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Further Results on Arithmetic Filters for Geometric Predicates
An efficient technique to solve precision problems consists in using exact
computations. For geometric predicates, using systematically expensive exact
computations can be avoided by the use of filters. The predicate is first
evaluated using rounding computations, and an error estimation gives a
certificate of the validity of the result. In this note, we studies the
statistical efficiency of filters for cosphericity predicate with an assumption
of regular distribution of the points. We prove that the expected value of the
polynomial corresponding to the in sphere test is greater than epsilon with
probability O(epsilon log 1/epsilon) improving the results of a previous paper
by the same authors.Comment: 7 pages 2 figures presented at the 15th European Workshop Comput.
Geom., 113--116, 1999 improve previous results (in other paper
A Quasi-Random Approach to Matrix Spectral Analysis
Inspired by the quantum computing algorithms for Linear Algebra problems
[HHL,TaShma] we study how the simulation on a classical computer of this type
of "Phase Estimation algorithms" performs when we apply it to solve the
Eigen-Problem of Hermitian matrices. The result is a completely new, efficient
and stable, parallel algorithm to compute an approximate spectral decomposition
of any Hermitian matrix. The algorithm can be implemented by Boolean circuits
in parallel time with a total cost of Boolean
operations. This Boolean complexity matches the best known rigorous parallel time algorithms, but unlike those algorithms our algorithm is
(logarithmically) stable, so further improvements may lead to practical
implementations.
All previous efficient and rigorous approaches to solve the Eigen-Problem use
randomization to avoid bad condition as we do too. Our algorithm makes further
use of randomization in a completely new way, taking random powers of a unitary
matrix to randomize the phases of its eigenvalues. Proving that a tiny Gaussian
perturbation and a random polynomial power are sufficient to ensure almost
pairwise independence of the phases is the main technical
contribution of this work. This randomization enables us, given a Hermitian
matrix with well separated eigenvalues, to sample a random eigenvalue and
produce an approximate eigenvector in parallel time and
Boolean complexity. We conjecture that further improvements of
our method can provide a stable solution to the full approximate spectral
decomposition problem with complexity similar to the complexity (up to a
logarithmic factor) of sampling a single eigenvector.Comment: Replacing previous version: parallel algorithm runs in total
complexity and not . However, the depth of the
implementing circuit is : hence comparable to fastest
eigen-decomposition algorithms know
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