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The (sequential) algorithm of Multiple Relatively Robust Representations, MRRR, is a more efficient variant of inverse iteration that does not require reorthogonalization. It solves the eigenproblem of an unreduced symmetric tridiagonal matrix...
Certification of bounds on expressions involving rounded operators
Marc Daumas, Guillaume Melquiond
Article No.: 2
Gappa is a tool designed to formally verify the correctness of numerical software and hardware. It uses interval arithmetic and forward error analysis to bound mathematical expressions that involve rounded as well as exact operators. It then...
Design patterns for multiphysics modeling in Fortran 2003 and C++
Damian W. I. Rouson, Helgi Adalsteinsson, Jim Xia
Article No.: 3
We present three new object-oriented software design patterns in Fortran 2003 and C++. These patterns integrate coupled differential equations, facilitating the flexible swapping of physical and numerical software abstractions at compile-time and...
Computing correctly rounded integer powers in floating-point arithmetic
Peter Kornerup, Christoph Lauter, Vincent Lefèvre, Nicolas Louvet, Jean-Michel Muller
Article No.: 4
We introduce several algorithms for accurately evaluating powers to a positive integer in floating-point arithmetic, assuming a fused multiply-add (fma) instruction is available. For bounded, yet very large values of the exponent, we aim at...
Singularity-free evaluation of collapsed-coordinate orthogonal polynomials
Robert C. Kirby
Article No.: 5
The L2-orthogonal polynomials used in finite and spectral element methods on nonrectangular elements may be defined in terms of collapsed coordinates, wherein the shapes are mapped to a square or cube by means of a...
On the efficiency of symbolic computations combined with code generation for finite element methods
Martin Sandve Alnæs, Kent-André Mardal
Article No.: 6
Efficient and easy implementation of variational forms for finite element discretization can be accomplished with metaprogramming. Using a high-level language like Python and symbolic mathematics makes an abstract problem definition possible, but...
Today computers have several levels of memory hierarchy. To obtain good performance on these processors it is necessary to design algorithms that minimize I/O traffic to slower memories in the hierarchy. In this article, we study the computation...
Optimizations for quadrature representations of finite element tensors through automated code generation
Kristian B. Ølgaard, Garth N. Wells
Article No.: 8
We examine aspects of the computation of finite element matrices and vectors that are made possible by automated code generation. Given a variational form in a syntax that resembles standard mathematical notation, the low-level computer code for...
Algorithm 898: Efficient multiplication of dense matrices over GF(2)
Martin Albrecht, Gregory Bard, William Hart
Article No.: 9
We describe an efficient implementation of a hierarchy of algorithms for multiplication of dense matrices over the field with two elements (F2). In particular we present our implementation—in the M4RI library—of...
Global polynomial approximation methods applied to piecewise continuous functions exhibit the well-known Gibbs phenomenon. We summarize known methods to remove the Gibbs oscillations and present a collection of Matlab programs that implement the...
Algorithm 900: A discrete time kalman filter package for large scale problems
Germ´n A. Torres
Article No.: 11
Data assimilation is the process of feeding a partially unknown prediction model with available information from observations, with the objective of correcting and improving the modeled results. One of the most important mathematical tools to...
Algorithm 901: LMEF—A program for the construction of linear multistep methods with exponential fitting for the numerical solution of ordinary differential equations
D. S. Vlachos, T. E. Simos
Article No.: 12
LMEF is a program written in MATLAB, to calculate the coefficients of a linear multi-step method (explicit, implicit or backward differentiation formulas) with algebraic and/or exponential fitting, for the numerical solution of first order...