For solving the linear algebraic equations Ax = b with the symmetric and positive definite matrix A, from elliptic equations, the traditional condition number in the 2-norm is defined by Cond. = lambda(1)/lambda(n) where lambda(1) and lambda(n) are the maximal and minimal eigenvalues of the matrix A, respectively. The condition number is used to provide the bounds of the relative errors from the perturbation of both A and b. Such a Cond. can only be reached by the worst situation of all rounding errors and all b. For the given b, the true relative errors may be smaller, or even much smaller than the Cond., which is called the effective condition number in Chan and Foulser [Effectively well-conditioned linear systems, SIAM J. Sci. Statist. Comput. 9 (1988) 963-969] and Christiansen and Hansen [The effective condition number applied to error analysis of certain boundary collocation methods, J. Comput. Appl. Math. 54(1) (1994) 15-36]. In this paper, we propose the new computational formulas for effective condition number Cond_eff, and define the new simplified effective condition number Cond-E. For the latter, we only need the eigenvector corresponding to the minimal eigenvalue of A, which can be easily obtained by the inverse power method. In this paper, we also apply the effective condition number for the finite difference method for Poisson's equation. The difference grids are not supposed to be quasiuniform. Under a non-orthogonality assumption, the effective condition number is proven to be O(1) for the homogeneous boundary conditions. Such a result is extraordinary, compared with the traditional Cond. = O(h(min)(-2)), where h(min) is the minimal meshspacing of the difference grids used. For the non-homogeneous Neumann and Dirichlet boundary conditions, the effective condition number is proven to be O(h(-1/2)) and O(h(-1/2)h(min)(-1)), respectively, where It is the maximal meshspacing of the difference grids. Numerical experiments are carried out to verify the analysis made. (c) 2006 Elsevier B.V. All rights reserved

We study efficient spectral-collocation and continuation methods (SCCM) for rotating two-component Bose-Einstein condensates (BECs) and rotating two-component BECs in optical lattices, where the second kind Chebyshev polynomials are used as the basis functions for the trial function space. A novel two-parameter continuation algorithm is proposed for computing the ground state and first excited state solutions of the governing Gross-Pitaevskii equations (GPEs), where the classical tangent vector is split into two constraint conditions for the bordered linear systems. Numerical results on rotating two-component BECs and rotating two-component BECs in optical lattices are reported. The results on the former are consistent with the published numerical results.

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We investigate bifurcation scenario of the von Karman equations with partially clamped boundary conditions defined in rectangular domains. First, we study how the (preconditioned) Block GMRES method can be used in the context of continuation methods for tracing solution curves of large systems of nonlinear equations. Next, we discuss linear stabilities of the von Karman equations with partially clamped boundary conditions. In particular, we calculate the first seven eigenvalues and its associated eigenfunctions of the linearized von Karman equations via computer algebra. The Block GMRES method is used to solve linear systems and to detect singularity along solution paths of the discrete problem. Sample numerical results are reported. (C) 2001 Elsevier Science B.V. All rights reserved

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