588 research outputs found
A Measure of Space for Computing over the Reals
We propose a new complexity measure of space for the BSS model of
computation. We define LOGSPACE\_W and PSPACE\_W complexity classes over the
reals. We prove that LOGSPACE\_W is included in NC^2\_R and in P\_W, i.e. is
small enough for being relevant. We prove that the Real Circuit Decision
Problem is P\_R-complete under LOGSPACE\_W reductions, i.e. that LOGSPACE\_W is
large enough for containing natural algorithms. We also prove that PSPACE\_W is
included in PAR\_R
A Numerical Algorithm for Zero Counting. I: Complexity and Accuracy
We describe an algorithm to count the number of distinct real zeros of a
polynomial (square) system f. The algorithm performs O(n D kappa(f)) iterations
where n is the number of polynomials (as well as the dimension of the ambient
space), D is a bound on the polynomials' degree, and kappa(f) is a condition
number for the system. Each iteration uses an exponential number of operations.
The algorithm uses finite-precision arithmetic and a polynomial bound for the
precision required to ensure the returned output is correct is exhibited. This
bound is a major feature of our algorithm since it is in contrast with the
exponential precision required by the existing (symbolic) algorithms for
counting real zeros. The algorithm parallelizes well in the sense that each
iteration can be computed in parallel polynomial time with an exponential
number of processors.Comment: We made minor but necessary improvements in the presentatio
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