34,038 research outputs found
Algorithm, Proof and Performances of a new Division of Floating Point Expansions
We present in this work a new algorithm for the division of floating point expansions. Floating expansion is a multiple precision data type developped with arithmetic operators that use the processor floating point unit for core computations instead of the integer unit. Researches on this subject have arised recently from the observation that the floating point unit becomes a more and more efficient part of modern computers. Many simple arithmetic operators and some very usefull geometric operators have already been presented on expansions. Yet previous work presented only a very simple division algorithm. We present in this work a new algorithm. We take this opportunity to extend the set of geometric operators with Bareiss' determinant on a matrix of size between 3 and 10. Running times with different determinant algorithms on different machines are compared with other multiprec- ision packages including GMP, CADNA and a computer geometry package working with modular arithmetic
Faster Geometric Algorithms via Dynamic Determinant Computation
The computation of determinants or their signs is the core procedure in many
important geometric algorithms, such as convex hull, volume and point location.
As the dimension of the computation space grows, a higher percentage of the
total computation time is consumed by these computations. In this paper we
study the sequences of determinants that appear in geometric algorithms. The
computation of a single determinant is accelerated by using the information
from the previous computations in that sequence.
We propose two dynamic determinant algorithms with quadratic arithmetic
complexity when employed in convex hull and volume computations, and with
linear arithmetic complexity when used in point location problems. We implement
the proposed algorithms and perform an extensive experimental analysis. On one
hand, our analysis serves as a performance study of state-of-the-art
determinant algorithms and implementations. On the other hand, we demonstrate
the supremacy of our methods over state-of-the-art implementations of
determinant and geometric algorithms. Our experimental results include a 20 and
78 times speed-up in volume and point location computations in dimension 6 and
11 respectively.Comment: 29 pages, 8 figures, 3 table
Efficient implementation of the Hardy-Ramanujan-Rademacher formula
We describe how the Hardy-Ramanujan-Rademacher formula can be implemented to
allow the partition function to be computed with softly optimal
complexity and very little overhead. A new implementation
based on these techniques achieves speedups in excess of a factor 500 over
previously published software and has been used by the author to calculate
, an exponent twice as large as in previously reported
computations.
We also investigate performance for multi-evaluation of , where our
implementation of the Hardy-Ramanujan-Rademacher formula becomes superior to
power series methods on far denser sets of indices than previous
implementations. As an application, we determine over 22 billion new
congruences for the partition function, extending Weaver's tabulation of 76,065
congruences.Comment: updated version containing an unconditional complexity proof;
accepted for publication in LMS Journal of Computation and Mathematic
Time- and Space-Efficient Evaluation of Some Hypergeometric Constants
The currently best known algorithms for the numerical evaluation of
hypergeometric constants such as to decimal digits have time
complexity and space complexity of or .
Following work from Cheng, Gergel, Kim and Zima, we present a new algorithm
with the same asymptotic complexity, but more efficient in practice. Our
implementation of this algorithm improves slightly over existing programs for
the computation of , and we announce a new record of 2 billion digits for
Modeling Algorithms in SystemC and ACL2
We describe the formal language MASC, based on a subset of SystemC and
intended for modeling algorithms to be implemented in hardware. By means of a
special-purpose parser, an algorithm coded in SystemC is converted to a MASC
model for the purpose of documentation, which in turn is translated to ACL2 for
formal verification. The parser also generates a SystemC variant that is
suitable as input to a high-level synthesis tool. As an illustration of this
methodology, we describe a proof of correctness of a simple 32-bit radix-4
multiplier.Comment: In Proceedings ACL2 2014, arXiv:1406.123
Secure Numerical and Logical Multi Party Operations
We derive algorithms for efficient secure numerical and logical operations
using a recently introduced scheme for secure multi-party
computation~\cite{sch15} in the semi-honest model ensuring statistical or
perfect security. To derive our algorithms for trigonometric functions, we use
basic mathematical laws in combination with properties of the additive
encryption scheme in a novel way. For division and logarithm we use a new
approach to compute a Taylor series at a fixed point for all numbers. All our
logical operations such as comparisons and large fan-in AND gates are perfectly
secure. Our empirical evaluation yields speed-ups of more than a factor of 100
for the evaluated operations compared to the state-of-the-art
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