IBM system/360 assembly language interval arithmetic software

Computer software designed to perform interval arithmetic is described. An interval is defined as the set of all real numbers between two given numbers including or excluding one or both endpoints. Interval arithmetic consists of the various elementary arithmetic operations defined on the set of all intervals, such as interval addition, subtraction, union, etc. One of the main applications of interval arithmetic is in the area of error analysis of computer calculations. For example, it has been used sucessfully to compute bounds on sounding errors in the solution of linear algebraic systems, error bounds in numerical solutions of ordinary differential equations, as well as integral equations and boundary value problems. The described software enables users to implement algorithms of the type described in references efficiently on the IBM 360 system

Quasilinear Structures in Stochastic Arithmetic and their Application

Stochastic arithmetic has been developed as a model for computing with imprecise numbers. In this model, numbers are represented by independent Gaussian variables with known mean value and standard deviation and are called stochastic numbers. The algebraic properties of stochastic numbers have already been studied by several authors. Anyhow, in most life problems the variables are not independent and a direct application of the model to estimate the standard deviation on the result of a numerical computation may lead to some overestimation of the correct value. In this work “quasilinear” algebraic structures based on standard stochastic arithmetic are studied and, from pure abstract algebraic considerations, new arithmetic operations called “inner stochastic addition and subtraction” are introduced. They appear to be stochastic analogues to the inner interval addition and subtraction used in interval arithmetic. The algebraic properties of these operations and the involved algebraic structures are then studied. Finally, the connection of these inner operations to the correlation coefficient of the variables is developed and it is shown that they allow the computation with non-independent variables. The corresponding methodology for the practical application of the new structures in relation to problems analogous to “dependency problems” in interval arithmetic is given and some numerical experiments showing the interest of these new operations are presented. ACM Computing Classification System (1998): D.2.4, G.3, G.4

A Mathematical Basis for an Interval Arithmetic Standard

Basic concepts for an interval arithmetic standard are discussed in the paper. Interval arithmetic deals with closed and connected sets of real numbers. Unlike floating-point arithmetic it is free of exceptions. A complete set of formulas to approximate real interval arithmetic on the computer is displayed in section 3 of the paper. The essential comparison relations and lattice operations are discussed in section 6. Evaluation of functions for interval arguments is studied in section 7. The desirability of variable length interval arithmetic is also discussed in the paper. The requirement to adapt the digital computer to the needs of interval arithmetic is as old as interval arithmetic. An obvious, simple possible solution is shown in section 8

Interval Arithmetic Using SSE-2

We present an implementation of double precision interval arithmetic using the single-instruction-multiple-data SSE-2 instruction and register set extensions. The implementation is part of a package for exact real arithmetic, which defines the interval arithmetic variation that must be used: incorrect operations such as division by zero cause exceptions, loose evaluation of the operations is in effect, and performance is more important than tightness of the produced bounds. The SSE2 extensions are suitable for the job, because they can be used to operate on a pair of double precision numbers and include separate rounding mode control and detection of the exceptional conditions. The paper describes the ideas we use to fit interval arithmetic to this set of instructions, shows a performance comparison with other freely available interval arithmetic packages, and discusses possible very simple hardware extensions that can significantly increase the performance of interval arithmetic

On the Arithmetic of Errors

An approximate number is an ordered pair consisting of a (real) number and an error bound, briefly error, which is a (real) non-negative number. To compute with approximate numbers the arithmetic operations on errors should be well-known. To model computations with errors one should suitably define and study arithmetic operations and order relations over the set of non-negative numbers. In this work we discuss the algebraic properties of non-negative numbers starting from familiar properties of real numbers. We focus on certain operations of errors which seem not to have been sufficiently studied algebraically. In this work we restrict ourselves to arithmetic operations for errors related to addition and multiplication by scalars. We pay special attention to subtractability-like properties of errors and the induced “distance-like” operation. This operation is implicitly used under different names in several contemporary fields of applied mathematics (inner subtraction and inner addition in interval analysis, generalized Hukuhara difference in fuzzy set theory, etc.) Here we present some new results related to algebraic properties of this operation.* The first author was partially supported by the Bulgarian NSF Project DO 02-359/2008 and NATO project ICS.EAP.CLG 983334

