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The Symmetric Level-Index System
Abstract
The purpose of this paper is to present the details of an arithmetic system which virtually abolishes the phenomena of computer
overflow and underflow in a logically symmetric manner. A generalized exponential function is used in such a way as to enable
very large numbers to be represented with a uniform precision, and very small numbers by reciprocation.
... The level-index number system for computer arithmetic was first suggested in Clenshaw and Olver [1], [2]. The scheme was extended to the symmetric level index, SLI, representation in [4] and has been studied in several further papers in the last few years. Much of the earlier work is summarized in the introductory survey [3]. ...
... Possible hardware implementations of SLI arithmetic were discussed in [18], [23], and [26] while a software implementation incorporating some extended arithmetic was described in [25]. The error analysis of SLI arithmetic is discussed in [2] and [4] and is extended in [11], [16], and [17]. Applications and software engineering aspects of the level-index system have been discussed in [5], [10], [12], and [24]. ...
... The freedom of this system from over-and underflow results from the fact that, working to a precision of no more than 5 million binary places in the index, the system is closed under the four basic arithmetic operations with only three bits allotted to the level. This is discussed briefly in [1], [4] and considered in some detail in [11]. ...
This paper begins with a general introduction to the symmetric level- index, SLI, system of number representation and arithmetic. This system provides a robust framework in which experimental computation can be performed without the risk of failure due to overflow/underflow or to poor scaling of the original problem. There follows a brief summary of some existing computational experience with this system to illustrate its strengths in numerical, graphical and parallel computational settings. An example of the use of SLI arithmetic to overcome graphics failure in the modeling of a turbulent combustion problem is presented. The main thrust of this paper is to introduce the idea of SLI-linear least squares data fitting. The use of generalized logarithm and exponential functions is seen to offer significant improvement over the more conventional linear regression tools for fitting data from a compound exponential decay such as the decay of radioactive materials.
... The details of LI and SLI number representation and their algorithms for arithmetic operations are described in [2], [4], [3], [1], and [7]. It is not surprising that to perform arithmetic operations would be more complicated in the SLI system than in FLP arithmetic. ...
... Since all the internal operations are performed in standard double precision FLP arithmetic, the error of the implemented SLI arithmetic operations is larger than what is predicted similarly as in [4]. ...
... Let z δ represent the error of z caused by inherent errors x δ in x and y δ in y. As stated in [4], z δ in a large addition satisfies Although the above error analysis is based on the full algorithm for pure SLI addition, the error bounds also work for the variations of the algorithm. The SLI-FLP mixed case is expected to have a better precision since the roundoff error in generating the b-sequence is avoided. ...
In this paper we describe a C++ implementation of a hybrid system combining SLI (symmetric level-index) arithmetic and FLP (floating-point) arithmetic. The principal motivation for the work to be presented is to promote the use of SLI arithmetic as a practical framework for scientific computing. This hybrid arithmetic is essentially overflow and underflow free, and its implementation has shown the suitability to be used in real life computations.
... Possible applications could be found in computer arithmetic, where research remains active in the search for alternatives to the IEEE floating-point to overcome numeric overflow/underflow [9,10]. In particular, it provides a formal background to proposals involving generalized exponentiation such as the Elias ω code [6] and the level-index codes [7,8]. It may also prove useful to the subfield of weighted automata, where semirings play a central role [11,12]. ...
New sequences of hyperoperations \cite{BE15,HI26,ACK28,GO47,TAR69} are presented together with their local algebraic properties. The commutative hyperoperations reported by Bennet \cite{BE15} are introduced as a sequence of monoids; after identifying semirings along the sequence, the corresponding fields are constructed via inverse completion.
... The resulting binary iterated-powers representation (BIPR) is a conceptually new approach to the problem of representing real numbers, not just an implementation variation of an existing one. For example, all the conventional representations such as IEEE 754 [5,6] with fixed or variable-width exponent, logarithmic number systems [7], level-index [8] and symmetric level-index [8,9] fall into a qualitatively different group based on variations of what we shall call the power-series representation (PSR or, in its binary version, BPSR). ...
Binary iterated powers (bips) are defined as functions of the type
f(x) = β^(e_0β^(e_1β^(e_2β^(...β^(e_n x)...)))) ≡ {b_0b_1b_2...b_n|x} ≡ {b|x},
where β is a non-negative base, e_k = 2*b_k - 1, and b = {b_0, b_1, b_2, ... , b_n} is a binary sequence of 0's and 1's.
