Five-Precision GMRES-Based Iterative Refinement
GMRES-based iterative refinement in three precisions (GMRES-IR3), proposed by Carson and Higham in 2018, uses a low precision LU factorization to accelerate the solution of a linear system without …
Probabilistic Rounding Error Analysis of Householder QR Factorization
The standard worst-case normwise backward error bound for Householder QR factorization of an matrix is proportional to , where is the unit roundoff. We prove that the bound can be replaced…
Computing the Square Root of a Low-Rank Perturbation of the Scaled Identity M…
We consider the problem of computing the square root of a perturbation of the scaled identity matrix, , where and are matrices with . This problem arises in various applications, including compute…
Arbitrary Precision Algorithms for Computing the Matrix Cosine and its Fréch…
Existing algorithms for computing the matrix cosine are tightly coupled to a specific precision of floating-point arithmetic for optimal efficiency so they do not conveniently extend to an arbitrar…
A Multiprecision Derivative-Free Schur--Parlett Algorithm for Computing Matri…
The Schur--Parlett algorithm, implemented in MATLAB as \textttfunm, evaluates an analytic function f at an n×n matrix argument by using the Schur decomposition and a block recurrence of Parlett. T…
Random Matrices Generating Large Growth in LU Factorization with Pivoting
We identify a class of random, dense, $n\times n$ matrices for which LU factorization with any form of pivoting produces a growth factor typically of size at least $n/(4 \log n)$ for large $n$. The…
Matrices with Tunable Infinity-Norm Condition Number and No Need for Pivoting…
We propose a two-parameter family of nonsymmetric dense $n\times n$ matrices $A(\alpha,\beta)$ for which LU factorization without pivoting is numerically stable, and we show how to choose $\alpha$ …
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