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** Up:** Stability and Accuracy Assessments
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** Contents**
** Index**

Computation of eigenvalues to high accuracy has been attracting lots of attention
over the last 10 years or so.
Tremendous progress has been made in both theoretical understanding
and numerical algorithms. But to give a detailed account is outside of the
scope of this book. Interested readers are referred to the literature.
On the algorithmic side there are the Demmel-Kahan QR method for
bidiagonal singular value computations [123] and (two-sided) Jacobi
methods for the eigenvalue problems of positive definite matrices.
For the singular value computations
[124,317,406], there are the bisection method for scaled diagonally
dominant matrices [40], and for matrices with acyclic graphs
[117,255], new implementations of the qd method
[168,360] and Demmel's algorithms for
structured matrices [115]. More recently,
[118] showed how to compute SVDs
to high relative accuracy for matrices that can be factored accurately
as , where is diagonal and and are any
well-conditioned matrices.
On the theoretical side,
analogous results to many celebrated theorems for absolute perturbations
are obtained for perturbations that are multiplicative,
( when is Hermitian)
[157,300,301,302,303,297].

** Next:** Generalized Hermitian Eigenvalue Problems
** Up:** Stability and Accuracy Assessments
** Previous:** Remarks on Clustered Eigenvalues.
** Contents**
** Index**
Susan Blackford
2000-11-20