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#### On Infinitely Many Algorithms for Solving Equations

(1998-10-15)

Translated by G. W. Stewart
This report contains a translation of ``Ueber unendlich viele
Algorithmen zur Aufl\"osung der Gleichungen,'' a paper by E.
Schr\"oder which appeared in {\it Mathematische Annalen\/} in 1870.
(Also ...

#### 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 ...

#### Lanczos and Linear Systems

(1998-10-15)

Lanczos's major contributions to the numerical solution of linear
equations are contained in two papers: ``An Iteration Method for the
Solution of the Eigenvalue Problem of Linear Differential and
Integral Operators'' and ...

#### Error Analysis of QR Updating with Exponential Windowing

(1998-10-15)

To appear in Mathematics of Computation
Exponential windowing is a widely used technique for suppressing the
effects of old data as new data is added to a matrix. Specifically,
given an $n\times p$ matrix $X_n$ and a ...

#### On the Convergence of Multipoint Iterations

(1998-10-15)

This note gives a new convergence proof for iterations based on
multipoint formulas. It rests on the very general assumption that if
the desired fixed point appears as an argument in the formula then
the the formula returns ...

#### An Updating Algorithm for Subspace Tracking

(1998-10-15)

In certain signal processing applications it is required to compute the
null space of a matrix whose rows are samples of a signal. The usual tool
for doing this is the singular value decomposition. However, the singular
value ...

#### Eigenvalues of Graded Matrices and the Condition Numbers of a Multiple Eigenvalue

(1998-10-15)

This paper concerns two closely related topics: the behavior of the
eigenvalues of graded matrices and the perturbation of a nondefective
multiple eigenvalue. We will show that the eigenvalues of a graded matrix
tend to ...

#### On the Convergence of a New Rayleigh Quotient Method with Applications to Large Eigenproblems

(1998-10-15)

In this paper we propose a variant of the Rayleigh quotient method to
compute an eigenvalue and corresponding eigenvectors of a matrix. It
is based on the observation that eigenvectors of a matrix with
eigenvalue zero are ...

#### An Elsner-Like Perturbation Theorem for Generalized Eigenvalues

(2003-12-18)

In 1985 Elsner established a general bound on the distance between an
eigenvalue of a matrix and the closest eigenvalue of a perturbation of
that matrix. In this note, we show that a similar result holds for
the ...