Mathematical values are usually computed using well-known mathematical formulas without thinking about their accuracy, which may turn awful with particular instances. This is the case for the computation of the area of a triangle. When the triangle is needle-like, the common formula has a very poor accuracy. Kahan proposed in 1986 an algorithm he claimed correct within a few ulps. Goldberg took...

Golub et al. (2001, BIT, 41, 71–85) gave a generalized successive over-relaxation method for the augmented systems. In this paper, the connection between the SOR-like method and the preconditioned conjugate gradient (PCG) method for the augmented systems is investigated. It is shown that the PCG method is at least as accurate (fast) as the SOR-like method. Numerical examples demonstrate that th...

M. Kohda, 2 V. Lechner, Y. Kunihashi, T. Dollinger, P. Olbrich, C. Schönhuber, I. Caspers, V.V. Bel’kov, L.E. Golub, D. Weiss, K. Richter, J. Nitta, and S.D. Ganichev ∗ Department of Materials Science, Tohoku University, Sendai 980-8579, Japan PRESTO, Japan Science and Technology Agency, Saitama 332-0012, Japan Institut für Experimentelle und Angewandte Physik, Universität Regensburg, D-93040 R...

We have developed a method to calculate the weak localization and antilocalization corrections based on the real-space simulation, where we provide 147 885 predetermined return orbitals of quasi-two-dimensional electrons with up to 5000 scattering events that are repeatedly used. Our model subsumes that of Golub [L. E. Golub, Phys. Rev. B 71, 235310 (2005)PRBMDO1098-012110.1103/PhysRevB.71.2353...

Golub and Loan (1980) presented a numerically-stable TLS algorithm which utilizes the singular value decomposition (SVD). Subsequent refinements to the method predominantly use SVD, and much of the current literature emphasizes stabilization of the inverse and implicit model regularization by SVD truncation (Fierro et al., 1997). Because it is numerically intensive, however, the SVD generally p...

In this paper we discuss algorithmic aspects of the computation of the Jordan canonical form. Inspired by the Golub & Wilkinson paper 9] on the computation of the Jordan canonical form, an O(n 3) algorithm was developed by Beelen & Van Dooren 3] for computing the Kronecker structure of an arbitrary pencil B ? A. Here we show how the ideas of this algorithm lead to a special algorithm for recons...

J. W. Cirtain, L. Golub, L. Lundquist, A. van Ballegooijen, A. Savcheva, M. Shimojo, E. DeLuca, S. Tsuneta, T. Sakao, K. Reeves, M. Weber, R. Kano, N. Narukage, K. Shibasaki 1 Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138 Nobeyama Solar Radio Observatory, Nobeyama, Nagano, 384-1305 National Astronomical Observatory of Japan, Mitaka, Tokyo, 181-8588 ISAS, Japan Aerospace Explora...

In considering the problem of binormalizing a real symmetric algebraic system of equations, an iterative algorithm (BIN) was proposed by Livne and Golub. Numerical experiments and theoretical results show that this algorithm is capable of significantly reducing the condition number relative to other available algorithms. The BIN algorithm is approximately two to four times more expensive than “...

Compensated summation algorithms are designed to improve the accuracy of ill-conditioned sums. They are based on algorithms, such as FastTwoSum, which are proved to provide, with rounding to nearest, the sum of two floating-point numbers and the associated rounding error. Discrete stochastic arithmetic enables one to estimate rounding error propagation in numerical codes. It requires a random r...