851 research outputs found
Polynomial approximation of functions of matrices and its application to the solution of a general system of linear equations
During the process of solving a mathematical model numerically, there is often a need to operate on a vector v by an operator which can be expressed as f(A) while A is NxN matrix (ex: exp(A), sin(A), A sup -1). Except for very simple matrices, it is impractical to construct the matrix f(A) explicitly. Usually an approximation to it is used. In the present research, an algorithm is developed which uses a polynomial approximation to f(A). It is reduced to a problem of approximating f(z) by a polynomial in z while z belongs to the domain D in the complex plane which includes all the eigenvalues of A. This problem of approximation is approached by interpolating the function f(z) in a certain set of points which is known to have some maximal properties. The approximation thus achieved is almost best. Implementing the algorithm to some practical problem is described. Since a solution to a linear system Ax = b is x= A sup -1 b, an iterative solution to it can be regarded as a polynomial approximation to f(A) = A sup -1. Implementing the algorithm in this case is also described
IllinoisGRMHD: An Open-Source, User-Friendly GRMHD Code for Dynamical Spacetimes
In the extreme violence of merger and mass accretion, compact objects like
black holes and neutron stars are thought to launch some of the most luminous
outbursts of electromagnetic and gravitational wave energy in the Universe.
Modeling these systems realistically is a central problem in theoretical
astrophysics, but has proven extremely challenging, requiring the development
of numerical relativity codes that solve Einstein's equations for the
spacetime, coupled to the equations of general relativistic (ideal)
magnetohydrodynamics (GRMHD) for the magnetized fluids. Over the past decade,
the Illinois Numerical Relativity (ILNR) Group's dynamical spacetime GRMHD code
has proven itself as a robust and reliable tool for theoretical modeling of
such GRMHD phenomena. However, the code was written "by experts and for
experts" of the code, with a steep learning curve that would severely hinder
community adoption if it were open-sourced. Here we present IllinoisGRMHD,
which is an open-source, highly-extensible rewrite of the original
closed-source GRMHD code of the ILNR Group. Reducing the learning curve was the
primary focus of this rewrite, with the goal of facilitating community
involvement in the code's use and development, as well as the minimization of
human effort in generating new science. IllinoisGRMHD also saves computer time,
generating roundoff-precision identical output to the original code on
adaptive-mesh grids, but nearly twice as fast at scales of hundreds to
thousands of cores.Comment: 37 pages, 6 figures, single column. Matches published versio
Improved Accuracy and Parallelism for MRRR-based Eigensolvers -- A Mixed Precision Approach
The real symmetric tridiagonal eigenproblem is of outstanding importance in
numerical computations; it arises frequently as part of eigensolvers for
standard and generalized dense Hermitian eigenproblems that are based on a
reduction to tridiagonal form. For its solution, the algorithm of Multiple
Relatively Robust Representations (MRRR) is among the fastest methods. Although
fast, the solvers based on MRRR do not deliver the same accuracy as competing
methods like Divide & Conquer or the QR algorithm. In this paper, we
demonstrate that the use of mixed precisions leads to improved accuracy of
MRRR-based eigensolvers with limited or no performance penalty. As a result, we
obtain eigensolvers that are not only equally or more accurate than the best
available methods, but also -in most circumstances- faster and more scalable
than the competition
Newton's Method in Three Precisions
We describe a three precision variant of Newton's method for nonlinear
equations. We evaluate the nonlinear residual in double precision, store the
Jacobian matrix in single precision, and solve the equation for the Newton step
with iterative refinement with a factorization in half precision. We analyze
the method as an inexact Newton method. This analysis shows that, except for
very poorly conditioned Jacobians, the number of nonlinear iterations needed is
the same that one would get if one stored and factored the Jacobian in double
precision. In many ill-conditioned cases one can use the low precision
factorization as a preconditioner for a GMRES iteration. That approach can
recover fast convergence of the nonlinear iteration. We present an example to
illustrate the results.Comment: 10 page
- …