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We propose a new nonlinear preconditioned conjugate gradient (PCG) method in real arithmetic for computing the ground states of rotational Bose--Einstein condensate, modeled by the Gross--Pitaevskii equation. Our algorithm presents a few improvements of the PCG method in complex arithmetic studied by Antoine, Levitt, and Tang [J. Comput. Phys., 343 (2017), pp. 92--109]. We show that the special structure of the energy functional $$E(\phi)$$ and its gradient with respect to $$\phi$$ can be fully exploited in real arithmetic to evaluate them more efficiently. We propose a simple approach for fast evaluation of the energy functional, which enables exact line search. Most importantly, we derive the discrete Hessian operator of the energy functional and propose a shifted Hessian preconditioner for PCG, with which the ideal preconditioned Hessian has favorable eigenvalue distributions independent of the mesh size. This suggests that PCG with our ideal Hessian preconditioner is expected to exhibit mesh size-independent asymptomatic convergence behavior. In practice, our preconditioner is constructed by incomplete Cholesky factorization of the shifted discrete Hessian operator based on high-order finite difference discretizations. Numerical experiments in two-dimensional (2D) and three-dimensional (3D) domains show the efficiency of fast energy evaluation, the robustness of exact line search, and the improved convergence of PCG with our new preconditioner in iteration counts and runtime, notably for more challenging rotational BEC problems with high nonlinearity and rotational speed. 

A flexible extended Krylov subspace method (F-EKSM) is considered for numerical approximation of the action of a matrix function f(A) to a vector b, where the function f is of Markov type. F-EKSM has the same framework as the extended Krylov subspace method (EKSM), replacing the zero pole in EKSM with a properly chosen fixed nonzero pole. For symmetric positive definite matrices, the optimal fixed pole is derived for F-EKSM to achieve the lowest possible upper bound on the asymptotic convergence factor, which is lower than that of EKSM. The analysis is based on properties of Faber polynomials of A and (I−A/s)−1. For large and sparse matrices that can be handled efficiently by LU factorizations, numerical experiments show that F-EKSM and a variant of RKSM based on a small number of fixed poles outperform EKSM in both storage and runtime, and usually have advantages over adaptive RKSM in runtime. 

This paper concerns the theory and development of inexact rational Krylov subspace methods for approximating the action of a function of a matrix f(A) to a column vector b. At each step of the rational Krylov subspace methods, a shifted linear system of equations needs to be solved to enlarge the subspace. For large-scale problems, such a linear system is usually solved approximately by an iterative method. The main question is how to relax the accuracy of these linear solves without negatively affecting the convergence of the approximation of f(A)b. Our insight into this issue is obtained by exploring the residual bounds for the rational Krylov subspace approximations of f(A)b, based on the decaying behavior of the entries in the first column of certain matrices of A restricted to the rational Krylov subspaces. The decay bounds for these entries for both analytic functions and Markov functions can be efficiently and accurately evaluated by appropriate quadrature rules. A heuristic based on these bounds is proposed to relax the tolerances of the linear solves arising in each step of the rational Krylov subspace methods. As the algorithm progresses toward convergence, the linear solves can be performed with increasingly lower accuracy and computational cost. Numerical experiments for large nonsymmetric matrices show the effectiveness of the tolerance relaxation strategy for the inexact linear solves of rational Krylov subspace methods. 

An inexact rational Krylov subspace method is studied to solve large-scale nonsymmetric eigenvalue problems. Each iteration (outer step) of the rational Krylov subspace method requires solution to a shifted linear system to enlarge the subspace, performed by an iterative linear solver for large-scale problems. Errors are introduced at each outer step if these linear systems are solved approx- imately by iterative methods (inner step), and they accumulate in the rational Krylov subspace. In this article, we derive an upper bound on the errors intro- duced at each outer step to maintain the same convergence as exact rational Krylov subspace method for approximating an invariant subspace. Since this bound is inversely proportional to the current eigenresidual norm of the target invariant subspace, the tolerance of iterative linear solves at each outer step can be relaxed with the outer iteration progress. A restarted variant of the inexact rational Krylov subspace method is also proposed. Numerical experiments show the effectiveness of relaxing the inner tolerance to save computational cost.