9,821 research outputs found
Fast and accurate evaluation of Wigner 3j, 6j, and 9j symbols using prime factorisation and multi-word integer arithmetic
We present an efficient implementation for the evaluation of Wigner 3j, 6j,
and 9j symbols. These represent numerical transformation coefficients that are
used in the quantum theory of angular momentum. They can be expressed as sums
and square roots of ratios of integers. The integers can be very large due to
factorials. We avoid numerical precision loss due to cancellation through the
use of multi-word integer arithmetic for exact accumulation of all sums. A
fixed relative accuracy is maintained as the limited number of floating-point
operations in the final step only incur rounding errors in the least
significant bits. Time spent to evaluate large multi-word integers is in turn
reduced by using explicit prime factorisation of the ingoing factorials,
thereby improving execution speed. Comparison with existing routines shows the
efficiency of our approach and we therefore provide a computer code based on
this work.Comment: 7 pages, 2 figures. Accepted for publication in SIAM Journal on
Scientific Computing (SISC
Faster all-pairs shortest paths via circuit complexity
We present a new randomized method for computing the min-plus product
(a.k.a., tropical product) of two matrices, yielding a faster
algorithm for solving the all-pairs shortest path problem (APSP) in dense
-node directed graphs with arbitrary edge weights. On the real RAM, where
additions and comparisons of reals are unit cost (but all other operations have
typical logarithmic cost), the algorithm runs in time
and is correct with high probability.
On the word RAM, the algorithm runs in time for edge weights in . Prior algorithms used either time for
various , or time for various
and .
The new algorithm applies a tool from circuit complexity, namely the
Razborov-Smolensky polynomials for approximately representing
circuits, to efficiently reduce a matrix product over the algebra to
a relatively small number of rectangular matrix products over ,
each of which are computable using a particularly efficient method due to
Coppersmith. We also give a deterministic version of the algorithm running in
time for some , which utilizes the
Yao-Beigel-Tarui translation of circuits into "nice" depth-two
circuits.Comment: 24 pages. Updated version now has slightly faster running time. To
appear in ACM Symposium on Theory of Computing (STOC), 201
Rational semimodules over the max-plus semiring and geometric approach of discrete event systems
We introduce rational semimodules over semirings whose addition is
idempotent, like the max-plus semiring, in order to extend the geometric
approach of linear control to discrete event systems. We say that a
subsemimodule of the free semimodule S^n over a semiring S is rational if it
has a generating family that is a rational subset of S^n, S^n being thought of
as a monoid under the entrywise product. We show that for various semirings of
max-plus type whose elements are integers, rational semimodules are stable
under the natural algebraic operations (union, product, direct and inverse
image, intersection, projection, etc). We show that the reachable and
observable spaces of max-plus linear dynamical systems are rational, and give
various examples.Comment: 24 pages, 9 postscript figures; example in section 4.3 expande
The cavity method at zero temperature
In this note we explain the use of the cavity method directly at zero
temperature, in the case of the spin glass on a Bethe lattice. The computation
is done explicitly in the formalism equivalent to 'one step replica symmetry
breaking'; we compute the energy of the global ground state, as well as the
complexity of equilibrium states at a given energy. Full results are presented
for a Bethe lattice with connectivity equal to three.Comment: 22 pages, 8 figures; Some minor correction
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