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Implicit standard Jacobi gives high relative accuracy
TLDR
We prove that the Jacobi algorithm applied implicitly on a decomposition A = XDXT of the symmetric matrix A, where D is diagonal, and X is well conditioned, computes all eigenvalues of A to high relative accuracy. Expand
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Weyl-type relative perturbation bounds for eigensystems of Hermitian matrices
We present a Weyl-type relative bound for eigenvalues of Hermitian perturbations A+E of (not necessarily definite) Hermitian matrices A. This bound, given in function of the quantityExpand
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Accurate solution of structured linear systems via rank-revealing decompositions
Linear systems of equations Ax = b, where the matrix A has some particular structure, arise frequently in applications. Very often structured matrices have huge condition numbers κ(A) = ‖A−1‖‖A‖ and,Expand
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Accurate Solution of Structured Least Squares Problems via Rank-Revealing Decompositions
TLDR
In this work, we introduce a framework that allows us to compute minimum 2-norm solutions of many classes of structured least squares problems accurately, i.e., with errors $\widehat{x}_0 - x_0_2 / \|x_0 \|_2 = O(n^2 m)$ flops, roughly the same cost as standard algorithm... Expand
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Perturbation Theory for Factorizations of LU Type through Series Expansions
TLDR
Component- and normwise perturbation bounds for the block LU factorization and block LDL$^*$ factorization of Hermitian matrices are presented. Expand
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An Orthogonal High Relative Accuracy Algorithm for the Symmetric Eigenproblem
TLDR
We propose a new algorithm for the symmetric eigenproblem that computes eigenvalues and eigenvectors with high relative accuracy for the largest class of symmetric, definite and indefinite matrices known so far. Expand
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MULTIPLICATIVE PERTURBATION THEORY AND ACCURATE SOLUTION OF LEAST SQUARES PROBLEMS
Least squares problems minx ‖b−Ax‖2 where the matrixA ∈ Cm×n (m ≥ n) has some particular structure arise frequently in applications. Polynomial data fitting is a well-known instance of problems thatExpand
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Multiplicative perturbation theory of the Moore–Penrose inverse and the least squares problem☆
Abstract Bounds for the variation of the Moore–Penrose inverse of general matrices under multiplicative perturbations are presented. Their advantages with respect to classical bounds under additiveExpand
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High relative accuracy algorithms for the symmetric eigenproblem
In this talk we will review the basic facts and results in the field of high relative accuracy. We will see which algorithms and for which classes of matrices give high relative accuracy. InExpand
Multiple LU factorizations of a singular matrix
A singular matrix A may have more than one LU factorizations. In this work the set of all LU factorizations of A is explicitly described when the lower triangular matrix L is nonsingular. To thisExpand
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