Level-index arithmetic operations

  title={Level-index arithmetic operations},
  author={C. W. Clenshaw and Frank W. J. Olver},
  journal={SIAM Journal on Numerical Analysis},
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 algorithms are accompanied by error analyses, which show that the algorithms can be executed with fixed-point arithmetic. Illustrative… Expand
A closed computer arithmetic
  • F. Olver
  • Computer Science
  • 1987 IEEE 8th Symposium on Computer Arithmetic (ARITH)
  • 1987
Two closely related new systems of computer arithmetic are proposed. It is shown that both are closed under arithmetic operations in finite-precision arithmetic, thereby offering a permanent solutionExpand
A software implementation of SLI arithmetic
  • P. Turner
  • Computer Science
  • Proceedings of 9th Symposium on Computer Arithmetic
  • 1989
The computational experiments reported show the great simplicity of program structure which this robust arithmetic permits and the ease of performing extended computational operations, such as scalar products and evaluation of polynomials, is evident from the package. Expand
Parallel and serial implementations of SLI arithmetic
This paper describes the various algorithms and software implementations of the Level-Index LI and Symmetric Level-Index SLI arithmetic schemes. After a brief introduction to the munberExpand
Basic Linear Algebra Operations in SLI Arithmetic
The main purpose of this paper is to present parallel SLI algorithms for arithmetic and basic linear algebra operations. Expand
Closure and precision in level-index arithmetic
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,Expand
Implementation of level-index arithmetic using partial table look-up
  • F. Olver, P. Turner
  • Computer Science
  • 1987 IEEE 8th Symposium on Computer Arithmetic (ARITH)
  • 1987
The approach used combines the advantages of parallel processing with the use of table look-up and the result is a potential implementation with ℓi operation times comparable with floating-point long multiplications. Expand
Towards a fast and reliable software implementation of SLI-FLP hybrid computer arithmetic
A C++ implementation of a hybrid system combining SLI (symmetric level-index) arithmetic and FLP (floating-point) arithmetic that has shown the suitability to be used in real life computations is described. Expand
Implementation and analysis of extended SLI operations
  • P. Turner
  • Computer Science
  • [1991] Proceedings 10th IEEE Symposium on Computer Arithmetic
  • 1991
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. Expand
Taylor approximation for symmetric level-index arithmetic processing
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 ofExpand
A Hybrid Number Representation Scheme Based on Symmetric Level-Index Arithmetic
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. Expand


A Portable Fortran Program to Find the Euclidean Norm of a Vector
This paper describes a successful version of a subprogram to find the Euclidean norm of an n-vector which is accurate and efficient, and should avoid all overflows and underflows, and is also portable. Expand
A unified algorithm for elementary functions
This paper describes a single unified algorithm for the calculation of elementary functions including multiplication, division, sin, cos, tan, arctan, sinh, cosh, tanh, arctanh, In, exp andExpand
Computer architecture and parallel processing
  • K. Hwang, F. Briggs
  • Computer Science
  • McGraw-Hill Series in computer organization and architecture
  • 1986
The authors have divided the use of computers into the following four levels of sophistication: data processing, information processing, knowledge processing, and intelligence processing. Expand
Floating-point computation
(a) Write a function in a programming language of your choice that takes a (32-bit IEEE format) float and returns a float with the property that: given zero, infinity or a positive normalisedExpand
Beyond Floating Point
Un systeme numerique est proposed pour l'arithmetique d'ordinateur basee sur des fonctions exponentielles iterees sur le sous-passement and le depassement de capacite. Expand