## Big Integers and Complexity Issues in Exact Real Arithmetic (1998)

Venue: | In Third Comprox workshop |

Citations: | 4 - 3 self |

### BibTeX

@INPROCEEDINGS{Heckmann98bigintegers,

author = {Reinhold Heckmann},

title = {Big Integers and Complexity Issues in Exact Real Arithmetic},

booktitle = {In Third Comprox workshop},

year = {1998}

}

### OpenURL

### Abstract

One possible approach to exact real arithmetic is to use linear fractional transformations to represent real numbers and computations on real numbers. We show how to determine the digits that can be emitted from a transformation, and present a criterion which ensures that it is possible to emit a digit. Using these results, we prove that the obvious algorithm to compute n digits from the application of a transformation to a real number has complexity O(n 2 ), and present a method to reduce this complexity to that of multiplying two n bit integers. 1 Introduction Linear Fractional Transformations (LFT's) provide an elegant approach to real number arithmetic [5,14,9,12,10,4]. One-dimensional LFT's x 7! ax+c bx+d are used as digits and to implement basic unary functions, while two-dimensional LFT's (x; y) 7! axy+cx+ey+g bxy+dx+fy+h provide binary operations such as addition and multiplication, and can be combined to obtain infinite expression trees denoting transcendental functions...

### Citations

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Citation Context ...lta A = A \Delta As= E if det A 6= 0, whence hAi \Gamma1 = hA i. 2.2 Representing Reals by LFT's The set R ? can be visualised as a circle. Intervalss[u; v] are anti-clockwise arcs from u to v, e.g., =-=[0; 1]-=- = fx 2 R j 0sxs1g, and [1; 0] = fx 2 R j 1sx or xs0g [ f1g. &% '$ 0 1 1 \Gamma1 Non-singular LFT's map intervals to intervals: if det M ? 0, then hMi[u; v] is [hMiu; hMiv], while for det M ! 0, we ge... |

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Citation Context ...l Multiplication of two integers of bit size n requires more than O(n) basic arithmetical operations. Any straightforward algorithm takes time O(n 2 ). However, there are several faster algorithms in =-=[8]-=-, including one which needs only O(n log n log log n) basic arithmetical operations, and one that simulates the multiplication in O(n) operations on a pointer machine. In the sequel, let us assume a f... |

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Citation Context ...nd present a method to reduce this complexity to that of multiplying two n bit integers. 1 Introduction Linear Fractional Transformations (LFT's) provide an elegant approach to real number arithmetic =-=[5,14,9,12,10,4]-=-. One-dimensional LFT's x 7! ax+c bx+d are used as digits and to implement basic unary functions, while two-dimensional LFT's (x; y) 7! axy+cx+ey+g bxy+dx+fy+h provide binary operations such as additi... |

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Citation Context ...as list [k 1 ; : : : ; km ] of digits, store it as the number K together with the length m. This brings the digit matrix approach in close relationship with Boehm and Cartwright's functional approach =-=[2,3]-=-. 6.4 Mass Emission Mass absorption may bring down the cost of real number computation, but for a real gain, also mass emission is needed. For, no matter how quick M 0 = M \Delta D 2 k 1 \Delta \Delta... |

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Citation Context ...as list [k 1 ; : : : ; km ] of digits, store it as the number K together with the length m. This brings the digit matrix approach in close relationship with Boehm and Cartwright's functional approach =-=[2,3]-=-. 6.4 Mass Emission Mass absorption may bring down the cost of real number computation, but for a real gain, also mass emission is needed. For, no matter how quick M 0 = M \Delta D 2 k 1 \Delta \Delta... |

42 | A new representation for exact real numbers
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Citation Context ...nd present a method to reduce this complexity to that of multiplying two n bit integers. 1 Introduction Linear Fractional Transformations (LFT's) provide an elegant approach to real number arithmetic =-=[5,14,9,12,10,4]-=-. One-dimensional LFT's x 7! ax+c bx+d are used as digits and to implement basic unary functions, while two-dimensional LFT's (x; y) 7! axy+cx+ey+g bxy+dx+fy+h provide binary operations such as additi... |

29 | Semantics of exact real arithmetic
- Potts, Edalat, et al.
- 1997
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Citation Context ... Fractional Transformations In this section, we present the framework of exact real arithmetic via LFT's [5,14,9], specialised to the version used by the group of Edalat and Potts at Imperial College =-=[12,10,11,13,4]-=-. 2.1 LFT's and Matrices General Linear Fractional Transformations (LFT's) are functions x 7! ax+c bx+d from reals to reals, parameterised by real numbers a, b, c, and d. In this paper, we shall only ... |

19 |
Continued Fraction Arithmetic
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(Show Context)
Citation Context ...nd present a method to reduce this complexity to that of multiplying two n bit integers. 1 Introduction Linear Fractional Transformations (LFT's) provide an elegant approach to real number arithmetic =-=[5,14,9,12,10,4]-=-. One-dimensional LFT's x 7! ax+c bx+d are used as digits and to implement basic unary functions, while two-dimensional LFT's (x; y) 7! axy+cx+ey+g bxy+dx+fy+h provide binary operations such as additi... |

18 | MSB-First Digit Serial Arithmetic
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Citation Context |

14 | Exact Real Computer Arithmetic
- Potts, Edalat
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Citation Context ... Fractional Transformations In this section, we present the framework of exact real arithmetic via LFT's [5,14,9], specialised to the version used by the group of Edalat and Potts at Imperial College =-=[12,10,11,13,4]-=-. 2.1 LFT's and Matrices General Linear Fractional Transformations (LFT's) are functions x 7! ax+c bx+d from reals to reals, parameterised by real numbers a, b, c, and d. In this paper, we shall only ... |

9 | Computable Real Arithmetic using Linear Fractional Transformations
- Potts
- 1996
(Show Context)
Citation Context |

8 | Contractivity of linear fractional transformations
- Heckmann
- 1998
(Show Context)
Citation Context ...0 i, hM 1 i, . . . are sufficiently contractive, then the intersection in (3) shrinks to a singleton set. In this case, the stream of matrices or LFT's denotes a unique real number (it converges). In =-=[7]-=-, some 3 Heckmann sufficient criteria for convergence are presented. Because of the usage of matrix multiplication in (3), we consider a stream of matrices converging to a real number x as a (formal) ... |

8 |
Exact Real Arithmetic based on Linear Fractional Transformations
- Potts, Edalat
- 1996
(Show Context)
Citation Context |

7 | The appearance of big integers in exact real arithmetic based on linear fractional transformations
- Heckmann
- 1998
(Show Context)
Citation Context ...rix multiplication applied to a transformer and a digit matrix. Usually, all the transformers employed in real number arithmetic have integer components. In Section 3, we reiterate the main result of =-=[6]-=-: if the difference of the column sums of a transformer is not zero, at least one entry of the transformer has bit size\Omega\Gamma n) after n digits have been emitted (law of big numbers). In Section... |