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On the Early History of the Singular Value Decomposition
(1998-10-15)
This paper surveys the contributions of five
mathematicians\,---\,Eugenio Beltrami (1835--1899), Camille Jordan
(1838--1921), James Joseph Sylvester (1814--1897), Erhard Schmidt
(1876--1959), and Hermann Weyl (1885--1955 ...
Direction-of-Arrival Estimation Using the Rank-Revealing URV Decomposition
(1998-10-15)
Appeared in Proceedings of ACASSP-91.
An algorithm for updating the null space of a matrix is described.
The algorithm is based on a new decomposition, called the URV
decomposition, which can be updated in $O(N^2)$ and ...
A Two-Stage Iteration for Solving Nearly Uncoupled Markov Chains
(1995-02-06)
This paper is concerned with an iteration for determining the
steady-state probability vector of a nearly uncoupled Markov Chain.
The states of these chains can be partitioned into aggregates with
low probabilities of ...
Communication and Matrix Computations on Large Message Passing Systems
(1998-10-15)
This paper is concerned with the consequences for matrix computations
of having a rather large number of general purpose processors, say
ten or twenty thousand, connected in a network in such a way that a
processor can ...
On the Stability of Sequential Updates and Downdates
(1998-10-15)
The updating and downdating of QR decompositions has important
applications in a number of areas. There is essentially one standard
updating algorithm, based on plane rotations, which is backwards
stable. Three downdating ...
On the Perturbation of LU and Cholesky Factors*
(1998-10-15)
In a recent paper, Chang and Paige have shown that the
usual perturbation bounds for Cholesky factors can systematically
overestimate the errors. In this note we sharpen their results and extend
them to the factors of ...
Invariant Subspaces and Capital Punishment (A Participatory Paper)
(1995-02-06)
The notion of invariant subspaces is useful in a number of
theoretical and practical applications. In this paper we give an
elementary treatment of invariant subspaces that stresses their
connection with simple eigenvalues ...
Determining Rank in the Presence of Error
(1998-10-15)
The problem of determining rank in the presence of error occurs in a
number of applications. The usual approach is to compute a
rank-revealing decomposition and make a decision about the rank by
examining the small elements ...
Scaling for Orthogonality
(1998-10-15)
In updating algorthms where orthogonal transformations are
accumulated, it is important to preserve the orthogonality of the
product in the presence of rounding error. Moonen, Van Dooren, and
Vandewalle have pointed out ...
Incremental Condition Calculation and Column Selection
(1998-10-15)
This paper describes a method for calculating the condition number of
a matrix in the Frobenius norm that can be used to select columns in
the course of computing a QR decomposition. When the number of rows
of the matrix ...