This paper explores the properties of both finite and infinite binary iterated powers in the real domain. In particular, it analyses the convergence behavior of infinite bips and shows that, for a range of bases and any infinite binary sequence b, they converge to a real value which depends upon b but not upon the starting value of x. This establishes an interesting bijection between a subset of infinite bips and the set of non-negative real numbers.
Among potential applications of bips is a qualitatively novel real numbers representation which is also briefly discussed.
... In an effort to represent real numbers without overflow/underflow, Clenshaw and Olver [CO84] proposed representing numbers based on repeated exponen- The authors improved on LI to represent numbers in the range [0, 1) with symmetric level-index (SLI) [CT88]. SLI represents numbers in the range [0, 1) with the reciprocal of the general exponential function. ...
... The zeration graph (y = a ° x = x ° a) takes the form of a "broken" straight line, with a strong discontinuity at point x = a. 9. PRACTICAL APPLICATIONS OF ZERATION 9.1. ...
Ackermann's function implies the existence of an infinite spectrum of new arith- metical operations (hyperoperations) belonging to the Grzegorczyk Hierarchy. Two of these hyperoperations, tetration and zeration, together with their inverses are given special atten- tion. Tetration (power tower, or superpower) has recently been described in the scientific lit- erature and has attracted the attention of both career and amateur mathematicians; zeration is less well known. However, the new arithmetical operation, called zeration, follows naturally from generalizing formulas used in iterative calculations of both known and new inverse op- erations. Zeration expands our approach to mathematical concepts such as infinity and conti- nuity and its application for describing discontinuities is shown in practical examples. The possibility of using tetration to represent very large numbers is outlined. Questions requiring further research are also raised.
... This, too, will be added to the laboratory. Natural logarithmic arithmetic is a bridge to the implementation of the level-index, LI, and symmetric levelindex , SLI systems [60] [61] [62] [63] [64] [65] [66] [67] [68] [69] [70] [71] [72] [73] [74] [75] [76] [77] [78] [79]. The implementation of these systems is discussed in greater detail in the next section. ...
... The level-index (LI) number representation and its algorithms for performing the basic arithmetic operations are introduced by Clenshaw and Olver in [3] and [4]. The symmetric form (SLI) was developed by Clenshaw and Turner in [6]. Using repeated logarithms, symmetric levelindex arithmetic essentially allows infinite number range, with a mathematical elegance and simplicity. ...
Symmetric level-index arithmetic was introduced to overcome the problems of overflow and underflow in scientific computations. A hybrid SLI-FLP number system, together with some recent improvements of SLI arithmetic can result in a sound implementation of over/underflow free computer arithmetic. The hybrid arithmetic automatically switches between FLP and SLI, in order to achieve both efficiency and robustness for the system. The number representation scheme and algorithms will be discussed briefly in this paper, followed by the description of a software implementation and its successful application to a turbulent combustion problem.
... Then we de ne the generalized error function generr(X;X) = jf(X) ? f(X)j (3) whereX = g(x) (4) is our external approximation to X. ...
. This paper proposes the use of level-index (LI) and symmetric level-index (SLI) computer arithmetic for practical computation with error bounds. Comparisons are made with floating-point and several advantages are identified. 1 Introduction Any approach to the general problem of assessing the total error in the output of computer programs depends on a detailed understanding of the computer arithmetic. The finite precision of the arithmetic gives rise to rounding errors that can be an important component of the total error. Accordingly, much effort has gone into refining the algorithms and circuitry that carry out floatingpoint arithmetic. One goal of this effort has been to minimize rounding errors. Another was to ensure that exceptional conditions, such as underflow and overflow, are detected and reported because their occurrence can completely invalidate the results of a computation. The present state of floating-point hardware design [5] is close to optimal, and so the question a...
... Usually, they amount to giving the set of representable numbers an even greater dynamic range. The most notable of these is level-index arithmetic [23] (or, a variant called symmetric level-index arithmetic [24]). Usually, these systems are developed within the fixed-precision framework; in this case, it must eventually face same problems of round-off error. ...
