system of k polynomial equations, which can be efficiently implemented by the fast Newton-Raphson algorithm with a Monte Carlo sampling procedure for the initial guesses. By extension, the present algorithm can also be used to solve a large class of Gaudin-type quantum many-body problems, including an efficient angular momentum projection method for multi-particle systems. Program summary Program title: exactPairingHS Catalogue identifier: AETD-v1-0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AETD-v1-0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 5411 No. of bytes in distributed program, including test data, etc.: 39758 Distribution format: tar.gz Programming language: Mathematica. Computer: Laptop, workstation. Operating system: Tested with MATHEMATICA version 9.0 on Mac OS X and Windows 7. RAM: Less than 10 MB Classification: 17.15. Nature of problem: The program calculates exact pairing energies based on the Heine-Stieltjes polynomial approach. Existing conventional exact-pairing approaches require solving systems of highly nonlinear equations, which are difficult and often impossible to solve beyond the simplest of the quantum-mechanical many-particle systems. In this study, the Heine-Stieltjes polynomial approach is employed to provide solutions for more than one or two pairs of particles residing in many energy levels. Solution method: The new Heine-Stieltjes polynomial approach transforms the pairing problem to one that involves the handling of only two matrix equations. This, combined with an efficient numerical algorithm implemented by the fast Newton-Raphson method with a Monte Carlo sampling procedure for the initial guesses, makes exact pairing solutions feasible even when more energy levels or heavy nuclei (many pairs) are considered. Running time: Less than a hundred seconds using a 2.80 GHz processor. The notebook takes approximately 23 min to complete. © 2014 Elsevier B.V. All rights reserved.

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