## On the Efficiency of Interval Multiplication Algorithms

Citations: | 1 - 0 self |

### BibTeX

@MISC{Popova_onthe,

author = {Evgenija D. Popova},

title = {On the Efficiency of Interval Multiplication Algorithms},

year = {}

}

### OpenURL

### Abstract

In this paper we present the theoretical base for some modifications in interval multiplication algorithms. A diversity of proposed implementation approaches is summarized along with a discussion on their costefficiency. It is shown that some improvements can be achieved by utilizing some properties of interval multiplication formulae and no special hardware support. Both conventional and extended interval multiplication operations are considered.

### Citations

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- Kulisch, Miranker
- 1981
(Show Context)
Citation Context ...c [2] in round-to-negative-infinity mode is symmetric with the arithmetic in round-to-positive-infinity mode. The reader is supposed to be familiar with the definition of computer interval arithmetic =-=[8]-=-. On the example of the classical algorithm, based on nine sign-dependent cases, we demonstrate how some properties of interval multiplication formulae can be exploited to gain an increased efficiency... |

16 |
Extended interval arithmetics: new results and applications
- Dimitrova, Markov, et al.
- 1992
(Show Context)
Citation Context ...striction of this formula to the conventional interval space IR results in the well-known classical interval arithmetic formula [1]. Details about generalized interval arithmetic can be found e.g. in =-=[5]-=-, [10]. The occurrence of min and max functions at the end-points of the result on multiplication of two zero-involving intervals hampers not only analytical derivations in interval analysis but affec... |

10 | Interval Operations Involving NaNs
- Popova
(Show Context)
Citation Context ...tives that support efficient implementation of max satisfying max(x, NaN) = NaN for any x, or max, based on IEEE comparisons, should be implemented by one comparison more using the unordered paradigm =-=[9]-=-. The alternative (“algebraic”) approach, used by Algorithm 3.2, provides efficiency in this situation, too, because no Invalid Operation exception 0 × (±∞) can arise in the branch {0 ◦ ∈ A, 0 ◦ ∈ B},... |

7 |
Einfuhrung in die Intervallrechnung
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(Show Context)
Citation Context ...+ b − }], A, B ∈ T , τ(A) = τ(B) = −; 0, A, B ∈ T , τ(A) �= τ(B). The restriction of this formula to the conventional interval space IR results in the well-known classical interval arithmetic formula =-=[1]-=-. Details about generalized interval arithmetic can be found e.g. in [5], [10]. The occurrence of min and max functions at the end-points of the result on multiplication of two zero-involving interval... |

7 | E.E.: Software and hardware techniques for accurate, self-validating arithmetic
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(Show Context)
Citation Context ...erval arithmetic software tools. The necessity of interval software performance improvements for numerical intensive applications has led to a number of hardware designs involving interval arithmetic =-=[12]-=-. Recent efforts in establishing several interval conventions and standards facilitate the development of commercial hardware and compiler support for interval arithmetic. However, speed remains the m... |

4 |
An Improved Algorithm for Computing the Product of Two
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(Show Context)
Citation Context ... + ≤ a + b − and ν(A) = ν(B) = −. The possibility for using only three floating-point multiplications in the worst case of multiplication of two zero involving normal intervals was first mentioned in =-=[6]-=-. Next analysis gives the theoretical basis for the algorithms considered in [4], [6] and Section 3.4. 120sReal Numbers and Computers 3 April 28,1998 For A ∈ D denote � σ(A), if A ∈ D \ T ; µ(A) = ν(A... |

4 |
D.: Generalized Interval Distributive Relations. Preprint No 2
- Popova
- 1997
(Show Context)
Citation Context ...tion of this formula to the conventional interval space IR results in the well-known classical interval arithmetic formula [1]. Details about generalized interval arithmetic can be found e.g. in [5], =-=[10]-=-. The occurrence of min and max functions at the end-points of the result on multiplication of two zero-involving intervals hampers not only analytical derivations in interval analysis but affects the... |

2 |
Hardware Support for Interval Arithmetic Extended Version, Report No
- GUDENBERG, J
- 1995
(Show Context)
Citation Context ...multiplier that could initiate a new operation on every cycle might suffice.” Further discussion on this approach and the design of special hardware, dedicated for interval operations, is provided in =-=[13]-=-. Based on the assumption that one arithmetic operation as well as a floating-point comparison lasts one unit of time while switches controlled by the sign bit are free of charge, the Algorithm 3.3 wa... |

1 |
Interval Arithmetic Specification”, Draft revised
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- 1997
(Show Context)
Citation Context ...ndent, although the first alternative is often preferred in low-level implementations designed for efficiency. As a typical representative of the first alternative we consider Algorithm 3.1, given in =-=[3]-=-. This algorithm computes the product A × B for A, B ∈ Z by finding maximum of four floating-point products and requires four floating-point multiplications and two floating-point comparison operation... |

1 |
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(Show Context)
Citation Context ...2. p = a − ▽ b + ; q = a + ▽ b − 3. r − = min(p, q); p = a − △ b − ; q = a + △ b + 4. r + = max(p, q) A completely different approach for the implementation of interval multiplication was proposed in =-=[4]-=- (see also [1]) and modified in [6] to reduce the number of floating-point products in multiplication of two zero containing intervals from four to three. This approach uses an initial transformation ... |

1 |
Hardware Support for Interval Arithmetic”, posts to numericinterest@validgh.com mailing list
- Hough
- 1994
(Show Context)
Citation Context ...rating to fully utilize the floating-point pipeline to exploit instructionlevel parallelism. The idea for a brute force approach in implementation of interval multiplication operation was proposed in =-=[7]-=-. It was suggested that an implementation of interval multiplication be based on the usual definition for A, B ∈ IR A × B = [min{a − b − , a − b + , a + b − , a + b + }, max{a − b − , a − b + , a + b ... |

1 | S.: “Towards Credible Implementation of Inner Interval Operations
- Popova, Markov
- 1997
(Show Context)
Citation Context ... = −(−a − b − ) 3. Restore−Rounding−Mode if µA = µB then R = [r − , r + ] else R = [−r + , −r − ] 128sReal Numbers and Computers 3 April 28,1998 4 Nonstandard Interval Products It was demonstrated in =-=[11]-=-, that a credible implementation of inner interval multiplication × − requires three floating-point multiplications for each signdependent case. The algorithm, given in [11], can be modified to proces... |