We describe a paradigm for numerical computing, based on exact computation. This emerging paradigm has many advantages compared to the standard paradigm which is based on fixed-precision. We first survey the literature on multiprecision number packages, a prerequisite for exact computation. Next we survey some recent applications of this paradigm. Finally, we outline some basic theory and techniques in this paradigm. 1 This paper will appear as a chapter in the 2nd edition of Computing in Euclidean Geometry, edited by D.-Z. Du and F.K. Hwang, published by World Scientific Press, 1994. 1 1 Two Numerical Computing Paradigms Computation has always been intimately associated with numbers: computability theory was early on formulated as a theory of computable numbers, the first computers have been number crunchers and the original mass-produced computers were pocket calculators. Although one's first exposure to computers today is likely to be some non-numerical application, numeri...
... Alternatives to the usual floating-point format have been proposed that provide so much range as to preclude any possibility of overflow or underflow. These include the symmetric level-index representation [Clenshaw and Olver 1984;Clenshaw and Turner 1988;Olver 1987;Turner 1989; and the so-called universal representation of real numbers [Hamada 1987], along with an older proposal by Matsui and Iri [1981]. Of course, ordinary floating point can be given arbitrary range by increasing the size of the exponent field (recall Section 3.1); however, these proposals purport to offer extraordinary range within the confines of a standard 32or 64-bit format (corresponding to IEEE Standard single and double precision, respectively). ...
Language Constructs Termination exception mechanisms like in Ada and C++ are supposed to terminate an unsuccessful computation as soon as possible after an exception occurs. However, none of the examples of numeric exception handling presented earlier depends ACM Transactions on Programming Languages and Systems, Vol. 18, No. 2, March 1996. Handling Floating-Point Exceptions 167 on the immediate termination of a calculation signaling an exception. The IEEE exception flags scheme actually takes advantage of the fact that an immediate jump is not necessary; by raising a flag, making a substitution, and continuing, the IEEE Standard supports both an attempted/alternate form and a default substitution with a single, simple reponse to exceptions. A detraction of the IEEE flag solution, though, is its obvious lack of structure. Instead of being forced to set and reset flags, one would ideally have available a language construct that more directly reflected the attempted/alternate algorit...
SLI (symmetric level-index) arithmetic was introduced to overcome the problems of over/underflow in scientific computing, but it is not widely adopted due to its complexity of performing arithmetic operations. In this paper we describe the implementation of a hybrid system, which takes both the advantages of SLI arithmetic and FLP (floating-point) arithmetic. The implementation has been carefully optimized and thoroughly tested. The aim of our work is a practical framework for over/ underflow-free computation, and an application of a condition number problem for tridiagonal matrices shows that the purpose is fulfilled.
This paper is a blueprint for the use of a massively parallel SIMD computer architecture for the simulation of various forms of computer arithmetic. The particular system used is a DEC/MasPar MP-1 with 4096 processors in a square array. This architecture has many advantages for such simulations due largely to the simplicity of the individual processors. Arithmetic operations can be spread across the processor array to simulate a hardware chip. Alternatively they may be performed on individual processors to allow simulation of a massively parallel implementation of the arithmetic. Compromises between these extremes permit speed-area tradeoffs to be examined. The paper includes a description of the architecture and its features. It then summarizes some of the arithmetic systems which have been, or are to be, implemented. The implementation of the level-index and symmetric level-index, LI and SLI, systems is described in some detail. An extensive bibliography is included.
In order to render the level-index, li, system of number representation and computer arithmetic a realistic scientific computing environment, it is necessary to develop and analyse algorithms for the (approximate) evaluation of the elementary functions with li arguments and/or outputs. In this paper the basic approaches to this problem are outlined and analysed. For some of the routines, operation times will turn out to be faster than for the basic arithmetic algorithms. It is shown that (as for those arithmetic operations) the internal calculations can be performed with fixed absolute precisions. We also discuss the appropriate system of representation for the arguments and values of the functions and the regions in which their evaluation is meaningful.
In this paper we present a brief summary of the Symmetric Level Index, SLI, system of number representation and computer arithmetic with only enough detail to allow the reader to understand the remaining discussion. This discussion is focused on an elementary example of chaos, the logistic map or shift map. This map, which was discussed by Ulam and von Neumann in the context of random number generators, is one of the classical one-dimensional examples of a chaotic map. However, in the conventional realm of floating-point computing, the chaotic behavior is completely lost. To a great extent the correct behavior of this map is recovered if the arithmetic is performed in the SLI system. This statement is valid for all three of the distinguished cases, where the process is seeded with an irrational, a rational, or a rational with a power of 2 denominator.
