## Lazy Functional Algorithms for Exact Real Functionals (1998)

Venue: | Lec. Not. Comput. Sci |

Citations: | 23 - 0 self |

### BibTeX

@INPROCEEDINGS{Simpson98lazyfunctional,

author = {Alex K. Simpson},

title = {Lazy Functional Algorithms for Exact Real Functionals},

booktitle = {Lec. Not. Comput. Sci},

year = {1998},

pages = {456--464},

publisher = {Springer}

}

### Years of Citing Articles

### OpenURL

### Abstract

. We show how functional languages can be used to write programs for real-valued functionals in exact real arithmetic. We concentrate on two useful functionals: definite integration, and the functional returning the maximum value of a continuous function over a closed interval. The algorithms are a practical application of a method, due to Berger, for computing quantifiers over streams. Correctness proofs for the algorithms make essential use of domain theory. 1 Introduction In exact real number computation, infinite representations of reals are employed to avoid the usual rounding errors that are inherent in floating point computation [4--6, 17]. For certain real number computations that are highly sensitive to small variations in the input, such rounding errors become inordinately large and the use of floating-point algorithms can lead to completely erroneous results [1, 14]. In such situations, exact real number computation provides guaranteed correctness, although at the (probably...

### Citations

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(Show Context)
Citation Context ...pecially in the light of their possible applications. The work of Berger, Di Gianantonio, Escard'o and Edalat, referred to above, was carried out in the context of the minimal functional language PCF =-=[15]-=- (and extensions of it). It would be fully possible to write this paper in the same setting, but we prefer instead to adopt a less spartan approach. The goal of this paper is to describe and verify pa... |

162 |
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Citation Context ...analysis of this situation will appear in the full version of the paper. The intrinsic intractibility of the operations of integration and finding maximum values is to be expected from the work of Ko =-=[13]-=-. Acknowledgements I have benefited from discussions with Pietro Di Gianantonio, Gordon Plotkin and, especially, Mart'in Escard'o. I thank Ieke Moerdijk, Jaap van Oosten and Harold Schellinx for their... |

49 |
Exact Real Arithmetic: Formulating Real Numbers as Functions. Chapter 3 of
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Citation Context ...ramming naturally supports the recursive definition of functions, which is the most useful method of defining exact functions on real numbers. Such considerations were important motivating factors in =-=[4, 5, 17, 6, 7, 10]-=-. One principal distinguishing feature of functional languages is their acceptance of functions as first-class values, and the associated possibility of passing functions as arguments to other functio... |

46 | PCF extended with real numbers
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Citation Context ...ramming naturally supports the recursive definition of functions, which is the most useful method of defining exact functions on real numbers. Such considerations were important motivating factors in =-=[4, 5, 17, 6, 7, 10]-=-. One principal distinguishing feature of functional languages is their acceptance of functions as first-class values, and the associated possibility of passing functions as arguments to other functio... |

42 |
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Citation Context ...ramming naturally supports the recursive definition of functions, which is the most useful method of defining exact functions on real numbers. Such considerations were important motivating factors in =-=[4, 5, 17, 6, 7, 10]-=-. One principal distinguishing feature of functional languages is their acceptance of functions as first-class values, and the associated possibility of passing functions as arguments to other functio... |

27 | Properly injective spaces and function spaces
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Citation Context ...In the full version of the paper the definitions and results in this section will be related to work on totality in domain theory [2, 3, 16], and to topological injectivity (and projectivity) results =-=[11]-=-. 4 Moduli of Continuity and Stream Quantifiers Consider any continuous function OE : 2 1 ! X? where X is any set. We say that f is total if, for all ff 2 2 ! , it holds that OE(ff) 2 X. Proposition 2... |

21 |
A Functional Approach to Computability on Real Numbers
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Citation Context |

19 | Arbitrary precision real arithmetic: design and algorithms
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Citation Context ...mputations that are highly sensitive to small variations in the input, such rounding errors become inordinately large and the use of floating-point algorithms can lead to completely erroneous results =-=[1, 14]-=-. In such situations, exact real number computation provides guaranteed correctness, although at the (probably inevitable) price of a loss of efficiency. How to improve efficiency is a field of active... |

10 |
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Citation Context |

8 |
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Citation Context ...n ! 3 ! there exists a total OE : (3 1 ) n ! 3 1 such that ` = �� OE. In the full version of the paper the definitions and results in this section will be related to work on totality in domain the=-=ory [2, 3, 16]-=-, and to topological injectivity (and projectivity) results [11]. 4 Moduli of Continuity and Stream Quantifiers Consider any continuous function OE : 2 1 ! X? where X is any set. We say that f is tota... |

8 |
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Citation Context |

4 |
Introduction to the Special Issue on "Real Numbers and Computers
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Citation Context ...mputations that are highly sensitive to small variations in the input, such rounding errors become inordinately large and the use of floating-point algorithms can lead to completely erroneous results =-=[1, 14]-=-. In such situations, exact real number computation provides guaranteed correctness, although at the (probably inevitable) price of a loss of efficiency. How to improve efficiency is a field of active... |

2 |
Total Objects and Sets in Domain Theory
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(Show Context)
Citation Context ...n ! 3 ! there exists a total OE : (3 1 ) n ! 3 1 such that ` = �� OE. In the full version of the paper the definitions and results in this section will be related to work on totality in domain the=-=ory [2, 3, 16]-=-, and to topological injectivity (and projectivity) results [11]. 4 Moduli of Continuity and Stream Quantifiers Consider any continuous function OE : 2 1 ! X? where X is any set. We say that f is tota... |

1 |
Totale Objecte und Mengen in Bereichstheorie
- Berger
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(Show Context)
Citation Context ...fic and useful functionals of integration and maximum. The algorithms rely on a clever, but little known, idea of Berger, who showed how to compute quantifiers over predicates on streams sequentially =-=[2]-=-. Berger's algorithms deserve to be better known, especially in the light of their possible applications. The work of Berger, Di Gianantonio, Escard'o and Edalat, referred to above, was carried out in... |

1 |
Exact Real Computer Arithmetic. Presented at workshop: New Paradigms for Computation on Classical Spaces
- Edalat, Potts
- 1997
(Show Context)
Citation Context ...situations, exact real number computation provides guaranteed correctness, although at the (probably inevitable) price of a loss of efficiency. How to improve efficiency is a field of active research =-=[9]-=-. Lazy functional programming provides a natural implementational style for exact real algorithms. One reason is that lazy functional languages support lazy infinite data structures, such as streams, ... |

1 |
Semantics of Programming
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(Show Context)
Citation Context ...otational semantics, we use directed-complete partial orders with least element (henceforth cpos) for interpreting datatypes, and continuous functions between them for interpreting programs (see e.g. =-=[12]-=-). In Sec. 3 we refer to cpos as topological spaces, understanding them as carrying the Scott topology. Given a set X, we write X? for the flat cpo with least element ? and with all other elements tak... |