Automated Dynamic Error Analysis Methods for Optimization of Computer Arithmetic Systems

Computer arithmetic is one of the more important topics within computer science and engineering. The earliest implementations of computer systems were designed to perform arithmetic operations and cost if not all digital systems will be required to perform some sort of arithmetic as part of their normal operations. This reliance on the arithmetic operations of computers means the accurate representation of real numbers within digital systems is vital, and an understanding of how these systems are implemented and their possible drawbacks is essential in order to design and implement modern high performance systems. At present the most widely implemented system for computer arithmetic is the IEEE754 Floating Point system, while this system is deemed to the be the best available implementation it has several features that can result in serious errors of computation if not implemented correctly. Lack of understanding of these errors and their effects has led to real world disasters in the past on several occasions. Systems for the detection of these errors are highly important and fast, efficient and easy to use implementations of these detection systems is a high priority. Detection of floating point rounding errors normally requires run-time analysis in order to be effective. Several systems have been proposed for the analysis of floating point arithmetic including Interval Arithmetic, Affine Arithmetic and Monte Carlo Arithmetic. While these systems have been well studied using theoretical and software based approaches, implementation of systems that can be applied to real world situations has been limited due to issues with implementation, performance and scalability. The majority of implementations have been software based and have not taken advantage of the performance gains associated with hardware accelerated computer arithmetic systems. This is especially problematic when it is considered that systems requiring high accuracy will often require high performance. The aim of this thesis and associated research is to increase understanding of error and error analysis methods through the development of easy to use and easy to understand implementations of these techniques

Automated Dynamic Error Analysis Methods for Optimization of Computer Arithmetic Systems

Computer arithmetic is one of the more important topics within computer science and engineering. The earliest implementations of computer systems were designed to perform arithmetic operations and cost if not all digital systems will be required to perform some sort of arithmetic as part of their normal operations. This reliance on the arithmetic operations of computers means the accurate representation of real numbers within digital systems is vital, and an understanding of how these systems are implemented and their possible drawbacks is essential in order to design and implement modern high performance systems. At present the most widely implemented system for computer arithmetic is the IEEE754 Floating Point system, while this system is deemed to the be the best available implementation it has several features that can result in serious errors of computation if not implemented correctly. Lack of understanding of these errors and their effects has led to real world disasters in the past on several occasions. Systems for the detection of these errors are highly important and fast, efficient and easy to use implementations of these detection systems is a high priority. Detection of floating point rounding errors normally requires run-time analysis in order to be effective. Several systems have been proposed for the analysis of floating point arithmetic including Interval Arithmetic, Affine Arithmetic and Monte Carlo Arithmetic. While these systems have been well studied using theoretical and software based approaches, implementation of systems that can be applied to real world situations has been limited due to issues with implementation, performance and scalability. The majority of implementations have been software based and have not taken advantage of the performance gains associated with hardware accelerated computer arithmetic systems. This is especially problematic when it is considered that systems requiring high accuracy will often require high performance. The aim of this thesis and associated research is to increase understanding of error and error analysis methods through the development of easy to use and easy to understand implementations of these techniques

Multi-attribute Group Decision Making of Internet Public Opinion Emergency with Interval Intuitionistic Fuzzy Number

In this paper, an emergency group decision method is presented to cope with internet public opinion emergency with interval intuitionistic fuzzy linguistic values. First, we adjust the initial weight of each emergency expert by the deviation degree between each expert\u27s decision matrix and group average decision matrix with interval intuitionistic fuzzy numbers. Then we can compute the weighted collective decision matrix of all the emergencies based on the optimal weight of emergency expert. By utilizing the interval intuitionistic fuzzy weighted arithmetic average operator one can obtain the comprehensive alarm value of each internet public opinion emergency. According to the ranking of score value and accuracy value of each emergency, the most critical internet public emergency can be easily determined to facilitate government taking related emergency operations. Finally, a numerical example is given to illustrate the effectiveness of the proposed emergency group decision method