509 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
Representations for generalized Drazin inverse of operator matrices over a Banach space
In this paper we give expressions for the generalized Drazin inverse of a
(2,2,0) operator matrix and a operator matrix under certain
circumstances, which generalizes and unifies several results in the literature
Supporting GENP with Random Multipliers
We prove that standard Gaussian random multipliers are expected to stabilize
numerically both Gaussian elimination with no pivoting and block Gaussian
elimination. Our tests show similar results where we applied circulant random
multipliers instead of Gaussian ones.Comment: 14 page
Nearly Optimal Computations with Structured Matrices
We estimate the Boolean complexity of multiplication of structured matrices
by a vector and the solution of nonsingular linear systems of equations with
these matrices. We study four basic most popular classes, that is, Toeplitz,
Hankel, Cauchy and Van-der-monde matrices, for which the cited computational
problems are equivalent to the task of polynomial multiplication and division
and polynomial and rational multipoint evaluation and interpolation. The
Boolean cost estimates for the latter problems have been obtained by Kirrinnis
in \cite{kirrinnis-joc-1998}, except for rational interpolation, which we
supply now. All known Boolean cost estimates for these problems rely on using
Kronecker product. This implies the -fold precision increase for the -th
degree output, but we avoid such an increase by relying on distinct techniques
based on employing FFT. Furthermore we simplify the analysis and make it more
transparent by combining the representation of our tasks and algorithms in
terms of both structured matrices and polynomials and rational functions. This
also enables further extensions of our estimates to cover Trummer's important
problem and computations with the popular classes of structured matrices that
generalize the four cited basic matrix classes.Comment: (2014-04-10
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