Optimal initial approximations for the Newton-Raphson division algorithm

Schulte, Michael; Omar, J.; Swartzlander, Earl E.

doi:10.1007/bf02307376

Cited by 31 publications

(8 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…To start the algorithm, an initial approximation to 1/Y is required. This approximation is made by a table-lookup on the most significand bits of Y [28]. ff the initial approximation has k bits of accuracy, then approximately [log2(p/k)] iterations are required to compute a reciprocal which is accurate to p bits.…”

Section: Multiplication Of the Intervals X ~-[A B] And Y -~ It D] Imentioning

confidence: 99%

A software interface and hardware design for variable-precision interval arithmetic

Schulte

Swartzlander

1995

Reliable Comput

View full text Add to dashboard Cite

This paper presents a software interface and hardware design for variable-precision, interval arithmetic. The software interface gives the programmer the ability to spedfy the predsion of the computation and determine the accuracy of the result Special instructions for vector and matrix operations are also provided. The hardware design directly supports variable-precision, interval arithmetic. This gready improves the accuracy of the computation and is much faster than existing software methods for controlling numerical error. Hardware algorithms are presented for the basic arithmetic operations, exact dot products, and elementary functions. Area and delay estimates indicate that the processor can be implemented on a single chip With a cycle time that is comparable to existing IEEE d0uble-predsion floating point processors. 1-IporpaMMHl IntroductionAdvances in VLSI technology and computer architecture have lead to increasingly faster digital computers. During each of the last three decades, the computational speeds of the fastest computers increased by a factor of roughly 100 [1]. This increase in computing power has lead to the development of computer systems which perform billions of arithmetic operations per second and has given researchers the ability to solve previously intractable problems. The large number of arithmetic operations, however, has made it extremely important to keep track of and control errors in numerical computations.

show abstract

Section: Multiplication Of the Intervals X ~-[A B] And Y -~ It D] Imentioning

confidence: 99%

A software interface and hardware design for variable-precision interval arithmetic

Schulte

Swartzlander

1995

Reliable Comput

View full text Add to dashboard Cite

show abstract

“…Many present studies for the method are mainly focused on the optimization of the algorithm and the applications in the new fields. Schulte, et al [6] developed methods for selecting initial approximation which minimized the absolute error of the final result reducing the number of cycles. Wang, et al [7] proposed a modified version of Newton-Raphson iteration, which worked well for decimal numbers.…”

Section: Introductionmentioning

confidence: 99%

Principle of MSD floating-point division based on Newton-Raphson method on ternary optical computer

Shen

Fan

2011

J. Shanghai Univ.(Engl. Ed.)

View full text Add to dashboard Cite

The division operation is not frequent relatively in traditional applications, but it is increasingly indispensable and important in many modern applications. In this paper, the implementation of modified signed-digit (MSD) floating-point division using Newton-Raphson method on the system of ternary optical computer (TOC) is studied. Since the addition of MSD floating-point is carry-free and the digit width of the system of TOC is large, it is easy to deal with the enough wide data and transform the division operation into multiplication and addition operations. And using data scan and truncation the problem of digits expansion is effectively solved in the range of error limit. The division gets the good results and the efficiency is high. The instance of MSD floating-point division shows that the method is feasible.

show abstract

“…In 1994 [16] developed explicit formulas for the optimal starting values for this iteration, as functions of the number n of iterations, and the interval (a, b)…”

Section: Introductionmentioning

confidence: 99%

Choosing starting values for certain Newton–Raphson iterations

Kornerup

Muller

2006

Theoretical Computer Science

View full text Add to dashboard Cite

We aim at finding the best possible seed values when computing a 1/p using the Newton-Raphson iteration in a given interval. A natural choice of the seed value would be the one that best approximates the expected result. It turns out that in most cases, the best seed value can be quite far from this natural choice. When we evaluate a monotone function f (a) in the interval [a min , a max ], by building the sequence x n defined by the Newton-Raphson iteration, the natural choice consists in choosing x 0 equal to the arithmetic mean of the endpoint values. This minimizes the maximum possible distance between x 0 and f (a). And yet, if we perform n iterations, what matters is to minimize the maximum possible distance between x n and f (a). In several examples, the value of the best starting point varies rather significantly with the number of iterations.

show abstract

Optimal initial approximations for the Newton-Raphson division algorithm

Cited by 31 publications

References 10 publications

A software interface and hardware design for variable-precision interval arithmetic

A software interface and hardware design for variable-precision interval arithmetic

Principle of MSD floating-point division based on Newton-Raphson method on ternary optical computer

Choosing starting values for certain Newton–Raphson iterations

Contact Info

Product

Resources

About