First it is proved that two recently introduced systems of computer arithmetic, the level-index (li) and symmetric level-index (sli) systems are closed under the four basic arithmetic operations, provided that division by zero is excluded and the operations are executed in finite precision. In consequence, the li and sli systems are free from the defects of overflow and underflow. Second, measures of precision are discussed and compared. Third, the ranges and local precisions of numbers stored in the li and sli systems are compared with corresponding ranges and local precisions of numbers stored in the floating-point system.
In a recent paper the authors described a system for the internal representation of numbers in a computer, based on repeated exponentiations. The main objective in introducing this system is to eradicate the problems of overflow and underflow. The present paper supplies algorithms for performing the four basic arithmetical operations in the new system. The algorithm are accompanied by error analyses, which show that the algorithms can be executed with fixed-point arithmetic. Illustrative examples are included.
We consider the parallel computation of vector norms and inner products in floating-point and a proposed new form of computer arithmetic, the symmetric level-index system. The vector norms provide an illuminating example of the contrast between the two arithmetic systems under discussion in terms of the ability to program for (complete) robustness and parallelizability. The conflict between robustness of the computation — in the sense of the dual requirements of accuracy and freedom from overflow and underflow — and easy parallelization of the algorithms within a floating-point environment is made plain. It is seen that this conflict disappears if the symmetric level-index system of arithmetic is used. The freedom from overflow and underflow offered by this system allows the programming of the straightforward definitions in a way which is simple, robust and immediately parallelizable. Numerical results are given to illustrate the fact that the symmetric level-index system yields results of comparable accuracy to those of floating-point in cases where the latter system works and still yields results of high accuracy when the floating-point system fails altogether.
During the past few years, two teams have been pursuing research into the development and application of level-index arithmetic, one in Maryland, USA, and one in Lancaster, UK. The long-term members of the Maryland team are F.W.J. Olver and D.W. Lozier, their counterparts at Lancaster being C.W. Clenshaw and P.R. Turner: several others have made contributions, including S. Cramer, F. Golam-Hossen, R.E. Kaylor, I. Reid and C. Sims.
The symmetric level-index,sli, system for representing numbers eases the monitoring of precision while avoiding the problems associated with overflow and underflow. In this paper the practical benefits of the system are displayed, using the root-squaring method of Graeffe as a vehicle.Das symmetrische Hhenindexsystem (sli) zur Zahlendarstellung vereinfacht die berwachung der Rechengenauigkeit und vermeidet overflow—oder underflow—Probleme. In dieser Arbeit werden die praktischen Vorzge des Systems am Beispiel des Graeffeverfahrens vorgefhrt.
A modified structure for floating-point representation, which can essentially eliminate overflow from floating-point calculations, is analyzed. The main part of the paper deals with a computational examination of a model for floating-point exponents and the probabilities of overflow and underflow. Earlier results in the absence of gradual underflow are presented for comparison but the primary focus is on the effect of gradual underflow, a version of gradual overflow, and of a proposal for an extended treatment of these exceptions which we call tapered overflow and underflow. The latter virtually eliminates the exceptions; its overflow threshold is . This paper reviews several models for the distribution of exponents of floating-point numbers and their evolution in the presence of repeated multiplicative operations. A continuous model, which closely approximates the discrete case, is seen to satisfy a functional differential equation with both delay and advance terms.
In this paper the distribution of floating-point exponents and its effect on the frequency of underflow and overflow is studied. Previous work suggested that the uniform distribution of exponents was the only continuous one consistent with the logarithmic distribution of floating-point fractions. This paper takes that as a starting point and analyzes, both mathematically and computationally, how the distribution evolves as a result of repeated multiplications and divisions. We see that continuous and discrete models exhibit very similar behavior, generating subsequent distributions which are splines of increasing degree with characteristics similar to those of a normal distribution. These distributions are related to a system of functional differential equations. The analysis leads to a simple computational model which is used as a basis for the experimental results. These confirm that the likelihood of exponent spill is substantially reduced if the initial distribution is restricted to a small range. Even better results are obtained if the initial range is symmetric.
Floating-point underflow is often regarded as either harmless or
as an indication that the computational algorithm is in need of scaling.
A counterexample to this view is given of a function for which contour
plotting is difficult due to floating-point underflow. The function
arose as an asymptotic solution to a model problem in turbulent
combustion in which two chemical species (fuel and oxidizer) mix and
react in a vortex field. Scaling is not a viable option because of
extreme sensitivity to a small physical parameter. Standard graphics
software packages produce erroneous contours without any indication of
difficulty. This example provides support for considering symmetric
level-index arithmetic, a new form of computer arithmetic which is
immune to underflow and overflow
Symmetric level-index arithmetic was introduced to overcome recognized limitations of floating-point systems, most notably overflow and underflow. The original recursive algorithms for arithmetic operations are parallelizable to some extent, particularly when applied to extended sums or products. The main purpose of this paper is to present parallel SLI algorithms for arithmetic and basic linear algebra operations.
