Sampling algebraic sets in local intrinsic coordinates

Numerical data structures for positive dimensional solution sets of polynomial systems are sets of generic points cut out by random planes of complimentary dimension. We may represent the linear spaces defined by those planes either by explicit linear equations or in parametric form. These descriptions are respectively called extrinsic and intrinsic representations. While intrinsic representations lower the cost of the linear algebra operations, we observe worse condition numbers. In this paper we describe the local adaptation of intrinsic coordinates to improve the numerical conditioning of sampling algebraic sets. Local intrinsic coordinates also lead to a better stepsize control. We illustrate our results with Maple experiments and computations with PHCpack on some benchmark polynomial systems.Comment: 13 pages, 2 figures, 2 algorithms, 2 table

Polynomial eigenvalue solver based on tropically scaled Lagrange linearization

We propose an algorithm to solve polynomial eigenvalue problems via linearization combining several ingredients: a specific choice of linearization, which is constructed using input from tropical algebra and the notion of well-separated tropical roots, an appropriate scaling applied to the linearization and a modified stopping criterion for the $QZ$ iterations that takes advantage of the properties of our scaled linearization. Numerical experiments suggest that our polynomial eigensolver computes all the finite and well-conditioned eigenvalues to high relative accuracy even when they are very different in magnitude.status: publishe

Random Matrices and the Convergence of Partition Function Zeros in Finite Density QCD

We apply the Glasgow method for lattice QCD at finite chemical potential to a schematic random matrix model (RMM). In this method the zeros of the partition function are obtained by averaging the coefficients of its expansion in powers of the chemical potential. In this paper we investigate the phase structure by means of Glasgow averaging and demonstrate that the method converges to the correct analytically known result. We conclude that the statistics needed for complete convergence grows exponentially with the size of the system, in our case, the dimension of the Dirac matrix. The use of an unquenched ensemble at $\mu=0$ does not give an improvement over a quenched ensemble. We elucidate the phenomenon of a faster convergence of certain zeros of the partition function. The imprecision affecting the coefficients of the polynomial in the chemical potential can be interpeted as the appearance of a spurious phase. This phase dominates in the regions where the exact partition function is exponentially small, introducing additional phase boundaries, and hiding part of the true ones. The zeros along the surviving parts of the true boundaries remain unaffected.Comment: 17 pages, 14 figures, typos correcte

On the completeness of solutions of Bethe's equations

We consider the Bethe equations for the isotropic spin-1/2 Heisenberg quantum spin chain with periodic boundary conditions. We formulate a conjecture for the number of solutions with pairwise distinct roots of these equations, in terms of numbers of so-called singular (or exceptional) solutions. Using homotopy continuation methods, we find all such solutions of the Bethe equations for chains of length up to 14. The numbers of these solutions are in perfect agreement with the conjecture. We also discuss an indirect method of finding solutions of the Bethe equations by solving the Baxter T-Q equation. We briefly comment on implications for thermodynamical computations based on the string hypothesis.Comment: 17 pages; 85 tables provided as supplemental material; v2: clarifications and references added; v3: numerical results extended to N=14, M=