Results 11  20
of
33
On Infinitely Precise Rounding for Division, Square Root, Reciprocal and Square Root Reciprocal
 PROCEEDINGS OF THE 14TH IEEE SYMPOSIUM ON COMPUTER ARITHMETIC
, 1999
"... Quotients, reciprocals, square roots and square root reciprocals all have the property that infinitely precise pbit rounded results for pbit input operands can be obtained from approximate results of bounded accuracy. We investigate lower bounds on the number of bits of an approximation accurate ..."
Abstract

Cited by 15 (2 self)
 Add to MetaCart
Quotients, reciprocals, square roots and square root reciprocals all have the property that infinitely precise pbit rounded results for pbit input operands can be obtained from approximate results of bounded accuracy. We investigate lower bounds on the number of bits of an approximation accurate to a unit in the last place sufficient to guarantee that correct round and sticky bits can be determined. Known lower bounds for quotients and square roots are given and/or sharpened, and a new lower bound for root reciprocals is proved. Specifically for reciprocals, quotients and square roots, tight bounds of order 2p + O(1) are presented. For infinitely precise rounding of the root reciprocal a lower bound can be found at 3p + O(1), but exhaustive testing for small sizes of the operand suggests that in practice (2+)p for small is usually sufficient. Algorithms can be designed for obtaining the round and sticky bits based on the bit pattern of an approximation computed to the required a...
VariablePrecision, Interval Arithmetic Processors
"... This chapter presents the design and analysis of variableprecision, interval arithmetic processors. The processors give the user the ability to specify the precision of the computation, determine the accuracy of the results, and recompute inaccurate results with higher precision. The processors sup ..."
Abstract

Cited by 12 (1 self)
 Add to MetaCart
This chapter presents the design and analysis of variableprecision, interval arithmetic processors. The processors give the user the ability to specify the precision of the computation, determine the accuracy of the results, and recompute inaccurate results with higher precision. The processors support a wide variety of arithmetic operations on variableprecision floating point numbers and intervals. Efficient hardware algorithms and specially designed functional units increase the speed, accuracy, and reliability of numerical computations. Area and delay estimates indicate that the processors can be implemented with areas and cycle times that are comparable to conventional IEEE doubleprecision floating point coprocessors. Execution time estimates indicate that the processors are two to three orders of magnitude faster than a conventional software package for variableprecision, interval arithmetic. 1.1 INTRODUCTION Floating point arithmetic provides a highspeed method for perform...
A comparison of three rounding algorithms for IEEE floatingpoint multiplication
, 1998
"... A new IEEE compliant floatingpoint rounding algorithm for computing the rounded product from a carrysave representation of the product is presented. The new rounding algorithm is compared with the rounding algorithms of Yu and Zyner [23] and of Quach et al. [18]. For each rounding algorithm, a log ..."
Abstract

Cited by 10 (2 self)
 Add to MetaCart
A new IEEE compliant floatingpoint rounding algorithm for computing the rounded product from a carrysave representation of the product is presented. The new rounding algorithm is compared with the rounding algorithms of Yu and Zyner [23] and of Quach et al. [18]. For each rounding algorithm, a logical description and a block diagram is given and the latency is analyzed. We conclude that the new rounding algorithm is the fastest rounding algorithm, provided that an injection (which depends only on the rounding mode and the sign) can be added in during the reduction of the partial products into a carrysave encoded digit string. In double precision the latency of the new rounding algorithm is 12 logic levels compared to 14 logic levels in the algorithm of Quach et al., and 16 logic levels in the algorithm of Yu and Zyner. 1. Introduction Every modern microprocessor includes a floatingpoint (FP) multiplier that complies with the IEEE 754 Standard [9]. The latency of the FP multiplier...
The CSD, GSVD, their Applications and Computations
 University of Minnesota
, 1992
"... Since the CS decomposition (CSD) and the generalized singular value decomposition (GSVD) emerged as the generalization of the singular value decomposition about fifteen years ago, they have been proved to be very useful tools in numerical linear algebra. In this paper, we review the theoretical and ..."
Abstract