Symmetric level-index arithmetic was introduced to overcome the problems of overflow and underflow in the floating-point system.
The purpose of this paper is to improve the algorithm performance of SLI arithmetic by introducing an approximation scheme
using Taylor's expansion. Following a brief summary of the SLI system and its algorithm for addition/subtraction, the deduction
and analysis of Taylor approximation are presented. Also some results from numerical experiments and timing comparisons on
possible implementations show the advantage of the approximation method.
The extension of the SLI (symmetric level index) system to complex
numbers and arithmetic is discussed. The natural form for representation
of complex quantities in SLI is in the modulus-argument form, and this
can be sensibly packed into a single 64-b word for the equivalent of the
32-b real SLI representation. The arithmetic algorithms prove to be very
slightly more complicated than for real SLI arithmetic. The
representation, the arithmetic algorithms, and the control of errors
within these algorithms are described
Extended arithmetic operations, such as forming scalar products,
in symmetric level index (SLI) arithmetic are considered. Schemes for
the implementation of such algorithms are described and analyzed in
terms of comparative timings for these operations and their
floating-point counterparts and in terms of the control of errors in the
computation. With sufficient parallelism available in the SLI processor,
the computation can be as fast as for floating-point operations. The SLI
operation can be modified to produce just a single rounding error from
extended operations very economically. The implementation details
suggest that any time-penalty associated with the use of SLI arithmetic
can be kept to a very small factor on highly parallel computers, perhaps
on the order of just two or three for typical scientific computing
programs
An implementation of the symmetric level-index (SLI) system, some
of its special features, and some computational experience with it are
described. The particular implementation discussed was developed for use
on IBM-compatible PC machines and is written in Turbo Pascal. This
allows many of the attractive features of a potential hardware
implementation of SLI arithmetic to be readily incorporated. The ease of
performing extended computational operations, such as scalar products
and evaluation of polynomials, is evident from the package. The
computational experiments reported also show the great simplicity of
program structure which this robust arithmetic permits
. Symmetric level-index arithmetic was introduced to overcome recognized limitations of floating-point systems, most notably overflow and underflow. The original recursive algorithms for arithmetic operations could be parallelized to some extent, particularly when applied to extended sums or products, and a SIMD software implementation of some of these algorithms is described. The main purpose of this paper is to present parallel SLI algorithms for arithmetic and basic linear algebra operations. 1. Introduction This paper reports on a continuing project to develop, implement and apply parallel algorithms for SLI (symmetric level-index) arithmetic. The algorithms are being developed with a view toward a possible future implementation in hardware but at this stage they are being coded for a particular SIMD (single instruction, multiple data) computer system, a DEC MasPar MP-1 1 . The algorithms cover individual arithmetic operations and extensions to the BLAs (basic linear alge...
In a recent paper the authors described a system for the internal representation of numbers in a computer, based on repeated exponentiations. The main objective in introducing this system is to eradicate the problems of overflow and underflow. The present paper supplies algorithms for performing the four basic arithmetical operations in the new system. The algorithm are accompanied by error analyses, which show that the algorithms can be executed with fixed-point arithmetic. Illustrative examples are included.
The symmetric level-index,sli, system for representing numbers eases the monitoring of precision while avoiding the problems associated with overflow and underflow. In this paper the practical benefits of the system are displayed, using the root-squaring method of Graeffe as a vehicle.Das symmetrische Hhenindexsystem (sli) zur Zahlendarstellung vereinfacht die berwachung der Rechengenauigkeit und vermeidet overflow—oder underflow—Probleme. In dieser Arbeit werden die praktischen Vorzge des Systems am Beispiel des Graeffeverfahrens vorgefhrt.
A new number system is proposed for computer arithmetic based on iterated exponential functions. The mare advantage is to eradicate overflow and undertow, but there are several other advantages and these are described and discussed.
Closure and precision in computer arithmetic
- D W Lozier
- F W J Olver
LOZIER, D. W., & OLVER, F. W. J. 1988 Closure and precision in computer arithmetic. (In preparation).