2,320 research outputs found
Uniform bounds on the 1-norm of the inverse of lower triangular Toeplitz matrices
A uniform bound on the 1-norm is given for the inverse of a lower triangular Toeplitz matrix with non-negative monotonically decreasing entries whose limit is zero. The new bound is sharp under certain specified constraints. This result is then employed to throw light upon a long standing open problem posed by Brunner concerning the convergence of the one-point collocationmethod for the Abel equation. In addition, the recent conjecture of Gauthier et al. is proved
Fast computation of the matrix exponential for a Toeplitz matrix
The computation of the matrix exponential is a ubiquitous operation in
numerical mathematics, and for a general, unstructured matrix it
can be computed in operations. An interesting problem arises
if the input matrix is a Toeplitz matrix, for example as the result of
discretizing integral equations with a time invariant kernel. In this case it
is not obvious how to take advantage of the Toeplitz structure, as the
exponential of a Toeplitz matrix is, in general, not a Toeplitz matrix itself.
The main contribution of this work are fast algorithms for the computation of
the Toeplitz matrix exponential. The algorithms have provable quadratic
complexity if the spectrum is real, or sectorial, or more generally, if the
imaginary parts of the rightmost eigenvalues do not vary too much. They may be
efficient even outside these spectral constraints. They are based on the
scaling and squaring framework, and their analysis connects classical results
from rational approximation theory to matrices of low displacement rank. As an
example, the developed methods are applied to Merton's jump-diffusion model for
option pricing
A weakly stable algorithm for general Toeplitz systems
We show that a fast algorithm for the QR factorization of a Toeplitz or
Hankel matrix A is weakly stable in the sense that R^T.R is close to A^T.A.
Thus, when the algorithm is used to solve the semi-normal equations R^T.Rx =
A^Tb, we obtain a weakly stable method for the solution of a nonsingular
Toeplitz or Hankel linear system Ax = b. The algorithm also applies to the
solution of the full-rank Toeplitz or Hankel least squares problem.Comment: 17 pages. An old Technical Report with postscript added. For further
details, see http://wwwmaths.anu.edu.au/~brent/pub/pub143.htm
Fast Algorithms for Displacement and Low-Rank Structured Matrices
This tutorial provides an introduction to the development of fast matrix
algorithms based on the notions of displacement and various low-rank
structures
Faster Inversion and Other Black Box Matrix Computations Using Efficient Block Projections
Block projections have been used, in [Eberly et al. 2006], to obtain an
efficient algorithm to find solutions for sparse systems of linear equations. A
bound of softO(n^(2.5)) machine operations is obtained assuming that the input
matrix can be multiplied by a vector with constant-sized entries in softO(n)
machine operations. Unfortunately, the correctness of this algorithm depends on
the existence of efficient block projections, and this has been conjectured. In
this paper we establish the correctness of the algorithm from [Eberly et al.
2006] by proving the existence of efficient block projections over sufficiently
large fields. We demonstrate the usefulness of these projections by deriving
improved bounds for the cost of several matrix problems, considering, in
particular, ``sparse'' matrices that can be be multiplied by a vector using
softO(n) field operations. We show how to compute the inverse of a sparse
matrix over a field F using an expected number of softO(n^(2.27)) operations in
F. A basis for the null space of a sparse matrix, and a certification of its
rank, are obtained at the same cost. An application to Kaltofen and Villard's
Baby-Steps/Giant-Steps algorithms for the determinant and Smith Form of an
integer matrix yields algorithms requiring softO(n^(2.66)) machine operations.
The derived algorithms are all probabilistic of the Las Vegas type
Multigrid waveform relaxation for the time-fractional heat equation
In this work, we propose an efficient and robust multigrid method for solving
the time-fractional heat equation. Due to the nonlocal property of fractional
differential operators, numerical methods usually generate systems of equations
for which the coefficient matrix is dense. Therefore, the design of efficient
solvers for the numerical simulation of these problems is a difficult task. We
develop a parallel-in-time multigrid algorithm based on the waveform relaxation
approach, whose application to time-fractional problems seems very natural due
to the fact that the fractional derivative at each spatial point depends on the
values of the function at this point at all earlier times. Exploiting the
Toeplitz-like structure of the coefficient matrix, the proposed multigrid
waveform relaxation method has a computational cost of
operations, where is the number of time steps and is the number of
spatial grid points. A semi-algebraic mode analysis is also developed to
theoretically confirm the good results obtained. Several numerical experiments,
including examples with non-smooth solutions and a nonlinear problem with
applications in porous media, are presented
Computational linear algebra over finite fields
We present here algorithms for efficient computation of linear algebra
problems over finite fields
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