Cited by 10 (0 self)
 Add to MetaCart
Since the CS decomposition (CSD) and the generalized singular value decomposition (GSVD) emerged as the generalization of the singular value decomposition about fifteen years ago, they have been proved to be very useful tools in numerical linear algebra. In this paper, we review the theoretical and numerical development of the decompositions, discuss some of their applications and present some new results and observations. We also point out some open problems. A Fortran 77 code has been written that computes the CSD and the GSVD. Keywords: singular value decomposition, CS decomposition, generalized singular value decomposition. Subject Classifications: AMS(MOS): 65F30; CR:G1.3 1 Introduction The singular value decomposition (SVD) of a matrix is one of the most important tools in numerical linear algebra. It has been widely used in scientific computing. Recently, Stewart [52] gave an excellent survey on the early history of the SVD back to the contributions of E. Beltrami and C. Jord...
The inherent inaccuracy of implicit tridiagonal QR
 University of Minnesota, Minneapolis
, 1992
"... Recently Demmel and Veselic showed that Jacobi's method has a tighter relative error bound for the computed eigenvalues of a symmetric positive de nite matrix than does QR iteration. Here we show the weaker error bound of QR as implemented in LAPACK's SSTEQR or EISPACK's IMTQL is unavoidable. We do ..."
Abstract

Cited by 9 (1 self)
 Add to MetaCart
Recently Demmel and Veselic showed that Jacobi's method has a tighter relative error bound for the computed eigenvalues of a symmetric positive de nite matrix than does QR iteration. Here we show the weaker error bound of QR as implemented in LAPACK's SSTEQR or EISPACK's IMTQL is unavoidable. We do this by presenting a particular symmetric positive de nite tridiagonal matrix for whichQRmust fail, given any reasonable shift strategy.
Software And Hardware Techniques For Accurate, SelfValidating Arithmetic
, 1996
"... The need for accurate and reliable numerical applications has led to the development of several software tools and hardware designs for accurate, selfvalidating arithmetic. Software tools include variableprecision software packages, interval arithmetic libraries, scientific programming languages, ..."
Abstract

Cited by 7 (1 self)
 Add to MetaCart
The need for accurate and reliable numerical applications has led to the development of several software tools and hardware designs for accurate, selfvalidating arithmetic. Software tools include variableprecision software packages, interval arithmetic libraries, scientific programming languages, computer algebra systems, and numerical problem solving environments. Hardware designs include coprocessors that support the directed rounding modes and exact dot products, variableprecision integer and floating point processors, and coprocessors for variableprecision, interval arithmetic. In this survey, we examine various software and hardware techniques for accurate, selfvalidating arithmetic and discuss their strengths and limitations. We also discuss numerical applications that employ these tools to produce accurate and reliable results. 1 INTRODUCTION Advances in VLSI technology, parallel processing, and computer architecture have led to increasingly faster digital computers. Duri...
High Performance Robust Computer Systems
, 2001
"... Although our society increasingly relies on computing systems for smooth, efficient operation; computer "errors" that interrupt our lives are commonplace. Better error and exception handling seems to be correlated with more reliable software systems[shelton00][koopman99]. Unfortunately, robust handl ..."
Abstract

