3,783 research outputs found
New formats for computing with real-numbers under round-to-nearest
An edited version of this work was accepted in IEEE Transactions on computers, DOI 10.1109/TC.2015.2479623In this paper, a new family of formats to deal with real number for applications requiring round to nearest is proposed.
They are based on shifting the set of exactly represented numbers which are used in conventional radix-R number systems.
This technique allows performing radix complement and round to nearest without carry propagation with negligible time and
hardware cost. Furthermore, the proposed formats have the same storage cost and precision as standard ones. Since conversion
to conventional formats simply require appending one extra-digit to the operands, standard circuits may be used to perform
arithmetic operations with operands under the new format. We also extend the features of the RN-representation system and
carry out a thorough comparison between both representation systems. We conclude that the proposed representation system
is generally more adequate to implement systems for computation with real number under round-to-nearest.Ministry of Education and Science of Spain under contracts TIN2013-42253-P
Computing with Exact Real Numbers in a Radix-r System
This paper investigates an arithmetic based upon the representation of computable exact real numbers by lazy infinite sequences of signed digits in a positional radix-r system. We discuss advantages and problems associated with this representation, and develop well-behaved algorithms for a comprehensive range of numeric operations, including the four basic operations of arithmetic
Unitary Representations of Wavelet Groups and Encoding of Iterated Function Systems in Solenoids
For points in real dimensions, we introduce a geometry for general digit
sets. We introduce a positional number system where the basis for our
representation is a fixed by matrix over \bz. Our starting point is a
given pair with the matrix assumed expansive, and
a chosen complete digit set, i.e., in bijective correspondence
with the points in \bz^d/A^T\bz^d. We give an explicit geometric
representation and encoding with infinite words in letters from .
We show that the attractor for an affine Iterated Function
System (IFS) based on is a set of fractions for our digital
representation of points in \br^d. Moreover our positional "number
representation" is spelled out in the form of an explicit IFS-encoding of a
compact solenoid \sa associated with the pair . The intricate
part (Theorem \ref{thenccycl}) is played by the cycles in \bz^d for the
initial -IFS. Using these cycles we are able to write down
formulas for the two maps which do the encoding as well as the decoding in our
positional -representation.
We show how some wavelet representations can be realized on the solenoid, and
on symbolic spaces
Rational series and asymptotic expansion for linear homogeneous divide-and-conquer recurrences
Among all sequences that satisfy a divide-and-conquer recurrence, the
sequences that are rational with respect to a numeration system are certainly
the most immediate and most essential. Nevertheless, until recently they have
not been studied from the asymptotic standpoint. We show how a mechanical
process permits to compute their asymptotic expansion. It is based on linear
algebra, with Jordan normal form, joint spectral radius, and dilation
equations. The method is compared with the analytic number theory approach,
based on Dirichlet series and residues, and new ways to compute the Fourier
series of the periodic functions involved in the expansion are developed. The
article comes with an extended bibliography
Shift Radix Systems - A Survey
Let be an integer and . The {\em shift radix system} is defined by has the {\em finiteness
property} if each is eventually mapped to
under iterations of . In the present survey we summarize
results on these nearly linear mappings. We discuss how these mappings are
related to well-known numeration systems, to rotations with round-offs, and to
a conjecture on periodic expansions w.r.t.\ Salem numbers. Moreover, we review
the behavior of the orbits of points under iterations of with
special emphasis on ultimately periodic orbits and on the finiteness property.
We also describe a geometric theory related to shift radix systems.Comment: 45 pages, 16 figure
Radix Conversion for IEEE754-2008 Mixed Radix Floating-Point Arithmetic
Conversion between binary and decimal floating-point representations is
ubiquitous. Floating-point radix conversion means converting both the exponent
and the mantissa. We develop an atomic operation for FP radix conversion with
simple straight-line algorithm, suitable for hardware design. Exponent
conversion is performed with a small multiplication and a lookup table. It
yields the correct result without error. Mantissa conversion uses a few
multiplications and a small lookup table that is shared amongst all types of
conversions. The accuracy changes by adjusting the computing precision
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