We consider a contour integral-based eigensolver that finds eigenvalues in a given domain and the corresponding eigenvectors of the generalized eigenvalue problem. In the contour integral-based eigensolver, quadrature points are placed in the complex plane in order to approximate the contour integral. When eigenvalues exist near a quadrature point, the accuracy of other eigenvalues is deteriorated. We herein propose a method by which to recover the accuracy of the eigenpairs when eigenvalues exist near a quadrature point. A numerical experiment is conducted in order to verify that the proposed method is efficient.
The Cholesky QR algorithm is an ideal QR decomposition algorithm for high performance computing, but known to be unstable. We present error analysis of the Cholesky QR algorithm in an oblique inner product defined by a positive definite matrix, and show that by repeating the algorithm twice (called CholeskyQR2), its stability is greatly improved.
In this paper, we discuss the optimal ate pairing over Barreto-Naehrig (BN) curves. First, we give an explicit formula for computing this pairing via elliptic nets associated to the twist curves. Second, we consider parallel algorithms to calculate elliptic nets for computing this pairing. Finally, we evaluate the costs of our parallel algorithms.
The present paper describes a method finding bead shapes in shell structure to decrease the absolute value of mean compliance under periodic loading by using a solution to shape optimization method. Variation of the shell structure in out-of-plane direction is chosen as a design variable. To create beads, the out-of-plane variation is restricted by using the sigmoid function. The integrated absolute value of mean compliance in target frequency range is used as objective function. An iterative algorithm based on the $H^1$ gradient method is used to solve the problem. The effectiveness of the method is confirmed by numerical example.
In this paper, an initial guess is proposed for Newton's method to compute the principal matrix square root, and a sufficient condition of the initial guess is shown for the quadratic convergence. Numerical examples indicate that the initial guess is promising for computing the principal square roots of overlap matrices arising from condensed matter physics.
It is well known that Strassen and Winograd algorithms can reduce the computational costs associated with dense matrix multiplications. We have already shown that they are also very effective for software-based multiple precision floating-point arithmetic environments such as the MPFR/GMP library. In this paper, we show that we can obtain the same effectiveness for double-double (DD) and quadruple-double (QD) environments supported by the QD library, and that parallelization can increase the speed of these multiple precision matrix multiplications. Finally, we demonstrate that our implemented parallelized Strassen and Winograd algorithms can increase the speed of parallelized LU decomposition.
Improved Stabilized Approximate Inverse (ISAINV) based on A-orthogonalization process is known as an effective preconditioning technique for the conjugate gradient (CG) method to solve highly ill-conditioned linear systems. This research aims to accelerate the convergence of the finite element analysis of shell structures by preserving the sparsity in the preconditioning matrix and by parallelizing the localized process of ISAINV preconditioning. In the numerical results, the proposed ISAINV preconditioner shows better convergence and faster computational time than the conventional preconditioning.
A truncation error of the interpolant is considered for a class of particle methods, which can describe Smoothed Particle Hydrodynamics (SPH). Owing to sufficient conditions of the weight function and a regularity of the family of discrete parameters, a truncation error estimate of the interpolant is established for a class of particle methods based on the Voronoi decomposition. Moreover, some numerical results are shown, which agree well with theoretical ones.