Cited by 6 (1 self)
 Add to MetaCart
Although our society increasingly relies on computing systems for smooth, efficient operation; computer "errors" that interrupt our lives are commonplace. Better error and exception handling seems to be correlated with more reliable software systems[shelton00][koopman99]. Unfortunately, robust handling of exceptional conditions is a rarity in modern software systems, and there are no signs that the situation is improving. This dissertation examines the central issues surrounding the reasons why software systems are, in general, not robust, and presents methods of resolving each issue. Although it is commonly held that building robust code is too impractical, we present methods of addressing common robustness failures in a simple, generic fashion. We develop uncomplicated checking mechanisms that can be used to detect and handle exceptional conditions before they can affect process or system state (preemptive detection). This gives a software system the information it needs to gracefully recover from the exceptional condition without the need for task restarts. The perception that computing systems can be either robust or fast (but not both) is a myth perpetuated by not only a dearth of quantitative data, but also an abundance of conventional wisdom whose truth is rooted in an era before modern superscalar processors. The advanced microarchitectural features of such processors are the key to building and understanding systems that are both fast and robust. This research provides an objective, quantitative analysis of the performance cost associated with making a software system highly robust. It develops methods by which the systems studied can be made robust for less than 5% performance overhead for nearly every case, and often much less. Studies indicate that most prog...
Integer Division Using Reciprocals
 In Proceedings of the Tenth Symposium on Computer Arithmetic
, 1991
"... As logic density increases, more and more functionality is moving into hardware. Several years ago, it was uncommon to find more than minimal support in a processor for integer multiplication and division. Now, several processors have multipliers included within the central processing unit on one in ..."
Abstract

Cited by 6 (0 self)
 Add to MetaCart
As logic density increases, more and more functionality is moving into hardware. Several years ago, it was uncommon to find more than minimal support in a processor for integer multiplication and division. Now, several processors have multipliers included within the central processing unit on one integrated circuit [8, 12]. Integer division, due to its iterative nature, benefits much less when implemented directly in hardware and is difficult to pipeline. By using a reciprocal approximation, integer division can be synthesized from a multiply followed by a shift. Without carefully selecting the reciprocal, however, the quotient obtained often suffers from offby one errors, requiring a correction step. This paper describes the design decisions we made when architecting integer division for a new 64 bit machine. The result is a fast and economical scheme for computing both unsigned and signed integer quotients that guarantees an exact answer without any correction. The reciprocal comput...
Interval Analysis in MATLAB
 Department of Mathematics, University of Manchester, Manchester
, 2002
"... The introduction of fast and ecient software for interval arithmetic, such as the MATLAB toolbox INTLAB, has resulted in the increased popularity of the use of interval analysis. We give an introduction to interval arithmetic and explain how it is implemented in the toolbox INTLAB. A tutorial is pro ..."
Abstract

Cited by 5 (0 self)
 Add to MetaCart
The introduction of fast and ecient software for interval arithmetic, such as the MATLAB toolbox INTLAB, has resulted in the increased popularity of the use of interval analysis. We give an introduction to interval arithmetic and explain how it is implemented in the toolbox INTLAB. A tutorial is provided for those who wish to learn how to use INTLAB. We then focus on the interval versions of some important problems in numerical analysis. A variety of techniques for solving interval linear systems of equations are discussed, and these are then tested to compare timings and accuracy. We consider univariate and multivariate interval nonlinear systems and describe algorithms that enclose all the roots. Finally, we give an application of interval analysis. Interval arithmetic is used to take account of rounding errors in the computation of Viswanath's constant, the rate at which a random Fibonacci sequence increases. 1
Computing the condition number of tridiagonal and diagonalplussemiseparable matrices in linear time
 Department of Mathematics, University of Manchester
, 2004
"... Abstract. For an n × n tridiagonal matrix we exploit the structure of its QR factorization to devise two new algorithms for computing the 1norm condition number in O(n) operations. The algorithms avoid underflow and overflow, and are simpler than existing algorithms since tests are not required for ..."
Abstract

Cited by 5 (0 self)
 Add to MetaCart
Abstract. For an n × n tridiagonal matrix we exploit the structure of its QR factorization to devise two new algorithms for computing the 1norm condition number in O(n) operations. The algorithms avoid underflow and overflow, and are simpler than existing algorithms since tests are not required for degenerate cases. An error analysis of the first algorithm is given, while the second algorithm is shown to be competitive in speed with existing algorithms. We then turn our attention to an n × n diagonalplussemiseparable matrix, A, for which several algorithms have recently been developed to solve Ax = b in O(n) operations. We again exploit the QR factorization of the matrix to present an algorithm that computes the 1norm condition number in O(n) operations.